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Contribution list

Contribution list

Below you can see a list of all accepted contributions to the 8th Congress of Mathematics. Use the search field to narrow down the display.

Plenary speakers

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The Beat of Math

Prof. Alfio Quarteroni, Politecnico di Milano and EPFL

Abstract

Mathematical models based on first principles are devised for the description of the blood motion in the human circulatory system, as well as for the simulation of the interaction between electrical, mechanical, and fluid-dynamical processes occurring in the heart. This is a classical environment where multi-physics and multi-scale processes have to be addressed.\\ Appropriate systems of nonlinear differential equations (either ordinary and partial) and efficient numerical strategies must be devised to allow for the analysis of both heart function and dysfunction, and the simulation, control, and optimization of therapy and surgery.\\ This presentation will address some of these issues and a few representative applications of clinical interest.\\ \noindent {\em Acknowledgment:} The work presented in this talk is part of the project iHEART that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 740132)

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Stable solutions to semilinear elliptic equations are smooth up to dimension 9

Prof. Xavier Cabré, ICREA and Universitat Politecnica de Catalunya

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The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist stable energy solutions which are singular. In this talk I will describe a recent work in collaboration with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to an open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.

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Torsion in algebraic groups and problems which arise

Prof. Umberto Zannier, Scuola Normale Superiore

Abstract

Since the investigations of Gauss on cyclotomy, the arithmetic of roots of unity and, more generally, of torsion points in (commutative) algebraic groups, has developed into rich and deep theories. Several problems were raised also concerning algebraic relations among torsion elements, as for instance in well-known (former) conjectures by Manin-Mumford and by Lang. In the talk we shall survey especially along this last topic, focusing on more recent finiteness results regarding torsion values attained by sections of families of abelian varieties. Finally, we shall briefly mention some applications.

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Minimal surfaces from a complex analytic viewpoint

Prof. Franc Forstnerič, University of Ljubljana

Abstract

In this talk, I will describe some recent developments in the theory of minimal surfaces in Euclidean spaces which have been obtained by complex analytic methods. After a brief history and background of the subject, I will present a new Schwarz-Pick lemma for minimal surfaces, describe approximation and interpolation results for minimal surfaces, and discuss the current status of the Calabi-Yau problem on the existence of complete conformal minimal surfaces with Jordan boundaries.

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Statistical Learning: Causal-oriented and Robust

Prof. Peter Bühlmann, ETH Zurich

Abstract

Reliable, robust and interpretable machine learning is a big emerging theme in data science and artificial intelligence, complementing the development of pure black box prediction algorithms. Looking through the lens of statistical causality and exploiting a probabilistic invariance property opens up new paths and opportunities for enhanced robustness, with wide-ranging prospects for various applications.

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Bernoulli Random Matrices

Ms. Alice Guionnet, ENS Lyon

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The study of large random matrices, and in particular the properties of their eigenvalues and eigenvectors, has emerged from the applications, first in data analysis and later as statistical models for heavy-nuclei atoms. It now plays an important role in many other areas of mathematics such as operator algebra and number theory. Over the last thirty years, random matrix theory became a field on its own, borrowing tools from different branches of mathematics. The purpose of this lecture is to illustrate this theory by focusing on the special case of Bernoulli random matrices. Such matrices are particularly interesting as they represent the adjacency matrix of Erdos-Renyi graphs.

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Escaping the curse of dimensionality in combinatorics

Prof. Janos Pach, Alfred Renyi Matematical Institute

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We discuss some notoriously hard combinatorial problems for large classes of graphs and hypergraphs arising in geometric, algebraic, and practical applications. These structures escape the ``curse of dimensionality'': they can be embedded in a bounded-dimensional space, or they have small VC-dimension or a short algebraic description. What are the advantages of low dimensionality? \begin{enumerate} \item With the help of suitable topological and algebraic separator theorems, large families of geometric objects embedded in a fixed-dimensional space can be split into subfamilies of roughly the same size, and then the smaller families can be analyzed recursively. \item Geometric objects in space can be compared by a number of naturally defined partial orders, so that one can utilize the theory of partially ordered sets. \item Graphs and hypergraphs of bounded VC-dimension admit very small epsilon-nets and can be particularly well approximated by random sampling. \item If the description complexity of a family of geometric objects is small, then the ``combinatorial complexity'' (number of various dimensional faces) of their arrangements is also small. This typically guarantees the existence of efficient algorithms to visualize, describe, control, and manipulate these arrangements. \end{enumerate}

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Geometric Valuation Theory

Prof. Monika Ludwig, TU Wien

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Valuations on compact convex sets in $\mathbb{R}^n$ play an active and prominent role in geometry. They were critical in Dehn's solution to Hilbert's Third Problem in 1901. They are defined as follows. A function $\operatorname{\rm Z}$ whose domain is a collection of sets $\mathcal{S}$ and whose co-domain is an Abelian semigroup is called a {\em valuation} if $$ \operatorname{\rm Z}(K) + \operatorname{\rm Z}(L) = \operatorname{\rm Z}(K\cup L) + \operatorname{\rm Z}(K\cap L),$$ whenever $K, L, K\cup L, K \cap L \in \mathcal{S}$. The first classification result for valuations on the space of compact convex sets, $\mathcal{K}^n$, in $\mathbb{R}^n$ (where $\mathcal{K}^n$ is equipped with the topology induced by the Hausdorff metric) was established by Blaschke. \begin{theorem}[Blaschke]\label{wb} A functional $\,\operatorname{\rm Z}:\mathcal{K}^n\to \mathbb{R}$ is a continuous, translation and SL$(n)$ invariant valuation if and only if there are $c_0, c_n\in\mathbb{R}$ such that $$\operatorname{\rm Z}(K) = c_0 V_0(K)+c_n V_n(K)$$ for every $K\in\mathcal{K}^n$. \end{theorem} Probably the most famous result in the geometric theory of valuations is the Hadwiger characterization theorem. \begin{theorem}[Hadwiger]\label{hugo} A functional $\,\operatorname{\rm Z}:\mathcal{K}^n\to \mathbb{R}$ is a continuous and rigid motion invariant valuation if and only if there are $c_0,\ldots, c_n\in\mathbb{R}$ such that $$\operatorname{\rm Z}(K) = c_0 V_0(K)+\dots+c_n V_n(K)$$ for every $K\in\mathcal{K}^n$. \end{theorem} Here $V_0(K),\ldots,V_n(K)$ are the intrinsic volumes of $K\in\mathcal{K}^n$. In particular, $V_0(K)$ is the Euler characteristic of $K$, while $2\, V_{n-1}(K)$ is the surface are of $K$ and $V_n(K)$ the $n$-dimensional volume of $K$. Hadwiger's theorem shows that the intrinsic volumes are the most basic functionals in Euclidean geometry. It finds powerful applications in Integral Geometry and Geometric Probability. The fundamental results of Blaschke and Hadwiger have been the starting point of the development of Geometric Valuation Theory. Classification results for valuations invariant (or covariant) with respect to important groups are central questions. The talk will give an overview of such results including recent extensions to valuations on function spaces. \begin{thebibliography}{10} \bibitem{BoeroeczkyLudwig} K.J. B\"or\"oczky, M. Ludwig, {\em Minkowski valuations on lattice polytopes}. J. Eur. Math. Soc. (JEMS) {\bf 21} (2019), 163–197. \bibitem{Colesanti-Ludwig-Mussnig-2} A. Colesanti, M. Ludwig, F. Mussnig, {\em Minkowski valuations on convex functions}, Calc. Var. Partial Differential Equations {\bf 56} (2017), 56:162. \bibitem{Colesanti-Ludwig-Mussnig-5} A.~Colesanti, M.~Ludwig, and F.~Mussnig, {\em The {H}adwiger theorem on convex functions.~{I}}, arXiv:2009.03702 (2020). \bibitem{Haberl:Parapatits_centro} C.~Haberl and L.~Parapatits, {\em The centro-affine {H}adwiger theorem}, J. Amer. Math. Soc. {\bf 27} (2014), 685--705. \bibitem{Hadwiger:V} H.~Hadwiger, {\em {V}or\-lesungen \"uber {I}nhalt, {O}ber\-fl\"ache und {I}so\-peri\-metrie}, Springer, Berlin, 1957. \bibitem{Klain:Rota} D.~A. Klain and G.-C. Rota, {\em Introduction to geometric probability}, Cambridge University Press, Cambridge, 1997. \bibitem{Ludwig:projection} M.~Ludwig, {\em Projection bodies and valuations}, Adv. Math. {\bf 172} (2002), 158--168. \bibitem{Ludwig:sobval} M.~Ludwig, {\em Valuations on {S}obolev spaces}, Amer. J. Math. {\bf 134} (2012), 827--842. \bibitem{Ludwig:Reitzner} M.~Ludwig and M.~Reitzner, {\em A characterization of affine surface area}, Adv. Math. {\bf 147} (1999), 138--172. \bibitem{Ludwig:Reitzner2} M.~Ludwig and M.~Reitzner, {\em A classification of SL$(n)$ invariant valuations}, Ann. of Math. (2) {\bf 172} (2010), 1219--1267. \end{thebibliography}

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The Mathematics of Deep Learning

Prof. Gitta Kutyniok, Technische Uninversität Berlin

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Despite the outstanding success of deep neural networks in real-world applications, ranging from science to public life, most of the related research is empirically driven and a comprehensive mathematical foundation is still missing. At the same time, these methods have already shown their impressive potential in mathematical research areas such as imaging sciences, inverse problems, or numerical analysis of partial differential equations, sometimes by far outperforming classical mathematical approaches for particular problem classes. The goal of this lecture is to first provide an introduction into this new vibrant research area. We will then survey recent advances in two directions, namely the development of a mathematical foundation of deep learning and the introduction of novel deep learning-based approaches to solve inverse problems and partial differential equations.

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Metric measure spaces and synthetic Ricci bounds

Prof. Karl-Theodor Sturm, Universität Bonn

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Metric measure spaces with synthetic Ricci bounds have attracted great interest in recent years, accompanied by spectacular breakthroughs and deep new insights. In this talk, I will provide a brief introduction to the concept of lower Ricci bounds as introduced by Lott-Villani and myself, and illustrate some of its geometric, analytic and probabilistic consequences, among them Li-Yau estimates, coupling properties for Brownian motions, sharp functional and isoperimetric inequalities, rigidity results, and structural properties like rectifiability and rectifiability of the boundary. In particular, I will explain its crucial interplay with the heat flow and its link to the curvature-dimension condition formulated in functional-analytic terms by Bakry-Émery. This equivalence between the Lagrangian and the Eulerian approach then will be further explored in various recent research directions: i) distribution-valued Ricci bounds which e.g. allow singular effects of non-convex boundaries to be taken into account, ii) time-dependent Ricci bounds which provide a link to (super-) Ricci flows for singular spaces, iii) upper curvature bounds.

Invited speakers

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A proof of the Erd\H{o}s-Faber-Lov\'asz conjecture

Prof. Daniela Kuhn, University of Birmingham

Abstract

Graph and hypergraph colouring problems are central to combinatorics, with applications and connections to many other areas, such as geometry, algorithm design, and information theory. However, for hypergraphs even basic problems have turned out to be rather challenging: in particular, the famous Erd\H{o}s-Faber-Lov\'asz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. (Here the chromatic index of a hypergraph $H$ is the smallest number of colours needed to colour the edges of $H$ so that any two edges that share a vertex have different colours.) There are also several other equivalent (dual) versions of this conjecture, e.g. in terms of colouring the vertices of nearly disjoint cliques. Erd\H{o}s considered this to be one of his three most favorite combinatorial problems and offered \$500 for the solution of the problem. In joint work with Dong-yeap Kang, Tom Kelly, Abhishek Methuku and Deryk Osthus, we prove this conjecture for every large $n$. We also provide `stability versions' of this result, which confirm a prediction of Kahn. In my talk, I will discuss some background, some of the ideas behind the proof as well as some related open problems.

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Subset products and derangements

Prof. Aner Shalev, Hebrew University of Jerusalem

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In the past two decades there has been intense interest in products of subsets in finite groups. Two important examples are Gowers' theory of Quasi Random Groups and its applications by Nikolov, Pyber, Babai and others, and the theory of Approximate Groups and the Product Theorem of Breuillard-Green-Tao and Pyber-Szabo on growth in finite simple groups of Lie type of bounded rank, extending Helfgott's work. These deep theories yield strong results on products of three subsets (covering, growth). What can be said about products of two subsets? I will discuss a recent joint work with Michael Larsen and Pham Tiep on this challenging problem, focusing on products of two normal subsets of finite simple groups, and deriving a number of applications. Our main application concerns derangements (namely, fixed-point-free permutations), studied since the days of Jordan. We show that every element of a sufficiently large finite simple transitive permutation group is a product of two derangements. Related results and problems will also be discussed. The proofs combine group theory, algebraic geometry and representation theory; it applies the proof by Fulman and Guralnick of the Boston-Shalev conjecture on the proportion of derangements.

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Regularization methods in inverse problems and machine learning

Prof. Martin Burger, FAU Erlangen-Nürnberg

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Regularization methods are at the heart of the solution of inverse problems and are of increasing importance in modern machine learning. In this talk we will discuss the modern theory of (nonlinear) regularization methods and some applications. We will put a particular focus on variational and iterative regularization methods and their connection with learning problems: we discuss the use of such regularization methods for learning problems on the one hand, but also the current route of learning regularization methods from data.

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Scaling Optimal Transport for High dimensional Learning

Mr. Gabriel Peyré, CNRS and ENS

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Optimal transport (OT) has recently gained lot of interest in machine learning. It is a natural tool to compare in a geometrically faithful way probability distributions. It finds applications in both supervised learning (using geometric loss functions) and unsupervised learning (to perform generative model fitting). OT is however plagued by the curse of dimensionality, since it might require a number of samples which grows exponentially with the dimension. In this talk, I will explain how to leverage entropic regularization methods to define computationally efficient loss functions, approximating OT with a better sample complexity. More information and references can be found on the website of our book ``Computational Optimal Transport''.

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Topology of symplectic fillings of contact $3$-manifolds

Prof. Burak Özbağcı, Koç University

Abstract

Ever since Donaldson showed that every symplectic $4$-manifold admits a Lefschetz pencil and Giroux proved that every contact $3$-manifold admits an adapted open book decomposition, at the turn of the century, Lefschetz fibrations and open books have been used fruitfully to obtain interesting results about the topology of symplectic fillings of contact $3$-manifolds. In this talk, I will present my contribution to the subject at hand based on joint work with several coauthors during the past 20 years.

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AAA-least squares rational approximation and solution of Laplace problems

Prof. Nick Trefethen, Oxford University

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In the past five years, computation with rational functions has advanced greatly with the introduction of the AAA algorithm for barycentric rational approximation and of lightning least-squares solvers for Laplace, Stokes, and Helmholtz problems. Here we combine these methods into a two-step method for solving planar Laplace problems. First, complex rational approximations to the boundary data are determined by AAA approximation, locally near each corner or other singularity. The poles of these approximations outside the problem domain are then collected and used for a global least-squares fit to the solution. Typical problems are solved in a second of laptop time to 8-digit accuracy, all the way up to the corners. This is joint work with Stefano Costa.

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Positive harmonic functions on the Heisenberg group

Mr. Yves Benoist, Université Paris-Sud

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A harmonic function on a group is a function which is equal to the average of its translates, average with respect to a finitely supported measure. First, we will survey the history of this notion. Then we will describe the extremal non-negative harmonic functions on the Heisenberg group. We will see that the classical partition function occurs as such a function and that this function is the only one beyond harmonic characters.

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Bayesian inverse problems, Gaussian processes, and PDEs

Prof. Richard Nickl, University of Cambridge

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The Bayesian approach to inverse problems has become very popular in the last decade after seminal work by A. Stuart (2010). Particularly in nonlinear applications with PDEs and when using Gaussian process priors, this can leverage powerful MCMC algorithms to tackle difficult high dimensional and nonconvex inference problems, with associated \textit{uncertainty quantification methodology}. We review the main ideas and then discuss recent progress on rigorous mathematical performance guarantees for such algorithms. We will touch upon issues such as how to prove posterior consistency theorems, how to objectively validate posterior based statistical uncertainty quantification, as well as the polynomial time computability of posterior measures in some nonconvex model examples arising with PDEs.

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Looking at Euler flows through a contact mirror: Universality and Turing completeness

Prof. Eva Miranda, Universitat Politecnica de Catalunya

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The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. Recently, Tao \cite{T1, T2, tao} launched a programme to address the global existence problem for the Euler and Navier-Stokes equations based on the concept of universality. Inspired by this proposal, we show that the stationary Euler equations exhibit several universality features, in the sense that, any non-autonomous flow on a compact manifold can be extended to a smooth stationary solution of the Euler equations on some Riemannian manifold of possibly higher dimension \cite{cmpparxiv}. A key point in the proof is looking at the h-principle in contact geometry through a contact mirror, unveiled by Etnyre and Ghrist in \cite{EG} more than two decades ago. We end up this talk addressing a question raised by Moore in \cite{Mo} : \emph{\lq\lq Is hydrodynamics capable of performing computations?"}. The universality result above yields the Turing completeness of the steady Euler flows on a 17-dimensional sphere. Can this result be improved? In \cite{cmppPNAS} we construct a Turing complete Euler flow in dimension 3. Time permitting, we discuss this and other generalizations contained in \cite{cmparxiv}. This talk is based on several joint works with Cardona, Peralta-Salas and Presas. \begin{thebibliography}{99} \bibitem{cmpparxiv} R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. \emph{Universality of Euler flows and flexibility of Reeb embeddings}, arXiv:1911.01963. \bibitem{cmppPNAS} R. Cardona, E. Miranda, D. Peralta-Salas, F. Presas. \emph{Constructing Turing complete Euler flows in dimension $3$}. PNAS May 11, 2021 118 (19) e2026818118; https://doi.org/10.1073/pnas.2026818118. \bibitem{cmparxiv} R. Cardona, E. Miranda and D. Peralta-Salas, \emph{Turing universality of the incompressible Euler equations and a conjecture of Moore}, arXiv:2104.04356 . \bibitem{EG} J. Etnyre, R. Ghrist. \emph{Contact topology and hydrodynamics I. Beltrami fields and the Seifert conjecture}. Nonlinearity 13 (2000) 441--458. \bibitem{Mo} C. Moore. \emph{Generalized shifts: unpredictability and undecidability in dynamical systems}. Nonlinearity 4 (1991) 199--230. \bibitem{T1} T. Tao. \emph{On the universality of potential well dynamics}. Dyn. PDE 14 (2017) 219--238. \bibitem{T2} T. Tao. \emph{On the universality of the incompressible Euler equation on compact manifolds}. Discrete Cont. Dyn. Sys. A 38 (2018) 1553--1565. \bibitem{tao} T. Tao. \emph{Searching for singularities in the Navier-Stokes equations}. Nature Rev. Phys. 1 (2019) 418--419. \end{thebibliography}

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Recent Results on Lieb-Thirring Inequalities

Prof. Rupert Frank, Caltech / LMU Munich

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Lieb-Thirring inequalities are a mathematical expression of the uncertainty and exclusion principles in quantum mechanics. Since their discovery in 1975 they have played an important role in several areas of analysis and mathematical physics. We provide a gentle introduction to classical aspects of this subject and present some recent progress. Finally, we discuss extensions, in the spirit of Lieb-Thirring inequalities, of several inequalities in harmonic analysis to the setting of families of orthonormal functions.

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Quadrature error estimates for layer potentials evaluated near curved surfaces in three dimensions

Prof. Anna-Karin Tornberg, KTH Royal Institute of Technology

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When numerically solving PDEs reformulated as integral equations, so called layer potentials must be evaluated. The quadrature error associated with a regular quadrature rule for evaluation of a layer potential increases rapidly when the evaluation point approaches the surface and the integral becomes nearly singular. Error estimates are needed to determine when the accuracy is insufficient and a more costly special quadrature method should be utilized. In this talk, we start by considering integrals over curves in the plane, using complex analysis involving contour integrals, residue calculus and branch cuts, to derive such error estimates. We first obtain error estimates for layer potentials in $\mathbb{R}^2$, for both complex and real formulations of layer potentials, both for the Gauss-Legendre and the trapezoidal rule. By complexifying the parameter plane, the theory can be used to derive estimates also for curves in in $\mathbb{R}^3$. These results are then used in the derivation of the estimates for integrals over surfaces. The estimates that we obtain have no unknown coefficients and can be efficiently evaluated given the discretization of the surface, invoking a local one-dimensional root-finding procedure. Numerical examples are given to illustrate the performance of the quadrature error estimates. The estimates for integration over curves are in many cases remarkably precise, and the estimates for curved surfaces in $\mathbb{R}^3$ are also sufficiently precise, with sufficiently low computational cost, to be practically useful.

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Nonstandard finite elements for wave problems

Prof. Ilaria Perugia, University of Vienna

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Finite elements are a powerful, flexible, and robust class of methods for the numerical approximation of solutions to partial differential equations. In their standard version, they are based on piecewise polynomial functions on a partition of the domain of interest. Continuity requirements are possibly dictated by the regularity of the exact solutions. By breaking these constraints, new methods that are specifically tailored to the problem at hand have been developed in order to better reproduce physical properties of the exact solutions, to enhance stability, and to improve accuracy vs. computational cost. Nonstandard finite element approximations of wave propagation problems based on the space-time paradigm will be the focus of this talk.

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Finite groups of birational transformations

Prof. Yuri Prokhorov, Steklov Mathematical Institute

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I survey the classification theory of finite groups of birational transformations of higher-dimensional algebraic varieties. This theory has been significantly developed during the last 10 years due to the success of the minimal model program. I concentrate on certain properties of these groups in arbitrary dimension. Also, I am going to discuss the three-dimensional case in more details.

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Modelling the genetics of spatially structured populations

Prof. Alison Etheridge, University of Oxford

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The last century has seen remarkable developments in the nature and scale of genetic data, and the tools with which to interrogate them. Nonetheless fundamental questions remain unanswered. The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic and evolutionary mechanisms. The effect of these mechanisms, and the way in which they interact with one another, is also influenced by the spatial structure of the population. The hunt for adequate mathematical models with which to investigate the interaction between the forces of evolution and spatial structure is still very much in progress. Capturing stochastic effects in a biologically meangingful way is particularly challenging. In this talk we shall explore some of the progress that has been made, and some of the remaining challenges.

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Propositional Proof Complexity

Mr. Alexander Razborov, U Chicago/Steklov Institute

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Propositional proof complexity studies efficient provability of statements that can be expressed in quantifier-free form, in various proof systems and under various notions of ``efficiency''. Statements of interest come from a variety of sources that, besides traditional combinatorial principles and other mathematical theorems, include areas like combinatorial optimization, practical SAT solving and operation research. While many proof systems considered in the modern proof complexity are still traditional, in the sense that they are Hilbert-style and logic-based, a considerable amount of attention has been in recent years paid to systems modelling algebraic and semi-algebraic reasoning, including elementary convex geometry. In this talk I will attempt to convey some basic ideas underlying this vibrant area, including a necessarily based sample of illustrating examples.

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Uniqueness results for discrete Schrodinger evolutions

Prof. Eugenia Malinnikova, NTNU/Stanford

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I will give a survey of some recent results on uniqueness for (semi)-discrete Schrödinger equation and their connections to classical uncertainty principle.

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The dawn of formalized mathematics

Prof. Andrej Bauer, University of Ljubljana

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When I was a student of mathematics I was told that someone had formalized an entire book on analysis just to put to rest the question whether mathematics could be completely formalized, so that mathematicians could proceed with business as usual. I subsequently learned that the book was Landau's “Grundlagen”~\cite{landau65:_grund_analy}, the someone was L.~S.~van Benthem Jutting~\cite{benthem79:_check_landau_grund_autom}, the tool of choice was Automath~\cite{daalen94:_descr_autom_some_aspec_languag_theor}, and that popular accounts of history are rarely correct. Formalized mathematics did not die out. Computer scientists spent many years developing proof assistants~\cite{nuprl,coq-manual,agda,lean} -- programs that help create and verify formal proofs and constructions -- until they became good enough to attract the attention of mathematicians who felt that formalization had a place in mathematical practice. The initial successes came slowly and took a great deal of effort. In the last decade, the complete formalizations of the odd-order theorem~\cite{gonthier13:_machin_check_proof_odd_order_theor} and the solution of Kepler's conjecture~\cite{hales17:_formal_proof_kepler} sparked an interest and provided further evidence of viability of formalized mathematics. The essential role of proof assistants in the development of homotopy type theory~\cite{hottbook} and univalent mathematics~\cite{UniMath} showed that formalization can be an inspiration rather than an afterthought to traditional mathematics. % Today the community gathered around Lean~\cite{lean}, the newcomer among proof assistants, has tens of thousands of members and is growing very rapidly thanks to the miracle of social networking. % The new generation is ushering in a new era of mathematics. Formalized mathematics, in tandem with other forms of computerized mathematics~\cite{CarFarKohRab:bmobb19}, provides better management of mathematical knowledge, an opportunity to carry out ever more complex and larger projects, and hitherto unseen levels of precision. However, its transformative power runs still deeper. % The practice of formalization teaches us that formal constructions and proofs are much more than pointless transliteration of mathematical ideas into dry symbolic form. Formal proofs have rich structure, worthy of attention by a mathematician as well as a logician; contrary to popular belief, they can directly and elegantly express mathematical insights and ideas; and by striving to make them slicker and more elegant, new mathematics can be discovered. Formalized mathematics is changing the role of foundations of mathematics, too. A good century ago, a philosophical crisis necessitated the development of logic and set theory, which served as the bedrock upon which the 20th century mathematics was built safely. However, most proof assistants shun logic and set theory in favor of type theory, the original resolution of the crisis given by Bertrand Russell~\cite{russell25} and reformulated into its modern form by Per Martin-Löf~\cite{Martin-Lof-1972}. The reasons for this phenomenon are yet to be fully understood, but we can speculate that type theory captures mathematical practice more faithfully because it directly expresses the structure and constructions of mathematical objects, whereas set theory provides plentiful raw material with little guidance on how mathematical objects are to be molded out if it. \begin{thebibliography}{10} \bibitem{nuprl} S.F. Allen, M.~Bickford, R.L. Constable, R.~Eaton, C.~Kreitz, L.~Lorigo, and E.~Moran. Innovations in computational type theory using {Nuprl}. {\em Journal of Applied Logic}, 4(4):428--469, 2006. \bibitem{CarFarKohRab:bmobb19} Jacques Carette, William~M. Farmer, Michael Kohlhase, and Florian Rabe. Big math and the one-brain barrier -- the tetrapod model of mathematical knowledge. {\em Mathematical Intelligencer}, 43(1):78--87, 2021. \bibitem{lean} Leo de~Moura, Sebastian Ullrich, and Dany Fabian. Lean theorem prover. \bibitem{gonthier13:_machin_check_proof_odd_order_theor} Georges Gonthier, Andrea Asperti, Jeremy Avigad, Yves Bertot, Cyril Cohen, François Garillot, Stéphane~Le Roux, Assia Mahboubi, Russell O'Connor, Sidi~Ould Biha, Ioana Pasca, Laurence Rideau, Alexey Solovyev, Enrico Tassi, and Laurent Théry. A machine-checked proof of the odd order theorem. In Sandrine Blazy, Christine Paulin{-}Mohring, and David Pichardie, editors, {\em Interactive Theorem Proving - 4th International Conference, {ITP} 2013, Rennes, France, July 22-26, 2013. Proceedings}, volume 7998 of {\em Lecture Notes in Computer Science}, pages 163--179. Springer, 2013. \bibitem{hales17:_formal_proof_kepler} Thomas Hales, Mark Adams, Gertrud Bauer, Tat~Dat Dang, John Harrison, Le~Truong Hoang, Cezary Kaliszyk, Victor Magron, Sean McLaughlin, Tat~Thang Nguyen, Quang~Truong Nguyen, Tobias Nipkow, Steven Obua, Joseph Pleso, Jason Rute, Alexey Solovyev, Thi Hoai~An Ta, Nam~Trung Tran, Thi~Diep Trieu, Josef Urban, Ky~Vu, and Roland Zumkeller. A formal proof of the {K}epler conjecture. {\em Forum of Mathematics, Pi}, 5, 2017. \bibitem{landau65:_grund_analy} E.~G. H.~Y. Landau. {\em Grundlagen der Analysis}. Chelsea Pub. Co., New York, NY, USA, 1965. \bibitem{Martin-Lof-1972} Per Martin-Löf. An intuitionistic theory of types. In Giovanni Sambin and Jan~M. Smith, editors, {\em Twenty-five years of constructive type theory ({V}enice, 1995)}, volume~36 of {\em Oxford Logic Guides}, pages 127--172. Oxford University Press, 1998. \bibitem{agda} Ulf Norell. {\em Towards a practical programming language based on dependent type theory}. PhD thesis, Chalmers University of Technology, 2007. \bibitem{hottbook} The Univalent~Foundations Program. {\em Homotopy Type Theory: Univalent Foundations for Mathematics} , Institute for Advanced Study, 2013. \bibitem{coq-manual} Coq~Development Team. The {Coq} proof assistant reference manual, version 8.7. \bibitem{benthem79:_check_landau_grund_autom} L.~S. van Benthem~Jutting. Checking {Landau’s} “{Grundlagen}” in the {Automath} system. In {\em Mathematical Centre Tracts}, volume~83. Mathematisch Centrum, The Netherlands, 1979. \bibitem{daalen94:_descr_autom_some_aspec_languag_theor} D.~T. van Daalen. A description of automath and some aspects of its language theory. In {\em Selected Papers on Automath}, pages 101--126. North-Holland Pub. Co., Amsterdam, The Netherlands, 1994. \bibitem{UniMath} Vladimir Voevodsky, Benedikt Ahrens, Daniel Grayson, et~al. {\em UniMath}: {Univalent} {Mathematics}, 2016. \bibitem{russell25} Alfred~North Whitehead and Bertrand Russell. {\em Principia Mathematica}. Cambridge University Press, 1925--1927. \end{thebibliography}

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From linear to nonlinear n-width : optimality in reduced modelling

Prof. Albert Cohen, Sorbonne Université

Abstract

The concept of n-width has been introduced by Kolmogorov as a way of measuring the size of compact sets in terms of their approximability by linear spaces. From a numerical perspective it may be thought as a benchmark for the performance of algorithms based on linear approximation. In recent years this concept has proved to be highly meaningful in the analysis of reduced modeling strategies for complex physical problems described by parametric problems. This lecture will first review two significant results in this area that concern (i) the practical construction of optimal spaces by greedy algorithms and (ii) the preservation of the rate of decay of widths under certain holomorphic transformation. It will then focus on recent attempts to propose non-linear version of n-widths, how these notions relate to metric entropies, and how they could be relevant to practical applications.

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Exponential sums over finite fields

Prof. Emmanuel Kowalski, ETH Zürich

Abstract

Exponential sums are among the simplest mathematical objects that one can imagine, but also among the most remarkably useful and versatile in number theory. This talk will survey the history, the mysteries and the surprises of such sums over finite fields, with a focus on questions related to the distribution of values of families of exponential sums. General principles and applications will be illustrated by concrete examples, where sums of two squares, Sidon sets, Larsen's Alternative, the variance of arithmetic functions over function fields and the lines on cubic threefolds will make appearances. (Based on joint work with A. Forey and J. Fresán)

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Convex integration and synthetic turbulence

Prof. László Székelyhidi Jr., University of Leipzig

Abstract

In the past decade convex integration has been established as a powerful and versatile technique for the construction of weak solutions of various nonlinear systems of partial differential equations arising in fluid dynamics, including the Euler and Navier-Stokes equations. The existence theorems obtained in this way come at a high price: solutions are highly irregular, non-differentiable, and very much non-unique as there is usually infinitely many of them. Therefore this technique has often been thought of as a way to obtain mathematical counterexamples in the spirit of Weierstrass' non-differentiable function, rather than advancing physical theory; "pathological", "wild", "paradoxical", "counterintuitive" are some of the adjectives usually associated with solutions obtained via convex integration. In this lecture I would like to draw on some recent examples to show that there are many more sides to the story, and that, with proper usage and interpretation, the convex integration toolbox can indeed provide useful insights for problems in hydrodynamics.

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Some Recent Developments on the Geometry of Random Spherical Eigenfunctions

Prof. Domenico Marinucci, University of Rome Tor Vergata

Abstract

A lot of efforts have been devoted in the last decade to the investigation of the high-frequency behaviour of geometric functionals for the excursion sets of random spherical harmonics, i.e., Gaussian eigenfunctions for the spherical Laplacian $\Delta_{S^2}$. In this talk we shall review some of these results, with particular reference to the asymptotic behaviour of variances, phase transitions in the nodal case (the Berry's Cancellation Phenomenon), the distribution of the fluctuations around the expected values, and the asymptotic correlation among different functionals. We shall also discuss some connections with the Gaussian Kinematic Formula, with Wiener-Chaos expansions and with recent developments in the derivation of Quantitative Central Limit Theorems (the so-called Stein-Malliavin approach).

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From K{\" a}hler-Einstein metrics to zeros of zeta functions

Prof. Robert Berman, Chalmers Univ. of Technology

Abstract

While the existence of a unique Kähler-Einstein metrics on a canonically polarized manifold X was established already in the seventies there are very few explicit formulas available (even in the case of complex curves!). In this talk I will give a non-technical introduction to a probabilistic construction of Kähler-Einstein metrics, which, in particular, yields canonical approximations of the Kähler-Einstein metric on X. The approximating metrics in question are expressed as explicit period integrals and the conjectural extension to the case of a Fano variety leads to some intriguing connections to zeros of some Archimedean zeta functions.

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Laplacians on infinite graphs

Prof. Aleksey Kostenko, University of Ljubljana

Abstract

There are two different notions of a Laplacian operator associated with graphs: discrete graph Laplacians and continuous Laplacians on metric graphs (widely known as quantum graphs). Both objects have a venerable history as they are related to several diverse branches of mathematics and mathematical physics. The existing literature usually treats these two Laplacian operators separately. In this talk, I will focus on the relationship between them (spectral, parabolic and geometric properties). One of our main conceptual messages is that these two settings should be regarded as complementary (rather than opposite) and exactly their interplay leads to important further insight on both sides. Based on joint work with Noema Nicolussi.

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Structure and classification of simple amenable $C^*$-algebras

Prof. Stuart White, University of Oxford

Abstract

In this talk I will give an overview of recent progress in the structure theory of simple amenable $C^*$-algebras and classification results. $C^*$-algebras are norm closed self-adjoint subalgebras of the bounded operators on a Hilbert space, with examples arising naturally from unitary representations of groups, and topological dynamics. They have a topological flavour, seen through the commutative algebras of continuous functions on locally compact Hausdorff spaces. The classification of $C^*$-algebras has its spiritual origins in the powerful structure and classification theorems for von Neumann algebras of Connes in the '70s. However, in the topological setting of $C^*$-algebras, higher dimensional phenomena can obstruct classification in general. Progress over the last decade has seen the identification of abstract structural conditions which give the maximal family of algebras which can be classified by $K$-theory and traces. These conditions now have equivalent formulations of very different natures, which can be used to bring naturally occurring examples within the scope of classification. The talk is based in part on joint works with Castillejos, Carri\'on, Evington, Gabe, Schafhauser, Tikuisis, and Winter.

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HMS categorical symmetries and hypergeometric systems

Špela Špenko, Université Libré de Bruxelles

Abstract

Hilbert's 21st problem asks about the existence of Fuchsian linear differential equations with a prescribed "monodromy representation" of the fundamental group. The first (slightly erroneous) solution was proposed by a Slovenian mathematician Plemelj. A suitably adapted version of this problem was solved, depending on the context, by Deligne, Mebkhout, Kashiwara-Kawai, Beilinson-Bernstein, ... The solution is now known as the Riemann-Hilbert correspondence. Homological mirror symmetry predicts the existence of an action of the fundamental group of the "stringy Kähler moduli space" (SKMS) on the derived category of an algebraic variety. This prediction was established by Halpern-Leistner and Sam for certain toric varieties. The decategorification of the action found by HLS yields a representation of the fundamental group of the SKMS and in joint work with Michel Van den Bergh we show that it is given by the monodromy of an explicit hypergeometric system of differential equations.

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Tackling discrete optimization problems by continuous methods

Prof. Mirjam Dür, Augsburg University

Abstract

Many NP-hard discrete and combinatorial optimization problems can be formulated with the help of quadratic expressions. These in turn can be linearized by lifting the problem from n-dimensional space to the space of n by n matrices. We show that this leads to a conic optimization problem, i.e., an optimization problem in matrix variables where a constraint requires the matrix to be in the cone of so called copositive or completely positive matrices. The complexity of the original problem is entirely shifted into the cone constraint. We discuss the pros and cons of this approach, and we review the state of the art in this area, covering both theory and numerical solution approaches.

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Fast algorithms from low-rank updates

Prof. Daniel Kressner, EPFL

Abstract

The development of efficient numerical algorithms for solving large-scale linear systems is one of the success stories of numerical linear algebra that has had a tremendous impact on our ability to perform complex numerical simulations and large-scale statistical computations. Many of these developments are based on multilevel and domain decomposition techniques, which are closely linked to Schur complements and low-rank updates of matrices. In this talk, we explain how these tools carry over to other important linear algebra problems, including matrix functions and matrix equations. Fast algorithms are derived from combining divide-and-conquer strategies with low-rank updates of matrix functions. The convergence analysis of these algorithms is built on a multivariate extension of the celebrated Crouzeix-Palencia result. The newly developed algorithms are capable of addressing a wide variety of matrix functions and matrix structures, including sparse matrices as well as matrices with hierarchical low rank and Toeplitz-like structures. Their versatility will be demonstrated with several applications and extensions. This talk is based on joint work with Bernhard Beckermann, Alice Cortinovis, Leonardo Robol, Stefano Massei, and Marcel Schweitzer.

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An invitation to Poisson Geometry

Prof. Marius Crainic, Utrecht University

Abstract

Poisson structures originate in the work of Lagrange and Poisson on the motion of planets in the solar system; the process of understanding them was long and it prompted the discovery of several fundamental concepts in mathematics, such as: Jacobi identity, Maurer-Cartan equations, etc. Poisson Geometry (the geometric study of Poisson structures) can be traced back to the work of Lie and Kirilov; the first systematic studies are found in the work of Lichnerowicz in the 1970s and Weinstein in the 1980s. Its remarkable development over the last few decades was driven by several problems (such as integrability or Conn's linearization theorem) and led to surprising new connections with various other fields. All these led to the present-day understanding of Poisson Geometry: it is an amalgam of Lie Theory, Symplectic Geometry and Foliation Theory, offering the framework for exciting interactions between these theories, as well as others. In the talk I will try to expand this abstract.

EMS Prize Winners

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Reciprocity laws for torsion classes

Ana Caraiani, Imperial College London

Abstract

The Langlands program is a vast network of conjectures that connect many areas of pure mathematics, such as number theory, representation theory, and harmonic analysis. At its heart lies reciprocity, the conjectural relationship between Galois representations and modular, or automorphic forms. A famous instance of reciprocity is the modularity of elliptic curves over the rational numbers: this was the key to Wiles’s proof of Fermat’s last theorem. I will give an overview of some recent progress in the Langlands program, with a focus on new reciprocity laws over imaginary quadratic fields.

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Elliptic curves and modularity

Jack Thorne, University of Cambridge

Abstract

The modularity conjecture for elliptic curves is most famous in its formulation for elliptic curves over the rational numbers -- indeed, Wiles proved the modularity of semistable elliptic curves over the rationals as part of his proof of Fermat's Last Theorem. Recently it has become possible to attack the problem of modularity of elliptic curves over more general number fields. I will explain what this means, what we know, and what kinds of consequences we can expect for the solution of Diophantine equations.

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Excitation spectrum of dilute trapped Bose gases

Phan Thanh Nam, LMU Munich

Abstract

We will discuss the low-lying eigenvalues of trapped Bose gases in the Gross-Pitaevskii regime. In particular, we will derive a nonlinear correction to the Bogoliubov approximation, thus capturing precisely the two-body scattering process of the particles. This extends a result of Boccato, Brennecke, Cenatiempo and Schlein on the homogeneous Bose gas. The talk is based on joint work with Arnaud Triay.

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Zero sets of Laplace eigenfunctions

Alexandr Logunov, Princeton University

Abstract

In the beginning of 19th century Napoleon set a prize for the best mathematical explanation of Chladni’s resonance experiments. Nodal geometry studies the zeroes of solutions to elliptic differential equations such as the visible curves that appear in Chladni's nodal portraits. We will discuss the geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interesting relation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. Topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE.

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On primes, almost primes and the Möbius function in short intervals

Kaisa Matomäki, University of Turku

Abstract

In this talk I will introduce some fundamental concepts in analytic number theory such as primes, the Riemann zeta function and the Möbius function. I will discuss classical results concerning them and connecting them to each other. Then I will move on to discussing the distribution of primes, almost primes and the Möbius function in short intervals. In particular I will describe some recent progress on the topic.

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From branching singularities of minimal surfaces to non-smoothness points on an ice-water interface

Joaquim Serra, ETH Zurich

Abstract

Stefan's problem, dating back to the XIX century, aims to describe the evolution of a block of ice melting in water. Its mathematical analysis experienced few progress until the 1970's, when Duvaut reformulated it as the gradient flow of a nice convex functional. In 1977, Caffarelli proved that the ice-water interface is an smooth surface outside of a certain closed set: the so-called singular set. This was as huge breakthrough. However, methods available back in the 1970's did not allow for a fine description of the structure of the singular set. During the following 20 years, Almgren developed his theory of branching singularities of minimal surfaces, and in 2008 these methods were applied to the not-so-well-known ``thin obstacle problem''. This induced, in recent years, a fruitful use of Almgren's methods to study singularities in Stefan's problem. But even with these powerful new tools in hand, we were only halfway to obtaining a fully satisfying description of the non-smoothness points on an ice-water interface...

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Lefschetz beyond positivity and its implications

Karim Adiprasito, University of Copenhagen

Abstract

I will present results of the Hard Lefschetz type that go beyond the classical case of Kaehler manifolds and projectivity, and survey their applications in combinatorics, geometry and topology.

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Discrete monodromy groups and Hodge theory

Simion Filip, University of Chicago

Abstract

The monodromy of differential equations has been studied for a long time, providing motivation for many of the concepts in complex analysis, algebraic geometry, and dynamical systems. After providing the necessary background, I will present some applications of ideas from dynamics to the study of monodromy groups of families of algebraic varieties, and explain how they relate to differential equations. I will also describe some connections between interesting classes of discrete subgroups of Lie groups and Hodge theory.

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The magic of dimensions 8 and 24

Prof. Maryna Viazovska, Ecole Polytechnique Federale de Lausanne

Abstract

This talk is about two exceptional mathematical objects: the E8 lattice and the Leech lattice. These two lattices are extraordinary tight and symmetric. They satisfy many extremal properties, in particular provide the densest sphere packings in their respective dimensions and are universally optimal. This lecture is an attempt to answer the question: what is so unusual about dimensions 8 and 24 and why they host these two "magic" lattices?

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Smooth compactifications of differential graded categories

Alexander Efimov, Steklov Mathematical Institute of RAS

Abstract

We will give an overview of results on smooth categorical compactifications, the questions of their existence and their construction. The notion of a categorical smooth compactification is a straightforward generalization of the corresponding usual notion for algebraic varieties. First, we will explain the result on the existence of smooth compactifications of derived categories of coherent sheaves on separated schemes of finite type over a field of characteristic zero. Namely, such a derived category can be represented as a quotient of the derived category of a smooth projective variety, by a triangulated subcategory generated by a single object. Then we will give an example of a homotopically finite DG category which does not have a smooth compactification: a counterexample to one of the Kontsevich's conjectures on the generalized Hodge to de Rham degeneration. If time permits, we will formulate a K-theoretic criterion for existence of a smooth categorical compactification, using a DG categorical analogue of Wall's finiteness obstruction from topology.

Felix Klein Prize Winner

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Overcoming the curse of dimensionality: from nonlinear Monte Carlo to deep learning

Prof. Arnulf Jentzen, University of Münster & The Chinese University of Hong Kong, Shenzhen

Abstract

Partial differential equations (PDEs) are among the most universal tools used in modelling problems in nature and man-made complex systems. For example, stochastic PDEs are a fundamental ingredient in models for nonlinear filtering problems in chemical engineering and weather forecasting, deterministic Schroedinger PDEs describe the wave function in a quantum physical system, deterministic Hamiltonian-Jacobi-Bellman PDEs are employed in operations research to describe optimal control problems where companys aim to minimise their costs, and deterministic Black-Scholes-type PDEs are highly employed in portfolio optimization models as well as in state-of-the-art pricing and hedging models for financial derivatives. The PDEs appearing in such models are often high-dimensional as the number of dimensions, roughly speaking, corresponds to the number of all involved interacting substances, particles, resources, agents, or assets in the model. For instance, in the case of the above mentioned financial engineering models the dimensionality of the PDE often corresponds to the number of financial assets in the involved hedging portfolio. Such PDEs can typically not be solved explicitly and it is one of the most challenging tasks in applied mathematics to develop approximation algorithms which are able to approximatively compute solutions of high-dimensional PDEs. Nearly all approximation algorithms for PDEs in the literature suffer from the so-called "curse of dimensionality" in the sense that the number of required computational operations of the approximation algorithm to achieve a given approximation accuracy grows exponentially in the dimension of the considered PDE. With such algorithms it is impossible to approximatively compute solutions of high-dimensional PDEs even when the fastest currently available computers are used. In the case of linear parabolic PDEs and approximations at a fixed space-time point, the curse of dimensionality can be overcome by means of Monte Carlo approximation algorithms and the Feynman-Kac formula. In this talk we prove that suitable deep neural network approximations do indeed overcome the curse of dimensionality in the case of a general class of semilinear parabolic PDEs and we thereby prove, for the first time, that a general semilinear parabolic PDE can be solved approximatively without the curse of dimensionality.

Otto Neugebauer Prize Winner

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On how mathematicians’ historical and philosophical reflections have been essential to the advancement of mathematics: A historical perspective

Dr. Karine Chemla, CNRS

Abstract

The presentation will focus on an episode in which the historical and philosophical work carried out by mathematicians played a decisive role in overcoming a mathematical difficulty and introducing an idea that had a major impact on future developments in mathematics. The episode in question is the introduction in 1845 by Ernst Eduard Kummer (1810-1893) of the "ideal factors" of what he called the "complex numbers". The first public presentation of this concept by Kummer in 1846 allows us to trace the impact on this breakthrough of the historical and philosophical reflections that Jean-Victor Poncelet (1788-1867) and Michel Chasles (1793-1880) developed while giving shape to what would become projective geometry. This episode suggests the benefits that could derive from a more systematic inclusion of historical and philosophical approaches in the practice of mathematics, as an integral part of it.

Public speakers

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European mathematics: a history in 200 stamps

Prof. Robin Wilson, Open University, UK (retired)

Abstract

In this talk I shall cover the entire history of European mathematics in around 50 minutes -- illustrating my talk with around 200 informative (and sometimes amusing) postage stamps from around the world featuring mathematics and mathematicians. This talk is aimed at an interested general audience and assumes no mathematical knowledge.

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Crossing Numbers: From Art and Circuit Design to Knots and Number Theory

Prof. Bojan Mohar, Simon Fraser University / IMFM

Abstract

In 1864, Sylvester asked what is the probability that four randomly chosen points in the plane form a convex quadrilateral. During World War II, Paul Tur\'an asked about an optimal design of railroads connecting $n$ factories with $m$ warehouses. In 1950s, the British painter Anthony Hill asked how to draw a network of $n$ interconnected nodes with fewest number of crossings. All these questions are still unresolved. The speaker will overview mathematical foundations of the common theme --- the theory of crossing numbers of graphs --- and will show some surprising relations with other branches of mathematics.

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Topological explorations in neuroscience

Prof. Kathryn Hess, EPFL

Abstract

The brain of each and every one of us is composed of hundreds of billions of neurons – often called nerve cells – connected by hundreds of trillions of synapses, which transmit electrical signals from one neuron to another. In reaction to stimulus, waves of electrical activity traverse the network of neurons, processing the incoming information. Tools provided by the field of mathematics called algebraic topology enable us to detect and describe the rich structure hidden in this dynamic tapestry. During this talk, I will guide you on a mathematical mystery tour of what topology has revealed about how the brain processes information.

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Mathematics: art or science?

Prof. Stanislav Smirnov, Université de Genève

Abstract

Mathematics is an amazing and mysterious science. Ever since the time of Plato, philosophers argue whether mathematical objects are imaginary, or whether they come from the real world, while mathematicians mostly prove theorems without even asking about their link to reality. On the other hand, the Pharaohs of Egypt and the Kings of Babylon had already grasped the practical power of mathematics, and of course the technological advances of the past two centuries are built on successful applications of our science. We will discuss the mathematician's perspective on\\ - where does mathematics come from?\\ - why is the "imaginary" science so useful in real life?\\ - how we choose problems to work on, and why do we find our science so fascinating? We will not be able to answer all these questions in our talk, but we will try to give a glimpse of how mathematicians work.

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Lie theory without groups

Prof. Andrei Okounkov, Columbia/Skoltech/HSE

Abstract

Group theory and, in particular, Lie group theory is central to many areas of mathematics and to many applications. Lie theory is a masterpiece of a theory and it may feel nearly complete. Because of this, and also because of the demand from applications, many Lie theorists have been exploring new worlds, in which groups yield the center stage to other geometric and algebraic structures.

Hirzebruch Lecture

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A mathematical journey through scales

Prof. Martin Hairer, Imperial College London

Abstract

The tiny world of particles and atoms and the gigantic world of the entire universe are separated by scales spanning about forty orders of magnitudes. As we move from one to the other, the laws of nature can behave in drastically different ways, going from quantum physics to general relativity through Newton’s classical mechanics, not to mention other intermediate "ad hoc" theories. Understanding the way in which the behaviour of mathematical models changes as we move from one scale to another is one of the great classical questions in mathematics and theoretical physics. The aim of this talk is to explore how these questions still inform and motivate interesting problems in probability theory and why so-called toy models, despite their superficially playful character, can sometimes lead to useful quantitative (and not just qualitative) predictions.

A game theory and its applications (MS - ID 56)

Probability

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How to beat the $1/e$-strategy of best choice (the random arrivals problem)

Prof. Alexander Gnedin, Queen Mary, University of London

Abstract

Elimination of dominated strategies is the fundamental technique used to reduce the size of a finite zero-sum game. For infinite games, however, the dominance phenomenon may occur within the set of minimax strategies. See [1] and references therein for the Blackwell-Hill-Cover paradox of this kind. In this talk we consider a more involved optimal stopping game. In the best choice problem with random arrivals, an unknown number $n$ of rankable items arrive at times sampled from the uniform distribution. As is well known, a real-time player can ensure stopping at the overall best item with probability at least $1/e$ by means of the strategy $\tau^*$ that waits until time $1/e$ then selects the first relatively best item to appear (if any). The number $1/e$ is also the value of the game against an adversary in charge of the variable $n$. We show that the adversary has no minimax strategy and\\ \noindent (i) for every $u$ there exist stopping strategies that strictly improve upon $\tau^*$ simultaneously for all $n\leq u$,\\ (ii) there exists a simple strategy outperforming $\tau^*$ simultaneously for all $n>1$ (strictly for $n>2$),\\ (iii) there exist more complex strategies strictly outperforming $\tau^*$ simultaneously for all $n>2$,\\ (iv) for every $\ell\geq 1$ there exist still more complex strategies that guarantee the winning probability at least $1/e$ for all $n$, and are outperforming $\tau^*$ simultaneously for all $n>\ell$.\\ (v) in the other direction (as the $\ell=\infty$ case of (iv)), there exist stopping strategies that guarantee the winning chance $1/e$, but are strictly dominated by $\tau^*$. The stopping strategies we employ are defined in terms of multiple time cutoffs, and rely decisions on both the arrival time of relatively best item in question and the number of arrivals seen so far. \\ %\begin{thebibliography}{99} %\bibitem{Guess} \noindent {\bf References}\\ Gnedin, A. (2016) Guess the larger number. {\it Math. Appl. (Warsaw)} {\bf 44}, 183--207. %\end{thebibliography}

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Schumpeterian Evolution of Consumers' Optima -- A Game Theory Insight

Dr. Marta Kornafel, Cracow University of Economics

Abstract

We will consider the behaviour of consumers' optimal allocations in the result of Schumpeter evolution of economy. The goal of consumer in general equilibrium model is to choose an optimal allocation that maximises his preferences over the budget set, given the price and initial allocation. The crucial point is to justify -- taking into account the changing preferences -- that the consumers choosing the optimal allocation at every stage of evolutionary process will end up at an optimal state of the final economy. In the talk we will provide the conditions, under which the positive answer is possible.

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Nash Equilibria in certain two-choice multi-player games played on the ladder graph

Ms. Victoria Sánchez Muñoz, National University of Ireland Galway

Abstract

We compute analytically the number of Nash Equilibria (NE) for a two-choice game played on a (circular) ladder graph with $2n$ players. We consider a set of games with generic payoff parameters, with the only requirement that a NE occurs if the players choose opposite strategies (anti-coordination game). The results show that for both, the ladder and circular ladder, the number of NE grows exponentially with (half) the number of players $n$, as $N_{NE}(2n)\sim C(\varphi)^n$, where $\varphi=1.618...$ is the golden ratio and $C_{circ}>C_{ladder}$. In addition, the value of the scaling factor $C_{ladder}$ depends on the value of the payoff parameters. However, that is no longer true for the circular ladder (3-degree graph), that is $C_{circ}$ is constant, which might suggest that the topology of the graph indeed plays an important role for setting the number of NE.

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Expected duration in multilateral selection problems

Prof. Krzysztof Szajowski, Wroclaw University of Science and Technology

Abstract

This paper treats the decision problem related to the observation of a Markov process by decision makers. The information delivered to the players is based on the aggregation of the high-frequency data by some functions. Admissible strategies are stopping moments related to the available information. The payments are defined by the state at the time of stopping. The players' decision to stop has various effects which depend on the decision makers' type (v.~\cite{SzaSka2019:Multilateral}). The knowledge about the type of the players is not public and in this way, the payers have also different information. The details of the description allow to formulate the problem as a Bayesian game with sets of strategies based on the stopping times. It is an extension of the Dynkin's game related to the observation of a Markov process with the random assignment mechanism of states to the players. The main question considered now is the expected duration of each DM in the game (v.~\cite{Dem2021:Duration}). Some examples related to the best choice problem (BCP) are analyzed (cf.~\cite{Yeo1997:Duration} and \cite{FerHarTam1992:Duration}). \begin{thebibliography}{4} \expandafter\ifx\csname urlstyle\endcsname\relax \providecommand{\doi}[1]{doi:#1}\else \providecommand{\doi}{doi:\begingroup\urlstyle{rm}\Url}\fi \bibitem{Dem2021:Duration} S.~Demers. \newblock Expected duration of the no-information minimum rank problem. \newblock \emph{Statist. Probab. Lett.}, 168:\penalty0 108950, 5, 2021. \newblock ISSN 0167-7152. \newblock \doi{10.1016/j.spl.2020.108950}. \bibitem{FerHarTam1992:Duration} T.~S. Ferguson, J.~P. Hardwick, and M.~Tamaki. \newblock Maximizing the duration of owning a relatively best object. \newblock In \emph{Strategies for sequential search and selection in real time ({A}mherst, {MA}, 1990)}, volume 125 of \emph{Contemp. Math.}, pages 37--57. Amer. Math. Soc., Providence, RI, 1992. \newblock \doi{10.1090/conm/125/1160608}. \bibitem{SzaSka2019:Multilateral} K.~Szajowski and M.~Skarupski. \newblock On multilateral incomplete information decision models. \newblock \emph{High Frequency}, 2\penalty0 (3-4):\penalty0 158--168, 2019. \newblock ISSN 2470-6981. \newblock \doi{10.1002/hf2.10047}. \bibitem{Yeo1997:Duration} G.~F. Yeo. \newblock Duration of a secretary problem. \newblock \emph{J. Appl. Probab.}, 34\penalty0 (2):\penalty0 556--558, 1997. \newblock ISSN 0021-9002. \newblock \doi{10.1017/s0021900200101184}. \end{thebibliography}

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Potential discrete-time dynamic games

Dr. Alejandra Fonseca-Morales, UNISON

Abstract

A potential dynamic game is a dynamic game for which one can find an optimal control problem whose optimal solutions are also Nash equilibria for the original dynamic game. We cover in this talk team games and action independent transition games in order to identify Markov-Nash equilibria via the potential approach.

A journey from pure to applied mathematics (MS - ID 53)

General Topics

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On the p-adic geometry of Shimura varieties

Ana Caraiani, Imperial College London

Abstract

The Langlands program is a vast network of conjectures that connect many areas of pure mathematics, such as number theory, representation theory, and harmonic analysis. Shimura varieties are certain algebraic-geometric objects that play an important role in the Langlands program. They have a nice moduli interpretation and they provide, in many cases, a geometric realisation of the global Langlands correspondence. I will illustrate the beautiful geometry of Shimura varieties using the simplest example, that of the modular curve. I will then mention some recent results that use p-adic geometry, specifically the theory of perfectoid spaces.

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Very weak solutions to PDEs in inhomogeneous and anisotropic spaces

Prof. Iwona Chlebicka, University of Warsaw

Abstract

I will discuss well-posedness and regularity to nonlinear PDEs of simple divergent form \[ - {\rm div} A(x,\nabla u)= f\quad \text{or}\quad \partial_t u- {\rm div} A(t,x,\nabla u)= f,\] where the datum $f$ is merely integrable or even is a measure, whereas the growth of $A$ is governed by an inhomogeneous and fully anisotropic $N$-function $M(x,\nabla u)$. Inhomogeneity means space-dependence and anisotropy yields dependence of $M$ on $\nabla u$ not necessarily via its length $|\nabla u|$. The datum is too poorly regular for weak solutions to exist, thus a more delicate notion of very weak solutions is necessary. Besides existence and uniqueness they share some regularity properties of the weak ones. The generality we admit covers classical linear and polynomial growth operators possibly with measurable coefficients, generalizations of $p$-Laplacian involving log-H\"older continuous variable exponent, as well as problems posed in fully anisotropic Orlicz spaces (under no growth conditions), and double-phase spaces within the range of parameters sharp for density of smooth functions. The conditions for the density and their importance will be stressed. The talk will be based on series of joint papers with: Youssuf Ahmida, Angela Alberico, Andrea Cianchi, Piotr Gwiazda, Ahmed Youssfi, and Anna Zatorska-Goldstein.

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On New Approaches for Nonsmooth Optimization

Ms. Andrea Walther, Humboldt Universität zu Berlin

Abstract

Numerous optimization tasks exhibit a nonsmooth behavior. In contrast to the classical smooth case, where optimality conditions are well studied and understood, criteria to determine whether a given point is optimal or even just stationary are still the subject of ongoing research for nonsmooth functions to be minimized. In this presentation, first we discuss new optimality conditions for a large class of piecewise smooth functions using so-called kink qualifications. Here, also the computational complexity to verify the new criteria is covered. Next, we present optimization algorithms resulting from these findings. Finally, we present some applications that fit into the considered problem class.

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Flows of nonsmooth vector fields: new results on non uniqueness and commutativity

Prof. Maria Colombo, EPFL Lausanne

Abstract

Given a vector field in $R^d$, the classical Cauchy-Lipschitz theorem shows existence and uniqueness of its flow (namely, the solution X(t) of the ODE $X'(t)=b(t,X(t))$ from any initial datum $x\in R^d$) provided the vector field is sufficiently smooth. The theorem looses its validity as soon as v is slightly less regular. However, in 1989, Di Perna and Lions introduced a generalized notion of flow, consisting of a suitable selection among the trajectories of the associated ODE, and they showed existence, uniqueness and stability for this notion of flow for much less regular vector fields. The talk presents an overview and new results in the context of the celebrated DiPerna-Lions and Ambrosio’s theory on flows of Sobolev vector fields, including a negative answer to the following long-standing open question: are the trajectories of the ODE unique for a.e. initial datum in $R^d$ for vector fields as in Di Perna and Lions theorem? We will exploit the connection between the notion of flow and an associated PDE, the transport equation, and combine ingredients from probability theory, harmonic analysis, and the “convex integration” method for the construction of nonunique solutions to certain PDEs.

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The fast p-Laplacian evolution equation. Global Harnack principle and fine asymptotic behaviour

Dr. Diana Stan, Universidad de Cantabria

Abstract

We study fine global properties of nonnegative solutions to the Cauchy Problem for the fast p-Laplacian evolution equation on the whole Euclidean space, in the so-called "good fast diffusion range". It is well-known that non-negative solutions behave for large times as B, the Barenblatt (or fundamental) solution, which has an explicit expression. We prove the so-called Global Harnack Principle (GHP), that is, precise global pointwise upper and lower estimates of nonnegative solutions in terms of B. This can be considered the nonlinear counterpart of the celebrated Gaussian estimates for the linear heat equation. To the best of our knowledge, analogous issues for the linear heat equation, do not possess such clear answers, only partial results are known. Also, we characterize the maximal (hence optimal) class of initial data such that the GHP holds, by means of an integral tail condition, easy to check. Finally, we derive sharp global quantitative upper bounds of the modulus of the gradient of the solution, and, when data are radially decreasing, we show uniform convergence in relative error for the gradients.

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Identifying 3D Genome Organization in Diploid Organisms via Euclidean Distance Geometry

Prof. Kaie Kubjas, Aalto University

Abstract

The 3D organization of the genome plays an important role for gene regulation. Chromosome conformation capture techniques allow one to measure the number of contacts between genomic loci that are nearby in the 3D space. In this talk, we study the problem of reconstructing the 3D organization of the genome from whole genome contact frequencies in diploid organisms, i.e. organisms that contain two indistinguishable copies of each genomic locus. In particular, we study the identifiability of the 3D organization of the genome and optimization methods for reconstructing it. This talk is based on joint work with Anastasiya Belyaeva, Lawrence Sun and Caroline Uhler.

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Linear non-degeneracy and uniqueness of the bubble solution for the critical fractional H\'enon equation in $\mathbb{R}^N$

Dr. Begoña Barrios, Universidad de La Laguna

Abstract

In this talk we show a linear non-degeneracy result of positive radially symmetric solutions of $$ (-\Delta)^s u = |x|^{\alpha} u^{\frac{N+2s+2\alpha}{N-2s}}\mbox{ in }\mathbb{R}^N, $$ where $(-\Delta)^s$ is the fractional Laplacian operator , $0 < s < 1$, $\alpha>-2s$ and $N>2s$. Moreover, as a consequence, a uniqueness result of those solutions with Morse index equal to one is obtained. In particular, we get that the ground state solution is unique. Our approach follows some ideas developed in the deep, and celebrated, papers done by R. Frank and E. Lenzmann (Acta Math. 2013) and R. Frank, E. Lenzmann, L. Silvestre (Comm. Pure Appl. Math. 2016) but, of course, our proofs are not based on ODE arguments as occurs in the local case. Our non-degeneracy result extends, in the radial setting, some known theorems proved by J. D\'avila, M. del Pino and Y. Sire (Proc. Amer. Math. Soc. 2013) and by F. Gladiali, M. Grossi and S.L.N. Neves (Adv. Math. 2013). However, due to the nature of the fractional operator and the weight in nonlinearity, we also argue in a different way than these authors do. The results presented in this talk have been obtained in collaboration with S. Alarc\'on and A. Quaas.

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On arithmetic functions orthogonal to deterministic sequences

Prof. Joanna Kułaga-Przymus, Nicolaus Copernicus University in Toruń

Abstract

The arithmetic M\"obius function $\mathbf{\mu}$ is believed to behave randomly. One way to express such a behaviour is the content of Sarnak's conjecture from 2010 on the M\"obius disjointness from all deterministic sequences. In 2015 Veech formulated a (dynamical) property of the M\"obius function itself and he conjectured that this property is equivalent to Sarnak's conjecture. My talk will be devoted to the strategy of the proof of Veech's conjecture and some consequences of this equivalence. The talk is based on a joint work with Adam Kanigowski, Mariusz Lema\'nczyk and Thierry de la Rue.

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Independent sets in random subgraphs of the hypercube

Dr. Gal Kronenberg, University of Oxford

Abstract

Independent sets in bipartite regular graphs have been studied extensively in combinatorics, probability, computer science and more. The problem of counting independent sets is particularly interesting in the $d$-dimensional hypercube $\{0,1\}^d$, motivated by the lattice gas hardcore model from statistical physics. Independent sets also turn out to be very interesting in the context of random graphs. In this talk we will review some fundamental results, and discuss new results on random subgraphs of the hypercube. This talk is based on joint work with Yinon Spinka.

Algorithmic Graph Theory (MS - ID 54)

Combinatorics and Discrete Mathematics

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The Slow-Coloring Game on a Graph

Prof. Douglas West, Zhejiang Normal University and University of Illinois

Abstract

The {\it slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a non-empty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it; the score under optimal play is the {\it cost} of the graph. A greedy strategy for Painter keeps the cost of $G$ to at most $\chi(G)n$ when $G$ has $n$ vertices, which is asyptotically sharp for Tur\'an graphs. On various classes Painter can do better. For $n$-vertex trees the maximum cost is $\lfloor 3n/2\rfloor$. There is a linear-time algorithm and inductive formula to compute the cost on trees, and we know all the extremal $n$-vertex trees. Also, Painter can keep the cost to at most $(1+3k/4)n$ when $G$ is $k$-degenerate, $7n/3$ when $G$ is outerplanar, $3.9857n$ when $G$ is acyclically $5$-colorable, and $3.4n$ when $G$ is planar. These bounds are not believed to be sharp. We will discuss strategies (algorithms) for Lister and Painter that establish various lower and upper bounds. The results appear in three papers with various subsets of Grzegorz Gutowski, Tomasz Krawczyk, Thomas Mahoney, Krzysztof Maziarz, Gregory J. Puleo, Michal Zajac, and Xuding Zhu.

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On Efficient Domination for $H$-free bipartite graphs

Prof. Andreas Brandstaedt, Institute of Informatics

Abstract

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} ({\em e.d.s.} for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (\emph{ED}) problem, asks for the existence of an e.d.s.\ in $G$; it is the Exact Cover problem for the closed neighborhood hypergraph of $G$. ED is known to be NP-complete even for very restricted $H$-free graph classes such as for $2P_3$-free chordal graphs (and thus, for $P_7$-free graphs) while it is solvable in polynomial time for $P_6$-free graphs. For $H$-free graphs, ED is either NP-complete or polynomial, i.e., a dichotomy. However, for $H$-free bipartite graphs, there is no such dichotomy. Lu and Tang showed that ED is NP-complete for chordal bipartite graphs and for planar bipartite graphs; actually, ED is NP-complete even for planar bipartite graphs with vertex degree at most 3 and girth at least $g$ for every fixed $g$. Thus, ED is NP-complete for $K_{1,4}$-free bipartite graphs and for $C_4$-free bipartite graphs. For classes of bounded clique-width, ED is solvable in polynomial time. Dabrowski and Paulusma published a dichotomy for clique-width of $H$-free bipartite graphs. For instance, clique-width of $S_{1,2,3}$-free bipartite graphs is bounded. In Discrete Applied Math. 270 (2019), we published a manuscript ``On efficient domination for some classes of $H$-free bipartite graphs''. We showed that (weighted) ED can be solved in polynomial time for $H$-free bipartite graphs when $H$ is $P_7$ or $\ell P_4$ for fixed $\ell$, and similarly for $P_9$-free bipartite graphs with vertex degree at most 3, and when $H$ is $S_{2,2,4}$. In this talk, we also mention a polynomial time solution for $P_8$-free bipartite graphs (which was an open problem in our publication).

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Shifting any path to an avoidable one

Matjaž Krnc, FAMNIT (University of Primorska)

Abstract

A vertex $v$ in a graph $G$ is avoidable if every induced path on three vertices with middle vertex $v$ is contained in an induced cycle. Dirac's classical result from 1961 on the existence of simplicial vertices in chordal graphs is equivalent to the statement that every chordal graph has an avoidable vertex. Beisegel, Chudovsky, Gurvich, Milani\v{c}, and Servatius (2019) generalized the notion of avoidable vertices to \emph{avoidable paths}, and conjectured that every graph that contains an induced $k$-vertex path also contains an avoidable induced $k$-vertex path; they proved the result for $k = 2$. The case $k = 1$ was known much earlier, due to a work of Ohtsuki, Cheung, and Fujisawa in 1976. The conjecture was proved for all $k$ in 2020 by Bonamy, Defrain, Hatzel, and Thiebaut. This result generalizes the mentioned results regarding avoidable vertices, as well as a result by Chv\'atal, Rusu, and Sritharan (2002) suggested by West on the existence of simplicial $k$-vertex paths in graphs excluding induced cycles of length at least $k+3$. In this talk we discuss an adaptation of the approach by Bonamy et al.~from a reconfiguration point of view. We say that two induced $k$-vertex paths are \emph{shifts} of each other if their union is an induced path with $k+1$ vertices and show that in every graph, every induced path can be shifted to an avoidable one. We also present an analogous result about paths that are not necessarily induced, where an efficient reconfiguration sequence relies on the properties of depth-first search trees.

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Firebreaking and Firefighting

Dr. Jess Enright, University of Glasgow

Abstract

In the classic firefighter game, a fire burns and firefighters defend in turns. Motivated by a practical epidemiological question, we defined a one-shot version of this game, and plan to discuss both some new progress on firefighting and this one-shot version: the {\sc Firebreak} problem. Suppose we have a network that is represented by a graph $G$. Potentially a fire (or other type of contagion) might erupt at some vertex of $G$. We are able to respond to this outbreak by establishing a firebreak at $k$ other vertices of $G$, so that the fire cannot pass through these. fortified vertices. The question that now arises is which $k$ vertices will result in the greatest number of vertices being saved from the fire, assuming that the fire will spread to every vertex that is not fully behind the $k$ vertices of the firebreak. This is the essence of the {\sc Firebreak} decision problem, which is the focus of this talk. We establish that the problem is intractable on the class of split graphs as well as on the class of bipartite graphs, but can be solved in linear time when restricted to graphs having constant-bounded treewidth, or in polynomial time when restricted to convenient classes of intersection graphs.

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Circularly compatible ones, $D$-circularity, and proper circular-arc bigraphs

Dr. Martín Safe, Universidad Nacional del Sur

Abstract

In 1969, Alan Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results allowed him to devise the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as Tucker himself remarks, he did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). In this work, we solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related $D$-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu, Das, Ghosh, and Sen [J. Graph Theory, 73(4):361--376, 2013]. Our findings generalize some known results about $D$-interval hypergraphs and proper interval bigraphs.

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Treewidth versus clique number: a complete dichotomy for one forbidden structure

Mr. Kenny Štorgel, Faculty of Information studies Novo mesto, and University of Primorska, Koper

Abstract

Treewidth is an important graph invariant, relevant for both structural and algorithmic reasons. A necessary condition for a graph class to have bounded treewidth is the absence of large cliques. We study graph classes closed under taking induced subgraphs in which this condition is also sufficient, which we call $(\textrm{tw},\omega)$-bounded. For six graph containment relations (the subgraph, topological minor, and minor relations, as well as their induced variants) we give a complete characterization of graphs $H$ for which the class of graphs excluding $H$ is $(\textrm{tw},\omega)$-bounded. Our results yield an infinite family of $\chi$-bounded induced-minor-closed graph classes and imply that the class of $1$-perfectly orientable graphs is $(\textrm{tw},\omega)$-bounded, leading to linear-time algorithms for $k$-coloring $1$-perfectly orientable graphs for every fixed~$k$. This answers a question of Bre\v sar, Hartinger, Kos, and Milani{\v c} (2018) and one of Beisegel, Chudnovsky, Gurvich, Milani{\v c}, and Servatius (2019), respectively. We also reveal some further algorithmic implications of $(\textrm{tw},\omega)$-boundedness related to list $k$-coloring and clique problems.

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Beyond treewidth: the tree-independence number

Prof. Martin Milanič, University of Primorska

Abstract

We introduce a new graph invariant called \emph{tree-independence number}. This is a common generalization of treewidth and independence number, in the sense that bounded treewidth or bounded independence number implies bounded tree-independence number. The tree-independence number of a graph $G$ is defined as the smallest positive integer $k$ such that $G$ has a tree decomposition whose bags induce subgraphs of $G$ with independence number at most $k$. While for $k = 1$ we obtain the well-known class of chordal graphs, we show that the problem of computing the tree-independence number of a graph is NP-hard in general. We consider six graph containment relations (the subgraph, topological minor, and minor relations, as well as their induced variants) and for each of them completely characterize graph classes of bounded tree-independence number defined by a single forbidden graph with respect to the relation. In each of these bounded cases, a tree decomposition with small independence number can be computed efficiently, which implies polynomial-time solvability of the Maximum Weight Independent Set (MWIS) problem. This in particular applies to an infinite family of generalizations of the class of chordal graphs, for which a polynomial-time algorithm for the MWIS problem was given by Frank in 1976.

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Graphs with two moplexes are more than perfect

Meike Hatzel, TU Berlin

Abstract

A well-known result by Dirac (1961) states that every chordal graph contains a simplicial vertex. This theorem proved to be very useful for structural and algorithmic applications. Moplexes are a generalisation of simplicial vertices in chordal graphs to the setting of general graphs, as Berry and Bordat (1998) proved that every non-complete graph contains at least two moplexes. There are results on the structure of chordal graphs with a bounded number of simplicial modules, for example the chordal graphs having at most two simplicial modules are interval. This motivates the research of graphs with a bounded number of moplexes. As only complete graphs have exactly one moplex, we consider the smallest interesting case: the class of graphs with at most two moplexes. Berry and Bordat (2001) proved that this class of graphs contains all connected proper interval graphs and is contained in the class of AT-free graphs. We strengthen the latter inclusion in two ways. First, we generalise it by proving that the asteroidal number yields a lower bound on the number of moplexes. Second, as our main structural result, we show that graphs with at most two moplexes are cocomparability. So, as the class of connected graphs with at most two moplexes is sandwiched between the connected proper interval graphs and cocomparability graphs, this leads to the natural question of whether the presence of at most two moplexes guarantees a sufficient amount of structure to efficiently solve problems that are known to be intractable on cocomparability graphs, but not on proper interval graphs. For two such problems, namely \textsc{Graph Isomorphism} and \textsc{Max-Cut}, we show that they stay hard on the graphs with two moplexes. On the other hand, we prove that every connected graph with two moplexes contains a Hamiltonian path.

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Clique-Width: Harnessing the Power of Atoms

Dr. Konrad Dabrowski, Durham University

Abstract

Many {\sf NP}-complete graph problems are polynomial-time solvable on graph classes of bounded clique-width. Several of these problems are polynomial-time solvable on a hereditary graph class ${\cal G}$ if they are so on the atoms (graphs with no clique cut-set) of ${\cal G}$. Hence, we initiate a systematic study into boundedness of clique-width of atoms of hereditary graph classes. A graph $G$ is {\it $H$-free} if $H$ is not an induced subgraph of $G$, and it is {\it $(H_1,H_2)$-free} if it is both $H_1$-free and $H_2$-free. A class of $H$-free graphs has bounded clique-width if and only if its atoms have this property. This is no longer true for $(H_1,H_2)$-free graphs, as evidenced by one known example. We prove the existence of another such pair $(H_1,H_2)$ and classify the boundedness of clique-width on $(H_1,H_2)$-free atoms for all but 18 cases. \begin{thebibliography}{1} \bibitem{DMNPR20} Konrad K. Dabrowski, Tom\'a\v{s} Masa\v{r}\'ik, Jana Novotn\'a, Dani\"el Paulusma, and Pawe{\l} Rzazewski. Clique-width: Harnessing the power of atoms. {\em Proc. WG 2020, LNCS}, 12301:119--133, 2020. Full version: arXiv:2006.03578. \end{thebibliography}

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Coloring $(4K_1,C_4,C_6)$-free graphs

Dr. Irena Penev, Charles University

Abstract

A graph is {\em even-hole-free} if it contains no induced cycles of even length. Even-hole-free graphs can be recognized in polynomial time, and furthermore, the \textsc{Maximum Clique} problem can be solved in polynomial time for such graphs. However, the time complexity of the \textsc{Vertex Coloring} problem is open for this class. The time complexity of \textsc{Vertex Coloring} is also open for graphs of stability number at most three. In this talk, we consider the intersection of these two classes: even-hole-free graphs of stability number at most three, or equivalently, $(4K_1,C_4,C_6)$-free graphs. We show that such graphs can be recognized and colored in cubic time.

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Computing Weighted Subset Transversals in $H$-free Graph

Prof. Matthew Johnson, Durham University

Abstract

In graph transversal problems, one seeks, given a graph, to find a small subset of the vertex set --- a \emph{transversal} --- that intersects every subgraph of a specified kind. Examples include vertex cover, feedback vertex set and odd cycle transversal. In \emph{subset} versions of the problem, a subset of the vertex set is also given and the transversal that is found must also intersect this subset. In \emph{weighted} versions, a vertex weighting is given and the aim is to find a transversal such that the sum of the weights of the transversal vertices is small. These problems are NP-hard in general. We discuss recent work on graph transversal problems for hereditary graph classes. We will focus in particular on the problems {\sc Weighted Subset Odd Cycle Transversal} and {\sc Weighted Subset Feedback Vertex Set} for graph classes that are $H$-free. We present new algorithms that lead us to classify the complexity of the problems except in the cases where $H$ is $2P_1+P_3$, $P_1+P_4$ or $2P_1+P_4$. We note that in the latter two cases the complexity remains open even for the unweighted non-subset versions of the problem. One interesting aspect of our classification is the discovery that the complexities of weighted and unweighted versions of {\sc Subset Odd Cycle Transversal} do not coincide for $H$-free graphs.

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Model-checking in multilayer graphs

Dr. Kitty Meeks, University of Glasgow

Abstract

Real-world networks often involve qualitatively different types of relationships between different objects: for example, in order to understand a social network, we can consider both online and face-to-face contact. Many computational questions we might want to answer about such systems are intractable unless we have some additional information about the structure. The different layers typically have different structural properties (for example, face-to-face contact will be much more influenced by geography than online contact) and this talk will address the following question: when can we make use of algorithmically useful structure in the individual layers to solve problems efficiently on the whole system?

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Extending Thomason's Algorithm

Prof. Kathie Cameron, Wilfrid Laurer University

Abstract

Carsten Thomassen and I proved that in any graph G, the number of cycles containing a specified edge as well as all the odd-degree vertices is odd if and only if G is eulerian. Where all vertices have even degree this is a theorem of Shunichi Toida and where all vertices have odd degree it is Andrew Thomason's extension of Smith's Theorem. Andrew Thomason proved his theorem by constructing a graph X(G) in which the odd-degree vertices correspond precisely to the things he wants to show there are an even number of, namely the hamiltonian cycles containing the specified edge. This provides an algorithm for given one of the objects, finding another. I have extended Thomason's algorithm to one which, in a non-eulerian graph, finds a second cycle containing a specified edge and all the odd-degree vertices. I will discuss some other parity theorems about paths, cycles, and trees in graphs; in particular, attempts to find proofs of them by showing that the objects of interest are the odd-degree vertices of an associated (generally large) graph.

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Approximate and Randomized algorithms for Computing a Second Hamiltonian Cycle

Dr. VIKTOR ZAMARAEV, University of Liverpool

Abstract

In 1946 Cedric Smith proved, using a non-constructive parity argument, that any cubic Hamiltonian graph contains at least two Hamiltonian cycles. This motivated the following computational problem, which is still largely open: given a Hamiltonian cycle C in a cubic Hamiltonian graph G, can we efficiently compute a second Hamiltonian cycle? In this talk, I will discuss various open questions surrounding this problem and present some efficient approximate and randomized algorithms for related problems. Joint work with Argyrios Deligkas, George B. Mertzios, and Paul G. Spirakis

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Width parameters and graph classes: the case of mim-width

Dr. Andrea Munaro, Queen's University Belfast

Abstract

A large number of $\mathsf{NP}$-hard graph problems become polynomial-time solvable on graph classes where the mim-width is bounded and quickly computable. Hence, when solving such problems on special graph classes, it is helpful to know whether the graph class under consideration has bounded mim-width. We extend the toolkit for proving (un)boundedness of mim-width of graph classes and initiate a systematic study into bounding mim-width from the perspective of hereditary graph classes. We present summary theorems of the current state of the art for the boundedness of mim-width for $(H_1,H_2)$-free graphs and observe several interesting consequences. We also study the mim-width of generalized convex graphs. This allows us to re-prove and strengthen a large number of known results.

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Colourful components in $k$-caterpillars and planar graphs

Mr. Clément Dallard, University of Primorska

Abstract

A connected component of a vertex-coloured graph is said to be \emph{colourful} if all its vertices have different colours. By extension, a graph is colourful if all its connected components are colourful. Given a vertex-coloured graph and an integer $p$, the \textsc{Colourful Components} problem asks whether there exist at most $p$ edges whose removal makes the graph colourful. Bulteau, Dabrowski, Fertin, Johnson, Paulusma, and Vialette (2019) proved that \textsc{Colourful Components} is NP-complete on trees with maximum degree at most $6$ and asked whether the same would hold if the maximum degree is at most~$5$. We show that the problem remains NP-complete on binary $4$-caterpillars, ternary $3$-caterpillars and quaternary $2$-caterpillars, where a \emph{$k$-caterpillar} is a tree containing a path $P$ such that every vertex is at distance at most $k$ from~$P$. On the other hand, we provide a linear-time algorithm for $1$-caterpillars (without restriction on the maximum degree), and thus almost settle the complexity dichotomy on $k$-caterpillars with respect to the maximum degree. \textsc{Colourful Components} has also been studied in vertex-coloured graphs with a bounded number of colours. Bruckner, H\"{u}ffner, Komusiewicz, Niedermeier, Thiel, and Uhlmann (2012) showed that the problem is NP-complete on $3$-coloured graphs with maximum degree $6$, and Bulteau et al. asked whether the problem would remain NP-complete on graphs with bounded number of colors and maximum degree~$3$. We answer their question in the affirmative by showing that the problem is NP-complete on $5$-coloured planar graphs with maximum degree $4$ and on $12$-coloured planar graphs with maximum degree~$3$.

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Interference-free Walks in Time: Temporally Disjoint Paths

Ms. Nina Klobas, Durham University

Abstract

We investigate the computational complexity of finding \emph{temporally disjoint} paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp.\ walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex or edge at the same time; otherwise, they interfere. We show that on general graphs the problem is computationally hard. The ``walk version'' is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The ``path version'' remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the underlying graph to be a path, we find a polynomial-time algorithm for a relevant special case while the problem remains NP-hard in general (for both paths and walks).

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Domination number of graphs with fixed minimum degree

Csilla Bujtás, University of Ljubljana

Abstract

We discuss upper bounds on the domination number of graphs with a fixed minimum degree. The main result states that for every graph $G$ on $n$ vertices and with minimum degree $5$, the domination number $\gamma(G)$ cannot exceed $n/3$. The proof is obtained via an algorithmic approach where a weighting method is combined with a discharging process.

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On 12-regular nut graphs

Prof. Riste Škrekovski, FMF & FIŠ

Abstract

A nut graph is a simple graph whose adjacency matrix is singular with $1$-dimensional kernel and corresponding eigenvector with non-zero elements. For each $d\in\{3,4,\dots,11\}$ are known all values $n$ for which there exists a $d$-regular nut graph of order $n$. In the talk, we consider all values $n$ for which there exists a $12$-regular nut graph of order $n$.

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On the treewidth of even-hole-free graphs

Ms. Ni Luh Dewi Sintiari, ENS de Lyon

Abstract

Even-hole-free graphs attracted attention lately (see~\cite{vuskovic:evensurvey}). However, many questions about them remain unanswered: polytime algorithm to color them, or to find a maximum stable set, despite the existence of several decomposition theorems. In general, the class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Nonetheless, bounding the size of the maximum clique does not turn the class into having bounded treewidth, because there exists a family of even-hole-free graphs with no 4-vertex clique which has unbounded tree-width~\cite{DBLP:journals/corr/abs-1906-10998}. We observe that the graph constructed in~\cite{DBLP:journals/corr/abs-1906-10998} has unbounded degree \emph{and} contains arbitrarily large clique-minors. We ask whether this is necessary. We prove that for every graph $G$, if $G$ excludes a fixed graph $H$ as a minor, then $G$ either has small tree-width, or $G$ contains a large wall or the line graph of a large wall as \emph{induced} subgraph. This can be seen as a strengthening of Robertson and Seymour's excluded grid theorem for the case of minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. Furthermore, we conjecture that even-hole-free graphs of bounded degree have bounded tree-width, and we prove it for subcubic graphs and give a partial result when the maximum degree is four. This conjecture is now proved in~\cite{maria:boundedDegree}. \medskip Based on a joint work with Pierre Aboulker, Isolde Adler, Eun Jung Kim, and Nicolas Trotignon. \smallskip \begin{thebibliography}{99} \bibitem{vuskovic:evensurvey} K. Vu{\v s}kovi{\'c}. \newblock {Even-hole-free graphs: a survey}. \newblock {Applicable Analysis and Discrete Mathematics, 2010}. \bibitem{DBLP:journals/corr/abs-1906-10998} N.~L.~D.~Sintiari and N.~Trotignon. \newblock {(Theta, triangle)-free and (even hole, $K_4$)-free graphs.~Part 1:~Layered wheels}. \newblock Journal of Graph Theory (early view), 2021. \bibitem{maria:boundedDegree} T. Abrishami, M. Chudnovsky, K. Vu{\v s}kovi{\'c}. \newblock {Even-hole-free graphs with bounded degree have bounded treewidth}. \newblock arXiv 2009.01297, 2020. \end{thebibliography}

Analysis of PDEs on Networks (MS - ID 26)

Partial Differential Equations and Applications

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Macroscopic traffic flow models on road networks

Dr. Paola Goatin, Inria

Abstract

The talk will review the main macroscopic models of vehicular traffic flow on networks, focusing on the description of dynamics at junctions. I will then show some optimal control application, as well as some recent developments accounting for the impact of modern navigation systems.

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Spectral minimal partitions on metric graphs, and applications

Prof. Delio Mugnolo, FernUniversität in Hagen

Abstract

There exist several possibilities of partitioning graphs or manifolds, including those based on Cheeger cuts or nodal domains. We present a different approach that elaborates on a theory developed in the last 15 years, among others, by Bonnaillie-Noël, Helffer, Hoffmann-Ostenhof, and Terracini. While these authors focus on domains, we are going to discuss the partitioning of metric graphs in terms of spectral quantities of the associated Laplacian. We introduce a well-defined class of spectral partitions of metric graphs and show some of their features. While the complicated topology of metric graphs prevents us from recovering all results that hold for domains, new remarkable features also arise. This is joint work with Matthias Hofmann, James Kennedy, Pavel Kurasov, Corentin Léna, Marvin Plümer.

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Initial-boundary value problems for transport equations in one space dimension with very rough coefficients

Dr. Laura Valentina Spinolo, CNR

Abstract

I will discuss new existence, uniqueness and regularity propagation results for solutions of transport equations defined in one-dimensional domains with boundaries. The only assumptions imposed on the coefficient are boundedness and near incompressibility, which means that the coefficient supports a nonnegative and bounded density. This analyis is motivated by applications to a source-destination model for traffic flows on road networks. The talk will be based on joint work with Simone Dovetta and Elio Marconi.

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The quintic NLS on the tadpole graph

Prof. Diego Noja, Università di Milano Bicocca

Abstract

The tadpole graph consists of a circle and a half-line attached at a vertex. We analyze standing waves of the nonlinear Schr\"{o}dinger equation with quintic power nonlinearity and Kirchhoff boundary conditions at the vertex. The profile of a standing wave with frequency $\omega\in (-\infty,0)$ is characterized as a global minimizer of the quadratic part of energy constrained to the unit sphere in $L^6$. The set of standing waves so defined strictly includes the set of ground states, i.e. the global minimizers of the energy at constant mass ($L^2$-norm), but it is actually wider. While ground states exist only for a certain interval of masses, the above standing waves exist for every $\omega \in (-\infty,0)$ and correspond to a bigger interval of masses. It is proven that there exist critical frequencies $\omega_1$ and $\omega_0$ with $-\infty < \omega_1 < \omega_0 < 0$ such that the standing waves are the ground state for $\omega \in [\omega_0,0)$, local constrained minima of the energy for $\omega \in (\omega_1,\omega_0)$ and saddle points of the energy at constant mass for $\omega \in (-\infty,\omega_1)$.\\ Joint work with D.E. Pelinovsky.

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First order Mean Field Games on networks

Prof. Claudio Marchi, University of Padova

Abstract

The theory of Mean Field Games studies the asymptotic behaviour of differential games (mainly in terms of their Nash equilibria) as the number of players tends to infinity. In these games, the players are rational and indistinguishable: each player aims at choosing its trajectory so to minimize a cost which depends on the trajectory itself and on the distribution of the whole population of agents. We focus our attention on deterministic Mean Field Games with finite horizon in which the states of the players are constrained in a network (in our setting, a network is given by a finite collection of vertices connected by continuous edges which cannot self-intersect). In these games, an agent can control its dynamics and has to pay a cost formed by a running cost depending on the evolution of the distribution of all agents and a terminal cost depending on the distribution of all agents at terminal time. As in the Lagrangian approach, we introduce a relaxed notion of Mean Field Games equilibria and we shall deal with probability measures on trajectories on the network instead of probability measures on the network. This is a joint work with: Y. Achdou (Univ. of Paris), P. Mannucci (Univ. of Padova) and N. Tchou (Univ. of Rennes).

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Towards nonlinear hybrids: the planar NLS with point interactions.

Prof. Riccardo Adami, Politecnico di Torino

Abstract

A natural, but not straightforward, generalization of networks is provided by hybrids, namely systems made by gluing together pieces of different dimension. The simplest hybrid is made of a plane connected to a halfline. Suitable matching conditions at the contact point are to be imposed in order to define a well-posed dynamics, and this procedure leads to studying propagation of waves in the presence of point interactions in the plane, subject to a possibly nonlinear dynamics. We discuss the case of the Nonlinear Schroedinger Equation, where the nonlinearity is either self-consistent, or external and concentrated at a point in the plane, morally the connection point of the plane and the halfline. This is a joint project with Filippo Boni, Raffaele Carlone, and Lorenzo Tentarelli.

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Dynamics and scattering of truncated coherent states on the star-graph in the semiclassical limit.

Dr. Claudio Cacciapuoti, Università dell'Insubria

Abstract

We consider the dynamics of a quantum particle of mass $m$ on the star-graph constituted by $n$ half-lines with a common origin. The generator of the dynamics is the Hamiltonian $H_K=-(2m)^{-1}\hbar^2 \Delta$ with Kirchhoff conditions in the vertex, $\hbar$ is the reduced Planck constant. Our aim is to obtain the semiclassical limit of the quantum evolution, generated by $H_K$, of an initial state resembling a coherent state (gaussian packet) concentrated on one of the edges of the graph. Due to the Kirchhoff conditions in the vertex, the corresponding classical dynamics on the graph cannot be described by Hamilton-Jacobi equations. For this reason, we define the classical dynamics through a Liouville operator on the graph, obtained by means of the Krein's theory of singular perturbations of self-adjoint operators. For the same class of initial states, we study also the semiclassical limit of the wave and scattering operators for the couple $H_K$ and $H_D$, where $H_D$ is the free Hamiltonian with Dirichlet conditions in the vertex.

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Discontinuous ground states for the NLSE on $\mathbb{R}$ with a \emph{F\"{u}l\"{o}p-Tsutsui $\boldsymbol\delta$} interaction

Ms. Alice Ruighi, Politecnico di Torino

Abstract

We analyse the existence and the stability of the ground states of the one-dimensional nonlinear Schr\"{o}dinger equation with a focusing power nonlinearity and a defect located at the origin. A ground state is intended as a global minimizer of the action functional on the Nehari's manifold and the defect considered is a F\"{u}l\"{o}p-Tsutsui $\delta$ type, namely a $\delta$ condition that allows discontinuities. The existence of ground states is proved by variational techniques, while the stability results follow from the Grillakis-Shatah-Strauss' theory. \\ This is a joint work with Riccardo Adami.

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Local minimizers in absence of ground states for the critical NLS energy on metric graphs

Dr. Nicola Soave, Politecnico di Milano

Abstract

We consider the mass-critical nonlinear Schr\"odinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, has been recently provided by R. Adami, E. Serra and P. Tilli (\emph{Comm. Math. Phys.} 352, no.1, 387-406, 2017). They proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this talk I present some results regarding cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

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Ground states of the NLSE with standard and delta nonlinearities on star graphs

Mr. Filippo Boni, Politecnico di Torino

Abstract

We study the existence of ground states at fixed mass of a Schr\"{o}dinger equation on star graphs with two subcritical power-type nonlinear terms: a pointwise one, located at the vertex of the graph, and a standard one. We show that existence and non-existence results strongly depend on the interplay between the two nonlinearities. In particular, we see that when one nonlinearity prevails the other, existence of ground states depends both on the mass and on the number of halflines in the graph, whereas if the two nonlinearities are in a specific balance, then existence of ground states is only determined by the number of halflines of the graph. This is a joint work with R. Adami and S. Dovetta.

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Bound states for nonlinear Dirac equations on metric graphs with localized nonlinearities

Dr. William Borrelli, Università Cattolica del Sacro Cuore

Abstract

The investigation of evolution equations on metric graphs has become very popular nowadays, as they represent effective models for the dynamics of physical systems confined in branched spatial domains. In particular, the Dirac operator \begin{equation} \label{eq-dirac_formal} \mathcal{D}:=-\imath c\sigma_{1}\frac{d}{dx}+mc^{2}\sigma_{3},\,,\qquad \sigma_{1}=\begin{pmatrix}0 &1 \\ 1 & 0 \end{pmatrix}\,,\sigma_{3}=\begin{pmatrix}1 &0 \\ 0 & -1 \end{pmatrix} \end{equation} on metric graphs has attracted a growing interest for the description of systems where confined particles exhibit a `relativistic behavior'. Here $m>0$ is the mass of the particle whose (effective) hamiltonian is given by \eqref{eq-dirac_formal} and $c>0$ is a phenomenological parameter, playing the role of the speed of light. In this talk, I will discuss nonlinear Dirac equations with localized nonlinearity, namely \begin{equation}\label{eq:nld} \mathcal{D}\psi-\chi_{_\mathcal{K}}|\psi|^{p-2}\,\psi=\omega \psi\,,\qquad \psi:\mathcal{G}\to\mathbb{C}^2\,,\quad p>2\,, \end{equation} where $\mathcal{G}$ is a metric graph with finitely many edges and $\mathcal{K}\neq\emptyset$ is its \emph{compact core}, i.e. the set of bounded edges, and $\chi_{\mathcal{K}}$ is its characteristic function. The reduction to this simplified model arises if one assumes that the nonlinearity affects only the compact core of the graph. This idea was originally exploited in the case of Schr\"odinger equation in and it represents a preliminary step toward the investigation of the case with the ``extended'' nonlinearity. Considering the operator endowed with \emph{Kirchoff-type} vertex conditions, we proved that \eqref{eq:nld} possesses infinitely many solutions with $\omega\in(-mc^2,mc^2)$, converging, after a suitable renormalization, in $H^1$-sense to solutions to the analogous Schr\"odinger equation $ -u''-\chi_{\mathcal{K}}\vert u\vert^{p-2}u=\lambda u$ for some $\lambda<0\,.$, in the \emph{non-relativistic limit} as $c\to+\infty$. \smallskip

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On the nonlinear Dirac equation on noncompact metric graphs

Dr. Raffaele Carlone, Università Federico II Napoli

Abstract

We discuss the Nonlinear Dirac Equation, with Kerr-type nonlinearity, on non-compact metric graphs with a finite number of edges, in the case of Kirchhoff-type vertex conditions. We will present results about the well-posedness for the associated Cauchy problem in the operator domain and, for infinite N-star graphs, and the existence of standing waves.

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On Pleijel's nodal domain theorem for quantum graphs

Mr. Marvin Plümer, FernUniversität in Hagen

Abstract

In this talk we present recent results on metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains $\nu_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schr\"{o}dinger operators with $L^1$-potentials as well as the $p$-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. We characterise the accumulation points of the sequence $(\frac{\nu_n}{n})_{n\in\mathbb N}$, which are shown to form a finite subset of $(0,1]$. This extends the previously known result that $\nu_n\sim n$ \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, we will see that the existence of accumulation smaller than $1$ is strictly related to the failure of the unique continuation principle on metric graphs. Finally we show that for (most) metric graphs -- metric trees and general metric graphs with at least one Dirichlet vertex -- there exists an infinite sequence of \textit{generic} eigenfunctions -- namely, eigenfunctions of multiplicity $1$ that do not vanish in the graph's vertices -- of the free Laplacian and infer that, in this case, $1$ is always an accumulation point of $\frac{\nu_n}{n}$. In order to do so we introduce a new type of secular function. The talk is based on joint work with Matthias Hofmann (Lisbon), James Kennedy (Lisbon), Delio Mugnolo (Hagen) and Matthias Täufer (Hagen).

Analysis on Graphs (MS - ID 48)

Mathematical Physics

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Universality of nodal count statistics for large quantum graphs.

Dr. Gregory Berkolaiko, Texas A&M University

Abstract

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s notrivial topology. This number, called the nodal surplus, is an integer between 0 and the first Betti number of the graph. We study the value distribution of the nodal surplus within the countably infinite spectrum of the graph. We conjecture that this distribution converges to Gaussian in any sequence of graphs of growing Betti number. We prove this conjecture for several special graph sequences and test it numerically for some other well-known types of graphs. An accurate computation of the distribution is made possible by a formula expressing the distribution as an integral over a high-dimensional torus with uniform measure.

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Solitons for the KdV equation on metric graphs

Dr. Christian Seifert, Technische Universität Hamburg

Abstract

We consider the Korteweg-de Vries equation on metric star graphs. After reviewing suitable coupling conditions at the vertex for the linearised equation, i.e. an Airy-type evolution equation, we investigate coupling conditions at the vertex such that the system allows for solitary waves. This is joint work with Delio Mugnolo (Hagen) and Diego Noja (Milan).

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Effects of time-reversal asymmetry in the vertex coupling of quantum graphs

Prof. Pavel Exner, Czech Academy of Sciences, Nuclear Physics Institute

Abstract

The talk concerns the effects coming from a violation of time-reversal symmetry in the vertex coupling of quantum graph, focusing on the situation when the asymmetry is maximal at a fixed energy. It is shown that such a coupling has a topological property: the transport behaviour of such a vertex in the high-energy regime depend substantially on the vertex parity. We explore consequences of this fact for several classes of graphs, both finite and infinite periodic ones. The results come from a common work with Marzieh Baradaran, Ji\v{r}\'{\i} Lipovsk\'{y}, and Milo\v{s} Tater.

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Multifractal eigenfunctions in a singular quantum billiard

Prof. Jon Keating, University of Oxford and London Mathematical Society

Abstract

Spectral statistics and questions relating to quantum ergodicity in star graphs are closely related to those of Šeba billiards (rectangular billiards with a singular point scatterer) and other intermediate systems. In intermediate systems, it has been suggested in the Physics literature that quantum eigenfunctions should exhibit mulifractal properties. I shall discuss the proof that this is the case for Šeba billiards. The work I shall report on was done jointly with Henrik Ueberschaer.

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Steklov eigenvalues on graphs

Prof. Bobo Hua, Fudan University

Abstract

The eigenvalues of the Dirichlet-to-Neumann operator are called Steklov eigenvalues, which are well studied in spectral geometry. In this talk, we introduce Steklov eigenvalues on graphs, and estimate them using geometric quantities, based on joint works with Wen Han, Zunwu He, Yan Huang, and Zuoqin Wang.

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Subordinacy theory on star-like graphs

Mr. Netanel Levi, Hebrew University of Jerusalem

Abstract

The notion of subordinacy was introduced by Gilbert and Pearson, and it enables one to separate the singular and absolutely continuous parts of the spectrum of Schroedinger operators on the line via asymptotic properties of solutions to the eigenvalue equation. Informally speaking, a solution is called subordinate if it decays faster than any other linearly independent solution. We present a generalization of the Gilbert-Pearson subordinacy theory to Schroedinger operators on star-like graphs, which are graphs that consist of a compact component C, to which a finite number of half-lines are attached. We use our result to draw conclusions on the multiplicity of the singular spectrum of such operators.

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Stochastic completeness and uniqueness class for graphs

Dr. Xueping Huang, Nanjing University of Information Science and Technology

Abstract

Uniqueness class for the heat equation on a weighted graph is closely related to stochastic completeness of the corresponding minimal continuous time random walk. For a class of so called globally local graphs, we obtain essentially sharp criteria which are in the same form as for manifolds. Sharp volume growth type criteria for stochastic completeness then follow, after a reduction to the globally local case. This talk is based on joint work with M. Keller and M. Schmidt.

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The density of states for periodic Jacobi matrices on trees

Prof. Jonathan Breuer, The Hebrew University of Jerusalem

Abstract

Periodic Jacobi matrices on trees are a natural generalization of one dimensional periodic Schroedinger operators. While the one dimensional theory is very well developed, very little is known about the general tree case. In this talk we will review some of the few known results with a focus on the question of convergence of the appropriately defined finite volume approximations. If time permits, we will also discuss some open problems. Based on joint works with Nir Avni, Gil Kalai and Barry Simon.

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Asymptotics of Green functions: Riemann surfaces and Graphs

Mrs. Noema Nicolussi, University of Vienna

Abstract

Analysis on graphs and Riemann surfaces admits many interesting parallels. Both settings admit a canonical measure (the Arakelov–Bergman and Zhang measures) and an associated canonical Green function reflecting crucial geometric information. Motivated by the question of describing the limit of the Green function on degenerating families of Riemann surfaces, we introduce new higher rank versions of metric graphs and their Laplace operators. We discuss how these limit objects describe the asymptotic of the Green function on metric graphs and Riemann surfaces close to the boundary of their respective moduli spaces. Based on joint work with Omid Amini (École Polytechnique).

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Neumann domains on metric graphs

Prof. Ram Band, Technion - Israel Institute of Technology

Abstract

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We present the following three main properties of Neumann domains: their count, wavelength capacity and spectral position. We study their bounds and probability distributions and use those to investigate inverse spectral problems. The relevant probability distributions are rigorously defined in terms of selected random variables for quantum graphs. To this end, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on $\mathbb{N}$. The talk is based on joint work with Lior Alon.

Analysis, Control and Inverse Problems for Partial Differential Equations (MS - ID 22)

Optimization and Control

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The Calderón problem with corrupted data

Dr. Pedro Caro, Basque Center for Applied Mathematics

Abstract

The inverse Calderón problem consists in determining the conductivity inside a medium by electrical measurements on its surface. Ideally, these measurements determine the Dirichlet-to-Neumann map and, therefore, one usually assumes the data to be given by such map. This situation corresponds to having access to infinite-precision measurements, which is unrealistic. In this talk, I will consider the Calderón problem assuming data to contain measurement errors and provide formulas to reconstruct the conductivity and its normal derivative on the surface (joint work with Andoni García). I will also present similar results for Maxwell's equations (joint work with Ru-Yu Lai, Yi-Hsuan Lin, Ting Zhou ). When modelling errors in these two different frameworks, one realizes the existence of certain freedom that yields different reconstruction formulas. To understand the whole picture of what is going on, we will rewrite the problem in a different setting, which will bring us to analyse the observational limit of wave packets with noisy measurements (joint work with Cristóbal J. Meroño).

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Strong unique continuation at the boundary in linear elasticity and its connection with optimal stability in the determination of unknown boundaries

Prof. Edi Rosset, Università di Trieste

Abstract

\date{} \textbf{Abstract:} Quantitative estimates of Strong Unique Continuation at the boundary for solutions to the isotropic Kirchhoff-Love plates subject to Dirichlet conditions, and for solutions to the Generalized plane stress problem subject to Neumann conditions are presented. These results have been applied to prove optimal stability estimates for the inverse problem of determining unknown boundaries. \medskip \noindent References: [1] G. Alessandrini, E. Rosset, S. Vessella, Optimal three spheres inequality at the boundary for the Kirchhoff-Love plate's equation with Dirichlet conditions, Arch. Rational Mech. Anal., 231 (2019), 1455--1486. [2] A. Morassi, E. Rosset, S. Vessella, Optimal stability in the identification of a rigid inclusion in an isotropic Kirchhoff-Love plate, SIAM J. Math. Anal., 51 (2019), 731--747. [3] A. Morassi, E. Rosset, S. Vessella, Optimal identification of a cavity in the Generalized Plane Stress problem in linear elasticity, J. Eur. Math. Soc., to appear. [4] A. Morassi, E. Rosset, S. Vessella, Doubling Inequality at the Boundary for the Kirchhoff-Love Plate's Equation with Dirichlet Conditions, Le Matematiche, LXXV (2020), 27--55, Open access.

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Exponential dynamical Luenberger observers for nonautonomous parabolic-like equations

Sergio Rodrigues, RICAM

Abstract

The estimation of the full state of a nonautonomous semilinear parabolic equation is achieved by a Luenberger type dynamical observer. The estimation is derived from an output given by a finite number of average measurements of the state on small regions. The state estimate given by the observer converges exponentially to the real state, as time increases. The result is semiglobal in the sense that the error dynamics can be made stable for an arbitrary given initial condition, provided a large enough number of measurements, depending on the norm of the initial condition, is taken. The output injection operator is explicit and involves a suitable oblique projection. The results of numerical simulations are presented showing the exponential stability of the error dynamics.

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Analysis of finite-element based discretizations in nonlinear acoustics

Dr. Vanja Nikolic, Radboud University

Abstract

\indent Nonlinear effects can be easily observed in sound waves with sufficiently large amplitudes. The nonlinearity will be apparent even sooner in high-frequency waves because these effects accumulate over the distance measured in wavelengths. This makes high-intensity ultrasonic waves inherently nonlinear. Their many applications range from non-invasive surgery to industrial welding and motivate the mathematical investigation into nonlinear acoustics. \\ \indent In this talk, we will discuss the a priori analysis of finite-element-based discretizations of nonlinear acoustic equations. In particular, we will focus on the conforming and (hybrid) discontinuous Galerkin discretizations in space for acoustic equations with nonlinearities of quadratic type, such as the Westervelt and Kuznetsov equations. These are quasilinear strongly damped wave equations that serve as classical models of sound propagation through thermoviscous fluids and gases. The general approach in the a priori error analysis combines the stability and convergence analysis of their linearizations with the Banach fixed-point theorem. Numerical experiments will illustrate the theoretical results.\\ \indent The talk is based on joint research with Paola F. Antonietti, Ilario Mazzieri (Politecnico di Milano), Markus Muhr, and Barbara Wohlmuth (TU Munich). \\

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Asymptotic behavior of dispersive electromagnetic waves in bounded domains

Prof. Cristina Pignotti, University of L'Aquila

Abstract

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. Then, a passitivity assumption guarantees the boundedness of the associated semigroup. Under suitable sufficient conditions, we prove the exponential or polynomial decay of the energy. Moreover, some illustrative examples are presented.

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Stable determination of polygonal and polyhedral interfaces from boundary measurements

Prof. Elisa Francini, Università di Firenze

Abstract

I will present some results concerning Lipschitz stability in the determination of polygonal and polyhedral inclusions from the Dirichlet to Neumann map.

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Weak Solutions for an Implicit, Degenerate Poro-elastic Plate System

Dr. Justin Webster, University of Maryland, Baltimore County

Abstract

We consider a recent plate model obtained as a scaled limit of the three dimensional quasi-static Biot system of poro-elasticity. The result is a ``2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. Motived by application, we allow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard theory. Weak solutions are defined and the problem is framed abstractly as an implicit, degenerate evolution problem: $$[Bp]_t+A(t)p=S.$$ Existence is obtained, and uniqueness follows under additional hypotheses on the temporal regularity of the permeability. Time permitting, we address the inertial case with constant permeability by way of semigroup theory.

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Existence and regularity of weak solutions for a fluid interacting with a non-linear shell in $3D$

Prof. Boris Muha, Faculty of Science, University of Zagreb

Abstract

We study the unsteady incompressible Navier-Stokes equations in three dimensions interacting with a non-linear flexible shell of Koiter type. We study weak solutions to the corresponding fluid-structure interaction (FSI) problem. The known existence theory for weak solutions is extended to non-linear Koiter shell models. We introduce a-priori estimates that reveal higher regularity of the shell displacement beyond energy estimates. These are essential for non-linear Koiter shell models, since such shell models are non-convex (w.r.t.\ terms of highest order). The estimates are obtained by introducing new analytical tools that allow to exploit dissipative effects of the fluid for the (non-dissipative) solid. The regularity result depends on the geometric constitution alone and is independent of the approximation procedure; hence it holds for arbitrary weak solutions. The developed tools are further used to introduce a generalized Aubin-Lions type compactness result suitable for fluid-structure interactions.

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Optimal Control in Poroelasticity

Dr. Lorena Bociu, NC State University

Abstract

In this talk we address optimal control problems subject to fluid flows through deformable, porous media. In particular, we focus on quadratic poroelasticity control problems, with both distributed and boundary controls, and prove existence and uniqueness of optimal control. Furthermore, we derive the first order necessary optimality conditions. These problems have important biological and biomechanical applications. For example, optimizing the pressure of the flow and investigating the influence and control of pertinent biological parameters are relevant in ocular tissue perfusion and its relation to the development of ocular neurodegenerative diseases such as glaucoma. 

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Boundedness in Total Variation regularization

Dr. Gwenael Mercier, University of Vienna

Abstract

In this talk, we investigate boundedness and convergence of total variation regularized linear inverse problems. We present a simple proof of boundedness of the minimizer for fixed regularization parameter, and derive the existence of uniform bounds for small enough noise under a source condition and adequate a priori parameter choices. We present a few (counter)examples. This is a joint work wirth K. Bredies (Graz) and José A. Iglesias (RICAM, Linz).

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Optimal control problem for a repulsive chemotaxis system

Dr. María Angeles Rodríguez-Bellido, Universidad de Sevilla

Abstract

The chemotaxis phenomenon can be understood as the directed movement of living organisms in response to chemical gradients. Keller and Segel \cite{keller-segel} proposed a mathematical model that describes the chemotactic aggregation of cellular slime molds. These molds move preferentially towards relatively high concentrations of a chemical substance secreted by the amoebae themselves. Such mechanism is called \textit{chemo-attraction} with production. However, when the regions of high chemical concentration generate a repulsive effect on the organisms, the phenomenon is called \textit{chemo-repulsion}. In this work, we want to study an optimal control problem for the (repulsive) Keller-Segel model and a bilinear control acting on the chemical equation in a $2D$ and $3D$ domains. The system can be written as: \begin{equation}\label{KS} \left\{\begin{array}{rcll} \partial_t u -\Delta u - \nabla \cdot (u \, \nabla v) &=& 0 & \mbox{in $\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ \partial_t v -\Delta v + v &=& u + f \, v\, 1_{\Omega_c}& \mbox{in $\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ \partial_{\textbf{n}} u= \partial_{\textbf{n}} v &=&0 & \mbox{on $\partial\Omega \times (0,T)$,}\\ \noalign{\vspace{-1ex}}\\ u(0,\cdot)=u_0\ge 0, \quad v(0,\cdot)&=&v_0\ge 0 & \mbox{in $\Omega$,} \end{array}\right. \end{equation} being $f:Q_c:=(0,T)\times \Omega_c\to \mathbb{R}$ (the control) with $\Omega_c\subset \Omega\subset \mathbb{R}^n$ ($n=2,3$) the control domain, and the state $u,v:Q:=(0,T)\times \Omega_c\to \mathbb{R}_+$ the celular density and chemical concentration, respectively. Here, $\textbf{n}$ is the outward unit normal vector to $\partial\Omega$. \medskip The existence and uniqueness of global in time weak solution $(u,v)$ for the uncontrolled system is known (see for instance \cite{cieslak,Diego4}). \medskip In this work we study an optimal control problem subject to a chemo-repulsion system with linear production term, and in which a bilinear control acts injecting or extracting chemical substance on a subdomain of control $\Omega_c\subset\Omega$. Existence of weak solutions are stablished (in the $3D$ case by using a regularity criterion), and, as a consequence, a global optimal solution together with first-order optimality conditions for local optimal solutions are deduced. \medskip The results presented in this talk are based on \cite{Exequiel1,Exequiel2}. \begin{thebibliography}{99} \markboth{}{} \bibitem{cieslak} Cieslak, T., Lauren\c cot, P., Morales-Rodrigo, C.: Global existence and convergence to steady states in a chemorepulsion system. Parabolic and Navier-Stokes equations. Part 1, 105-117, Banach Center Publ., 81, Part 1, Polish Acad. Sci. Inst. Math., Warsaw, 2008. \bibitem{Exequiel1} Guill\'en-Gonz\'alez, F.; Mallea-Zepeda, E.; Rodr\'{\i}guez-Bellido, M.A. {Optimal bilinear control problem related to a chemo-repulsion system in 2D domains.} ESAIM Control Optim. Calc. Var. 26 (2020), Paper No. 29, 21 pp. \bibitem{Exequiel2} Guill\'en-Gonz\'alez, F.; Mallea-Zepeda, E.; Rodr\'{\i}guez-Bellido, M.A. {A regularity criterion for a 3D chemo-repulsion system and its application to a bilinear optimal control problem.} SIAM J. Control Optim. 58 (2020), no. 3, 1457--1490. \bibitem{Diego4} Guill\'en-Gonz\'alez, F.; Rodr\'{\i}guez-Bellido, M. A.; Rueda-G\'omez, D. A. {Unconditionally energy stable fully discrete schemes for a chemo-repulsion model.} Math. Comp. 88 (2019), no. 319, 2069--2099. \bibitem{keller-segel} Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theo. Biol. Vol. 26, 399-415 (1970). \end{thebibliography}

Applied Combinatorial and Geometric Topology (MS - ID 34)

Combinatorics and Discrete Mathematics

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Hamiltonian complexes, interval graphs, determinantal ideals

Dr. Bruno Benedetti, University of Miami

Abstract

There are three mutually-related notions in graph theory, from three different centuries: Hamiltonian paths, (unit)-interval graphs, and binomial edge ideals. We study the d-dimensional generalization of these ideas. This is joint work with Matteo Varbaro and Lisa Seccia (arXiv:2101.09243).

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Invariants for tame parametrised chain complexes

Prof. Claudia Landi, Università di Modena e Reggio Emilia

Abstract

Data analysis is often about simplifying, ignoring most of the information available and extracting what might be meaningful to a task at hand. This strategy of making sense by disregarding some information and focusing on aspects that might be relevant is very common across mathematics. Colocalization in homotopy theory is an example of such a process. In colocalization, simplification is achieved by approximating arbitrary objects by other objects that one considers simpler and more manageable. For instance, by approximating a given space by n-connected spaces, one obtains its n-connected cover. The aim of our presentation is to explain why extracting persistence invariants in Topological Data Analysis (TDA) is an example of the homotopical colocalization process. This allows us to extract computable invariants also in certain cases, such as commutative ladders, that have not been covered by more standard approaches. Furthermore, it allows for a comprehensive theory including several cases that standard persistence theory handles separately, such as persistence modules, zigzag modules, and commutative ladders.

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On the Connectivity of Branch Loci of Spaces of Curves

Prof. Milagros Izquierdo, Linköping University

Abstract

Since the 19th century the theory of Riemann surfaces has a central place in mathematics putting together complex analysis, algebraic and hyperbolic geometry, group theory and combinatorial methods. Since Riemann, Klein and Poincar{'e} among others, we know that a compact Riemann surface is a complex curve, and also the quotient of the hyperbolic plane by a Fuchsian group. \smallskip In this talk we study the connectivity of the moduli spaces of Riemann surfaces (i.e in spaces of Fuchsian groups). Spaces of Fuchsian groups are orbifolds where the singular locus is formed by Riemann surfaces with automorphisms: {\it the branch loci}: With a few exceptions the branch loci is disconnected and consists of several connected components. This talk is a survey of the different methods and topics playing together in the theory of Riemann surfaces. \smallskip Joint wotk with Antonio F. Costa

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Periodic projections of alternating knots

Dr. Antonio F. Costa, UNED

Abstract

Let $K$ be an oriented prime alternating knot that is $q$-periodic with $q\geq 3$, i.e. $K$ admits a symmetry that is a rotation of order $q$. We present a proof that $K$ has an alternating $q$-periodic projection. The main tool is the Menasco-Thistlethwaite's Flyping theorem. We present also some results about the visibility on projections of $q$-freely periodicity of alternating knots.

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Classifying compact PL 4-manifolds according to generalized regular genus and G-degree

Prof. Paola Cristofori, University of Modena and Reggio Emilia

Abstract

{\it (d+1)-colored graphs}, that is $(d+1)$-regular graphs endowed with a proper edge-coloration, are the objects of a long-studied representation theory for closed PL $d$-manifolds, which has been recently extended to the whole class of compact PL $d$-manifolds. In this context, combinatorially defined PL invariants play a relevant role; in this talk we will focus on two of them: the {\it generalized regular genus} and the {\it G-degree}. The former extends to higher dimension the classical notion of Heegaard genus for 3-manifolds; the latter has arisen in connection with {\it Colored Tensor Models} (CTM), a particular kind of tensor models, that have been intensively studied in the last years, mainly as an approach to quantum gravity in dimension greater than two. CTMs established a link between colored graphs and tensor models, since the Feynman graphs of a $d$-dimensional CTM are precisely $(d+1)$-colored graphs. Furthermore, the G-degree of a colored graph is a crucial quantity driving the 1/N expansion of the free energy of a CTM. This talk will mainly concern recent results achieved in dimension 4: in particular, the classification of all compact PL 4-manifolds with generalized regular genus at most one or with G-degree at most 18. Furthermore, we will discuss interesting classes of 5-colored graphs ({\it semi-simple and weak semi-simple crystallizations}), representing compact PL 4-manifolds with empty or connected boundary and minimizing the invariants. In the simply-connected case they also belong to a wider class of 5-colored graphs which are proved to induce handle-decompositions of the represented 4-manifold lacking in 1- and/or 3-handles; therefore their study is strictly related to the problem, posed by Kirby, of the existence of special handle-decompositions for any simply-connected closed PL 4-manifold.

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A colored approach for the self-assembly of DNA structures

Dr. Simona Bonvicini, University of Modena and Reggio Emilia

Abstract

We study a graph theory problem related to the self-assembly of DNA structures. The self-assembly can be obtained by several methods that are based on the Watson-Crick complementary properties of DNA strands. We consider the method based on branched junction molecules, that is, star-shaped molecules whose arms have cohesive ends that allow the molecules to join together in a prescribed way and form a larger molecule (DNA complex). In the language of graphs, a branched junction molecule is called a tile and consists of a vertex with labeled half-edges; labels represent the cohesive ends and belong to a set $\{a, \hat a: a\in\Sigma\}$, where $\Sigma$ is a finite set of symbols; a tile is denoted by the multiset consisting of the labels of the half-edges; and two tiles are of the same tile type if they are denoted by the same multiset. We can create an edge between the vertices $u$, $v$ if and only if $u$ has a half-edge labeled by $a$ and $v$ has a half-edge labeled by $\hat a$; the edge thus obtained is said to be a bond-edge of type $a\hat a$; by connecting the vertices according to the labels, we can construct a graph $G$ representing a DNA complex. The following problem is considered: given a graph $G$, determine the minimum number of tile types and bond-edge types that are necessary to construct $G$. We describe the problem by edge-colored graphs and show some upper bounds for the number of bond-edge types that are necessary to construct an arbitrary graph.

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Explicit computation of some families of Hurwitz numbers

Prof. Carlo Petronio, Università di Pisa

Abstract

I will describe the computation (based on dessins d'enfant) of the number of equivalence classes of surface branched covers matching a given branch datum belonging to a certain very specific family.

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Shellings from relative shellings

Prof. Russ Woodroofe, University of Primorska

Abstract

It is frequently helpful to build complicated mathematical objects by breaking down into subobjects with simpler properties. In joint work with Andrés Santamaría-Galvis, we have shown how to usefully glue together shellings of relative simplicial complexes to construct a shelling of a large simplicial complex. Indeed, one of the main other ways of using the "divide and conquer" approach to build a shelling, vertex-decomposability, can be viewed as a consequence of our approach. One useful consequence is that our approach makes it relatively straightforward to find a shellable simplicial complex satisfying any of a variety of conditions on its facets that contain a free face. Another is an improved proof that the shellability decision problem is NP-complete.

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Partition extenders, skeleta of simplices, and Simon's conjecture

Bennet Goeckner, University of Washington

Abstract

If a pure simplicial complex is partitionable, then its $h$-vector has a combinatorial interpretation in terms of any partitioning of the complex. Such an interpretation does not exist for non-partitionable complexes. Given a non-partitionable complex, we will construct a relative complex---called a \emph{partition extender}---that allows us to write the $h$-vector of a non-partitionable complex as the difference of two $h$-vectors of partitionable complexes in a natural way. We will show that all pure complexes have partition extenders. A similar notion can be defined for Cohen--Macaulay and shellable complexes. We will show precisely which complexes have Cohen--Macaulay extenders, and we will discuss a connection to a conjecture of Simon on the extendable shellability of uniform matroids. This is joint work with Joseph Doolittle and Alexander Lazar.

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Transitive factorizations of pairs of permutations and three-dimensional constellations

Dr. Luca Lionni, Radboud University, Nijmegen

Abstract

Factorizations of pairs of permutations that generate the full symmetric group on n-symbols are shown to be in bijection with a certain family of three-dimensional 4-colored triangulations that generalize constellations. These spaces are ramified coverings of the three-dimensional sphere, branched over the n-unlink. We will present a generalization of the Riemann-Hurwitz formula, and identify the spaces that maximize the number of branching edges for a fixed number of sheets.

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Complexity of graph manifolds

Prof. Alessia Cattabriga, University of Bologna

Abstract

The notion of complexity  for compact 3-dimensional manifolds, was introduced by Matveev in the nineties as a way to measure how complicated a manifold is; indeed, for closed orientable irreducible  manifolds, the complexity coincides with the minimum number of tetrahedra needed to construct the manifold, with the only exceptions of the 3-sphere, the projective space and the  lens space L(3, 1)  all having complexity zero. There exists a census of manifolds according to increasing complexity that, for the orientable case, is Recognizer catalogue and  includes all manifolds up to complexity 12 (see http://matlas.math.csu.ru/?page=search). In this talk, after recalling some general result about complexity, I will focus on an important class of manifolds: graph manifolds. These manifolds have been introduced and classified by Waldhausen in the sixties   and are defined as compact 3-manifolds obtained by gluing Seifert fibre spaces along toric boundary components.  I will present an upper bound for their complexity  obtained in a joint work with Michele Mulazzani (University of Bologna)  that is sharp for all the 14502 graph manifolds included in the Recognizer catalogue

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Partitioning the projective plane and the dunce hat

Andrés David Santamaría-Galvis, University of Primorska

Abstract

The faces of a simplicial complex induce a partial order by inclusion in a natural way. We say that the complex is partitionable if its poset can be partitioned into boolean intervals, with a maximal face at the top of each. In this work we show that all the triangulations of the real projective plane, the dunce hat, and the open Möbius strip are partitionable. To prove that, we introduce simple yet useful gluing tools that allow us to abridge the discussion about partitionability of a given complex in terms of smaller constituent relative subcomplexes. The gluing process generates partitioning schemes with a distinctive shelling-like flavor.

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Kirby diagrams, edge-colored graphs and trisections of PL 4-manifolds

Mrs. Maria Rita Casali, University of Modena and Reggio Emilia

Abstract

It is well-known that any {\it framed link} $(L,c)$ uniquely represents the 3-manifold $M^3(L,c)$ obtained from $\mathbb S^3$ by Dehn surgery along $(L,c)$, as well as the PL 4-manifold $M^4(L,c)$ obtained from $\mathbb D^4$ by adding 2-handles along $(L,c)$. Moreover, if trivial dotted components are also allowed (i.e. in case of a {\it Kirby diagram} $(L^{(*)},c))$, the associated PL 4-manifold $M^4(L^{(*)},c)$ is obtained from $\mathbb D^4$ by adding 1-handles along the dotted components and 2-handles along the framed components. In the present talk we present the relationship between framed links and/or Kirby diagrams and the so called {\it crystallization theory}, which represents compact PL manifolds of arbitrary dimension by regular edge-colored graphs: in particular, we describe how to construct a 5-colored graph representing $M^4(L^{(*)},c)$, directly “drawn over” a planar diagram of $(L^{(*)},c))$. As a consequence, the combinatorial properties of Kirby diagrams yield upper bounds for both the graph-defined invariants {\it gem-complexity} and {\it generalized regular genus} of the associated 4-manifold. Further, the described relationship turns out to be strictly related to the possibility of studying {\it trisections} of 4-manifolds via edge-colored graphs, also in the extended case of compact PL 4-manifolds with connected boundary.

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Universality classes of triangulations in dimensions greater than 2

Prof. Valentin Bonzom, Universite Sorbonne Paris Nord

Abstract

Combinatorial maps are cellular topological models for surfaces, which include triangulations, quadrangulations, etc. of all genera. It is well-known that at fixed topology, all models lie in the same universality class. It means in particular that they have the same critical exponents in the enumeration formula and the same limit object at large scales. For instance, the universality class for planar maps is also known as the universality class of 2D pure quantum gravity and its scaling limit is known as the Brownian sphere. It is important to generalize this framework to greater dimensions, with motivations from combinatorics, topology, mathematical physics and quantum gravity. However, it has been a challenge due to the intricate nature of combinatorics and topology in dimensions 3 and greater. Here I will present results based on the model of colored triangulations for PL-manifolds. They admit a representation as edge-colored graphs which is amenable to combinatorial analysis. I will focus on a combinatorial generalization of Euler's relation for combinatorial maps, which is to bound the number of $(d-2)$-simplices linearly in the number of $d$-simplices, and identifying the triangulations which maximize the bound for different colored building blocks. In 3d, I prove that those triangulations, for any set of colored building blocks homeomorphic to the 3-ball, are in bijection with trees. In even dimensions, we have proved that several universality classes can be achieved. Both those results are in sharp contrast with the case of surfaces.

Approximation Theory and Applications (MS - ID 78)

Analysis and its Applications

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Information potential for some probability distributions

Prof. Ioan Rasa, Technical University of Cluj-Napoca, Romania

Abstract

This talk is devoted to some properties of the entropies of some discrete or continuous probability distributions. We will focus on the information potential (IP) associated with such distributions depending on a real parameter $x$. The R\'enyi entropy and the Tsallis entropy are naturally related to IP. Convexity properties and bounds of the associated IP, useful in Information Theoretic Learning, are discussed and translated as properties of the entropies. Several examples and numerical computations illustrate the general results as well as their relationship with positive linear operators.

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Discretisation of integrals on compact spaces using distance functions

Prof. Martin Buhmann, Justus-Liebig University

Abstract

For the purpose of partitioning compact sets, discretisation of integrals and finding quadrature rules on compact sets, it is important to have estimates for the ensuing error of the approximation. It is desirable to have estimates on the remainders that are independent of space dimension, and of course we wish the errors to decrease as fast as possible when the number of summands in the discretisation increases. In this joint work with Feng Dai and Yeli Niu (Edmonton) we find such error estimates using regular partitions with particular attention to (but not only to) discretisations on spheres. In fact, our estimates are quite general, they apply to compact pathconnected metric spaces, and we are able to improve several earlier error estimates from the literature.

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Gauss-Lucas theorem in polynomial dynamics

Dr. Margaret Stawiska-Friedland, American Mathematical Society/MathSciNet

Abstract

Using versions of the Gauss-Lucas theorem adapted to dynamics, we prove that for every complex polynomial $p$ of degree $d \geq 2$ the convex hull $H_p$ of the Julia set $J_p$ of $p$ satisfies $p^{-1}(H_p) \subset H_p$. This settles positively a conjecture by P. Alexandersson. We also characterize the families of polynomials for which the equality $p^{-1}(H_p) = H_p$ is achieved.

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Strict positive definiteness of non-radial kernels on $d$-dimensional spheres

Dr. Janin Jäger, Justus Liebig University

Abstract

Isotropic positive definite functions are used in approximation theory and are for example applied in geostatistics and physiology. They are also of importance in statistics where they occur as correlation functions of homogeneous random fields on spheres. We study a class of function applicable for interpolation of arbitrary scattered data on $\mathbb{S}^{d-1}$ by linear combinations of a kernel $K:\mathbb{S}^{d-1}\times \mathbb{S}^{d-1}\rightarrow \mathbb{C}$ evaluated at the interpolation points in the second argument. The isotropic kernels are a special case of this approach and we study kernels with more general properties like axial symmetry and invariance under parity. A class of kernels for which the resulting interpolation problem is uniquely solvable for any distinct point set $\Xi\subset \mathbb{S}^{d}$ are known strict positive definite isotropic functions. Using recent results of Bonfim and Menegatto \cite{Bonfim2016} and the famous representations of isotropic positive definite functions on $\mathbb{S}^{d-1}$ due to Schoenberg as starting point we derive new sufficient conditions for strict positive definiteness of axial symmetric and convolutional kernels. The results extend a necessary and sufficient characterisation of strict positive definite isotropic basis functions by Chen et al. proven in \cite{Chen2003} to a non-radial kernel class.% We also prove in which cases the reproducing kernel Hilbert spaces of these kernels coincide with Sobolev spaces and thereby extend existing error estimates to this class of kernels. %For this case we prove a characterisation of (conditionally) positive definite function on the Hilbert sphere in terms of monotonicity properties of $\phi$. We also prove such results for conditionally negative definite functions which are also admissible for interpolation. %As a result a theorem of Gneiting is generalised showing that all functions $\phi \in C([0,\pi])$ which are completely monotone on $(0,\pi)$ and no linear polynomials are strictly positive definite on all spheres. Similar results are presented for conditionally positive definite functions of order 1. %\bibitem{Schoenberg1942b} I.J. Schoenberg: {\it Positive definite functions on the sphere}, Duke Math. j., 9, 96-108, 1942. \bigskip\noindent \bibliographystyle{plain} \begin{thebibliography}{99} \bibitem{Chen2003}Chen, D. and Menegatto, V. A. and Sun, X.: {\it A necessary and sufficient condition for strictly positive definite functions on spheres}, Proceedings of the American Mathematical Society, 131, 2733-2740, 2003. \bibitem{Bonfim2016} Bonfim, R. and Menegatto, V., \textit{ Strict positive definiteness of multivariate covariance functions on compact two-point homogeneous spaces}, Journal of Multivariate Analysis, 152, 2016. \end{thebibliography}

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A complex funtion theory for Mellin Analysis and applications to sampling

Prof. Carlo Bardaro, University of Perugia

Abstract

The aim of this research is to extend in a simple way the well-known Paley--Wiener theorem of Fourier Analysis, which characterizes the so-called bandlimited functions, to the setting of Mellin transform. In order to do that, we introduced in these last years a notion of polar analytic function [1], which provides a simple way of describing functions that are analytic on a part of the Riemann surface of the logarithm. Applications to various topics of Mellin Analysis are developed, in particular sampling type theorems and quadrature formulae (for a survey see [2]). \vskip0,3cm \noindent [1] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Development of a new concept of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. {\bf 64}(12), 2040--2062 (2019) \\ \noindent [2] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Polar-analytic functions: old and new results, applications. Pre-print submitted, (2021)

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Integral-type operators on mobile intervals

Dr. Mirella Cappelletti Montano, Università degli Studi di Bari Aldo Moro

Abstract

In this talk, we present a sequence $(C_n)_{n \geq 1}$ of positive linear operators, introduced in \cite{cl} and acting on spaces of continuous functions as well as on spaces of integrable functions on $[0, 1]$. These operators represent a Kantorovich-type modification, on mobile intervals, of the ones discussed in \cite{cmr}. We state some qualitative properties of the sequence $(C_n)_{n \geq 1}$ and we prove that it is an approximation process both in $C([0, 1])$ and in $L^p([0, 1])$, also providing some estimates of the rate of convergence. Moreover, we determine an asymptotic formula and we prove that suitable iterates of the operators $C_n$ converge, both in $C([0, 1])$ and, under suitable assumptions, in $L^p([0, 1])$ to a limit semigroup. Finally, we compare our operators with other existing ones in the literature showing that they allow a lower approximating error estimate. \begin{thebibliography}{99} \bibitem{cl} M. Cappelletti Montano, Vita Leonessa, \emph{On a sequence of Kantorovich-type operators}, Constr. Math. Anal. \textbf{2} (3) (2019), 130-143. \bibitem{cmr} D. Cardenas-Morales, P. Garrancho, I. Ra\c{s}a, \emph{Bernstein-type operators which preserve polynomials}, Comput. Math. Appl. \textbf{62} (1) (2011), 158--163. \end{thebibliography}

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Approximation by Durrmeyer-Sampling Type Operators in Functional Spaces

Dr. Michele Piconi, University of Perugia, Perugia, Italy

Abstract

Sampling-type operators have been introduced in order to give an approximate version of the celebrated classical sampling theorem. Among these operators, we have studied the Durrmeyer-Sampling type operators (DSO) [4], (see also [6,\,2]), which represent a further generalization of the well-known Generalized [3] and Kantorovich-Sampling operators [1,\,5]. \\ In order to follow a unifying approach, we have provided a general result in terms of convergence, which is represented by a modular convergence theorem in Orlicz spaces. From the latter result, the convergence in $L^p$-spaces follows as particular case. This approximation result for DSO is important, in some particular case, from the applications point of view, e.g. in image processing, where we have to work with not-necessarily continuous signals. \\ For the sake of completeness of the theory, we have also studied the continuous case, providing a pointwise and uniform convergence theorem and quantitative estimates. \\ Moreover, all the above convergence results for DSO can also be extended in the multidimensional setting. \\ %----------------references------------------------------------------------------ \textbf{References} \begin{description} \item [[1]] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, Kantorovich-Type Generalized Sampling Series in the Setting of Orlicz Spaces, Sampling Theory in Signal and Image Processing, 9 (6) (2007), 29-52. \item [[2]] C. Bardaro, I. Mantellini, Asymptotic expansion of generalized Durrmeyer sampling type series, Jean J. Approx., 6 (2) (2014), 143-165. \item [[3]] P. L. Butzer, A. Fisher, R. L. Stens, Approximation of continuous and discontinuous functions by generalized sampling series, J. Approx. Theory, 50 (1987), 25-39. \item [[4]] D. Costarelli, M. Piconi, G. Vinti, On the convergence properties of Durrmeyer-Sampling type operators in Orlicz spaces, (2020), arXiv: 2007.02450v1, submitted. \item [[5]] D. Costarelli, G. Vinti, Approximation by Multivariate Generalized Sampling Kantorovich Operators in the Setting of Orlicz Spaces, Bollettino U.M.I., Special issue dedicated to Prof. Giovanni Prodi, 9 (IV) (2011), 445-468. \item [[6]] J. L. Durrmeyer, Une firmule d'inversion de la transformée de Laplace: applications à la théorie des moments, Thése de 3e cycle, Universitè de Paris, (1967). \end{description}

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Bernstein-Chlodovsky operators preserving exponentials

Dr. Vita Leonessa, University of Basilicata

Abstract

The aim of this talk is to illustrate a generalization of Bernstein-Chlodovsky operators, introduced and studied in \cite{1,2}, that preserves the exponential function $e^{-2x}$ ($x\geq 0$). In particular, in \cite{1} we studied its approximation properties in several function spaces, also evaluating the rate of convergence by means of suitable moduli of continuity. As a consequence, we proved better error estimation than the original operators on certain intervals. In \cite{2} we continued the study of such operators by proving some Voronovskaya type theorems and deducing saturation results. A comparison of this new class of operators with the classical Bernstein-Chlodovsky ones is also made, proving that the new operators provide better approximation results for certain functions on $[0,+\infty)$. \begin{thebibliography}{99} \bibitem{1} T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., \textit{On Bernstein-Chlodovsky operators preserving $e^{-2x}$}, Bull. Belg. Math. Soc. Simon Stevin \textbf{26} (2019), no. 5, 681--698. \bibitem{2} T. Acar, M. Cappelletti Montano, P. Garrancho, V. L., \textit{Voronovskaya type results for Bernstein-Chlodovsky operators preserving $e^{-2x}$}, J. Math. Anal. Appl. \textbf{491} (2020), no. 1, 124307, 14 pp. \end{thebibliography}

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An iso-geometric radial basis function partition of unity method for PDEs in thin structures

Prof. Elisabeth Larsson, Uppsala University

Abstract

The application that motivates this work is numerical simulation of the biomechanics of the respiratory system. The main respiratory muscle is the diaphragm, which is a thin structure. There are several challenges associated with the geometry, including its representation. Here, we first use a radial basis function partition of unity method (RBF-PUM) to make a smooth reconstruction of the geometry from noisy medical image data. Then we use RBF-PUM to approximate the solution of a PDE problem posed in this geometry. In a PUM, the global approximation is expressed as a weighted combination of local approximations over patches that form a cover of the domain. A particular benefit of RBF-PUM is that we can adapt each local approximation to the local properties of the problem. For this thin, curved, non-trivial geometry, we can scale the local problems to ensure sufficient local resolution of the thickness dimension. We show results for a simple Poisson test problem and show that we can achieve high-order convergence with an appropriate choice of method parameters.

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A general method to study the convergence of nonlinear operators in Orlicz spaces

Dr. Luca Zampogni, University of Perugia

Abstract

We introduce a general setting in which we define nets of nonlinear operators whose domains are some set of functions defined in a locally compact topological group $G$. These nets assume the form $$ T_wf:=z\mapsto\int_H\chi_w(z-h_w(t),L_{h_w(t)}(f))d\mu_H(t), \;x>0,$$ where $H$ is a topological group with (left-invariant) Haar measure $\mu_H$, $(\chi_w)_w$ is a net of Kernels functions defined on $G\times \mathbb R$, $h_w$ are homeomorphism from $H$ to $G$ and $L_{h_w(t)}:L(G)\rightarrow\mathbb R$ are linear operators. We analyze the behavior of such nets, and detect the fairest assumption which are needed for the nets to converge in Orlicz spaces. As a consequence, we give a result of convergence in a subspace of a Orlicz space.

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Bounds for Several Statistical Indicators

Dr. Augusta Ratiu, Lucian Blaga University of Sibiu

Abstract

In this paper, we study the problem of finding new bounds for some statistical indicators that characterize a data series. Using a refinement of Aczél's inequality, we find other bounds for dispersion, standard deviation and coefficient of variation.

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Inequalities for Legendre polynomials and applications in information potential

Dr. Daniel Florin Sofonea, Lucian Blaga University of Sibiu, Romania

Abstract

The classical orthogonal polynomials play an important role in applications of mathematical analysis, spectral method with applications in fluid dynamics and other areas of interest. This work is devoted to the orthogonal polynomials. Bounds of Legendre polynomials are obtained in terms of inequalities. Also, entropies associated with discrete probability distributions is a topic considered. Bounds of the entropies which improve some previously known results are obtained.

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Modification of exponential type operators preserving exponential functions connected with $x^3$

Dr. Carmen Violeta Popescu (Muraru), Vasile Alecsandri University of Bacau

Abstract

In the present paper, we propose modification of the exponential type operators, which are connected with $x^3$. Such operators are connected with exponential functions. We estimate moments and establish some direct results in terms of modulus of continuity.

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Sampling strategies for approximation in kernel spaces

Dr. Gabriele Santin, Bruno Kessler Foundation

Abstract

Kernel methods provide powerful and flexible techniques to approximate functions defined on general domains, with possible high-dimensional input and output dimension, and using samples at scattered locations. In this context, the problem of choosing the location of the sampling points is of great interest, both from a practical and a theoretical viewpoint. On one hand, it is of theoretical interest to know the limit and benefits of the choice of optimal point location, and to design feasible algorithms to select them. On the other hand, several applications are described by large datasets, and it may be interesting to select a possibly small portion of the data that allows an accurate reconstruction of the full problem. In this talk we will discuss some greedy methods and show that they are effective techniques in both scenarios. In particular, we will first introduce some results on the general structure and theory of kernel-based greedy methods, and describe their efficient implementation. We will then show that, in certain circumstances, they may be proven to be worst-case optimal. We will focus mainly on interpolation, and mention some application to quadrature. Moreover, we will discuss the use of these techniques on some real world applications.

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Kernel-based approximation methods on graphs

Wolfgang Erb, University of Padova

Abstract

We study how the concept of positive definite functions can be transfered to a graph setting in order to approximate graph signals with generalized shifts of a graph basis function (GBF). This concept merges kernel-based approximation with spectral theory on graphs and can be regarded as a graph analog of radial basis function methods in euclidean spaces or on the sphere. We provide several descriptions of positive definite functions on graphs, the most relevant one is a Bochner- type characterization in terms of positive Fourier coefficients. These descriptions allow us to design GBF’s and to study GBF approximation in more detail: we are able to characterize the native spaces of the interpolants, we give explicit estimates for the approximation error and provide ways on how to calculate the approximants in an efficient manner. As a final application, we show how GBFs can be used for classification tasks on graphs.

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On derivative sampling using Kantorovich-type sampling operators

Dr. Gert Tamberg, Tallinn University of Technology

Abstract

For $f\in C(\mathbb{R})$ the generalized sampling operators are given by ($t\in{\mathbb{R}}$; $W>0$) \begin{equation}\label{uldvrida} (S_Wf)(t) := \sum_{k=-\infty}^{\infty}f({k\over W})s(Wt-k), \end{equation} where $s$ is a certain kernel function, i.e. \[ s\in L^1({\mathbb{R}}),\hspace*{0.5cm} \sum_{k\in\mathbb{Z}}s(u-k)=1, \hspace*{0.5cm}(u\in\mathbb{R}). \] We show a connection between generalized sampling operators with averaged kernels and generalized Kantorovich-type sampling operators. Using this connection, we can estimate the order of approximation of derivatives.

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Variation diminishing type estimates for generalized sampling operators and applications

Prof. Laura Angeloni, University of Perugia

Abstract

The variation diminishing estimate is a classical result that is usually investigated working in BV spaces with some classes of operators: such result essentially ensures that the variation of the operator is not bigger than the variation of the function to which it is applied. We will present estimates of this kind, besides results about convergence in variation, for multivariate sampling-type operators. Differently from the one-dimensional frame, where variation diminishing type results are usually quite easy to be achieved, the multidimensional case is more delicate: nevertheless it is interesting, also from an applicative point of view, since it is connected to some problems of Digital Image Processing, in particular to smoothing procedures. \vskip1cm \noindent \textbf{References} \noindent [1] L. Angeloni, G. Vinti, \emph{Estimates in variation for multivariate sampling-type operators}, Dolomites Research Notes on Approximation, {\bf 14}(2) (2021), 1--9.\\ \noindent [2] L. Angeloni, D. Costarelli, G. Vinti, \emph{A characterization of the convergence in variation for the generalized sampling series}, Ann. Acad. Sci. Fenn. Math., {\bf 43} (2018), 755--767.\\ \noindent [3] L. Angeloni, D. Costarelli, G. Vinti, \emph{Convergence in variation for the multidimensional generalized sampling series and applications to smoothing for digital image processing}, Ann. Acad. Sci. Fenn. Math., \textbf{45} (2020), 751--770.\\ \noindent [4] L. Angeloni, D. Costarelli, M. Seracini, G. Vinti, L. Zampogni, \emph{Variation diminishing-type properties for multivariate sampling Kantorovich operators}, Boll. Unione Mat. Ital., Special Issue "Measure, Integration and Applications" dedicated to Prof. Domenico Candeloro, {\bf 13} (2020), 595--605.\\ \noindent [5] P.L. Butzer, A. Fisher, R.L. Stens, \emph{Generalized sampling approximation of multivariate signals: theory and applications}, Note Mat., \textbf{1} (10), 173--191 (1990).\\ \noindent [6] C. Bardaro, P.L. Butzer, R.L. Stens, G. Vinti, \emph{Prediction by Samples From the Past With Error Estimates Covering Discontinuous Signals}, IEEE Trans. Inform. Theory, \textbf{56} (1), 614--633 (2010).\\

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Piecewise-regular approximation of maps into real algebraic sets

Prof. Marcin Bilski, Jagiellonian University

Abstract

A real algebraic set $W$ of dimension $m$ is said to be uniformly rational if each of its points has a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of $\mathbb{R}^m.$ Let $l$ be any nonnegative integer. It turns out that every map of class $\mathcal{C}^l$ from a compact subset of a real algebraic set into a uniformly rational real algebraic set can be approximated in the $\mathcal{C}^l$ topology by piecewise-regular maps of class $\mathcal{C}^k$, where $k$ is an arbitrary integer greater than or equal to $l. $

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A generalization of extremal functions and polynomial inequalities

Prof. Mirosław Baran, Pedagogical University of Cracow

Abstract

Consider a normed space $(\mathcal{P}(\mathbb{C}^N),\mathcal{N})$ of polynomials of $N$ variables equi\-pped with a fixed norm $\mathcal{N},$ which can be arbitrary. We can define a {\it radial} version of a polynomial extremal function, which has a sense in a general situation. In the case of the supremum norm, $\mathcal{N}(P)=\sup\{|P(z)|:\ z\in E\}$ our extremal functions are a radial modification of the classical Siciak's extremal function $\Phi (E,z)$. In this special case we can also consider a local radialization of the Siciak's extremal function and its logarithm $V(E,z)$ (the pluricomplex Green function). We shall show connections between the behaviour of such extremal functions and polynomial inequalities of Markov's and Bernstein's type. In particular, there will obtained new results on Bernstein' inequality involving higher derivatives of polynomials at interior points of compact subsets of $\mathbb{R}^N$.

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Metric Fourier approximation of set-valued functions of bounded variation

Dr. Elena Berdysheva, Justus Liebig University Giessen

Abstract

We study set-valued functions (SVFs) mapping a real interval to compact sets in~${\mathbb R}^d$. Older approaches to the approximation investigated almost exclusively SVFs with convex images (values), the standard methods suffer from convexification. In this talk I will describe a new construction that adopts the trigonometric Fourier series to set-valued functions with general (not necessarily convex) compact images. Our main result is analogous to the classical Dirichlet-Jordan Theorem for real functions. It states the pointwise convergence in the Hausdorff metric of the metric Fourier partial sums of a multifunction of bounded variation to a set determined by the values of the metric selections of the function. In particular, if the multifunction $F$ is of bounded variation and continuous at a point $x$, then the metric Fourier partial sums of it at $x$ converge to $F(x)$.

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Admissible meshes on algebraic sets

Dr. Agnieszka Kowalska, Pedagogical University of Krakow

Abstract

Admissible meshes were formally introduced by J. P. Calvi and N. Levenberg in 2008. Such meshes are nearly optimal for uniform least squares approximation and contain interpolation sets nearly as good as Fekete points of the domain. Optimal admissible meshes have been constructed on many polynomially determining compact sets, e.g., sections of discs, ball, convex bodies, sets with regular boundary, by different analytical and geometrical techniques. Regarding subsets of algebraic varieties admissible meshes are known only for a few compacts like sections of a~sphere, a~torus, a~circle and curves in $\mathbb{C}$ with analytic parametrization. We construct polynomial weakly admissible meshes on compact subsets of algebraic hypersurfaces in $\mathbb{C}^{N+1}$. These meshes are optimal in some cases. We present also partial results for algebraic sets of codimension greater than one.

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Functional differential equations with maxima, via step by step contraction principle

Dr. Diana Otrocol, Technical University of Cluj-Napoca

Abstract

T.A. Burton presented in some examples of integral equations a notion of progressive contractions on $C([a,\infty \lbrack).$ In 2019, I.A. Rus formalized this notion (I.A. Rus, \textit{Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle}, Advances in the Theory of Nonlinear Analysis and its Applications 3 (2019) No. 3, 111-120), put "step by step" instead of "progressive" in this notion, and give some variant of step by step contraction principle in the case of operators with Volterra property on $C([a,b],\mathbb{B)}$ and $C([a,\infty \lbrack,\mathbb{B})$, where $\mathbb{B}$ is a Banach space. In this paper we use the abstract result given by I.A. Rus, to study some classes of functional differential equations with maxima.

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Best Ulam constant of a linear difference equation

Dr. Alina Ramona Baias, Technical University of Cluj Napoca

Abstract

An equation is called Ulam stable if for every approximate solution of it there exists an exact solution near it. We present some results on Ulam stability for some linear difference equations. In a Banach space $X$ the linear difference equation with constant coefficients $x_{n+p}=a_1x_{n+p-1}+\ldots+a_px_n,$ is Ulam stable if and only if the roots $r_k,$ $1\leq k\leq p,$ of its characteristic equation do not belong to the unit circle. If $|r_k|>1,$ $1\leq k\leq p,$ we prove that the best Ulam constant of this equation is $\frac{1}{|V|}\sum\limits_{s=1}^{\infty}\left|\frac{V_1}{r_1^s}-\frac{V_2}{r_2^s}+\ldots+\frac{(-1)^{p+1}V_p}{r_p^s}\right|,$ where $V=V(r_1,r_2,\ldots,r_p)$ and $V_k=V(r_1,\ldots,r_{k-1},r_{k+1}, \ldots, r_p),$ $1\leq k\leq p,$ are Vandermonde determinants.

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Best Ulam constant of a linear differential operator

Prof. Dorian Popa, Technical University of Cluj-Napoca,

Abstract

The Ulam stability of an operator $L$ acting in Banach spaces is equivalent with the stability of the associated equation $Lx=y.$ An equation is called Ulam stable if for every approximate solution of it there exists an exact solution near it. We present some results on Ulam stability for some linear differential operators. The linear differential operator with constant coefficients $$D(y)=y^{(n)}+a_1 y^{(n-1)}+\ldots+a_n y,\quad y\in \mathcal{C}^{n}(\mathbb{R}, X)$$ acting in a Banach space $X$ is Ulam stable if and only if its characteristic equation has no roots on the imaginary axis. We prove that if the characteristic equation of $D$ has distinct roots $r_k$ satisfying $\Re r_k>0,$ $1\leq k\le n,$ then the best Ulam constant of $D$ is $K_D=\frac{1}{|V|}\int_{0}^{\infty}\left|\sum\limits_{k=1}^n(-1)^kV_ke^{-r_k x}\right|dx,$ where $V=V(r_1,r_2,\ldots,r_n)$ and $V_k=V(r_1,\ldots,r_{k-1},r_{k+1}, \ldots, r_n),$ $1\leq k\leq n,$ are Vandermonde determinants.

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A Hermite-Hadamard type inequality with applications to the estimation of moments of convex functions of random variables

Mrs. PAȘCA RALUCA IOANA , TECHNICAL UNIVERSITY CLUJ-NAPOCA

Abstract

In this paper we present a Hermite-Hadamard type inequality with applications to the estimation of moments for convex functions of several random variables. A particular case of the derived result can be applied to estimate the sum of moments for identically distributed random variables. We were motivated in our research by the applications of moment estimation in fields such as probability theory, mathematical statistics and statistical learning.

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A generalization of a local form of the classical Markov inequality

Dr. Tomasz Beberok, University of Applied Sciences in Tarnow

Abstract

In this talk we introduce a generalization to compact subsets of certain algebraic varieties of the classical Markov inequality on the derivatives of a polynomial in terms of its own values. We also introduce an extension to such sets of a local form of the classical Markov inequality, and show the equivalence of introduced Markov and local Markov inequalities.

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H\"older continuity of the pluricomplex Green function

Prof. Rafał Pierzchała, Jagiellonian University

Abstract

I will discuss the following problem of Ple\'sniak (1988). Let $h:U\to\mathbb{C}^{N'}$, where $U\subset\mathbb{C}^N$ is an open set, be a holomorphic map $(N, N'\in\mathbb{N})$. Assume that a compact set $\emptyset\neq K\subset\mathbb{C}^N$ has the HCP property (that is, the pluricomplex Green function $V_K$ of $K$ is H\"older continuous) and $\hat{K}\subset U$. Under what conditions does it happen that $h(K)$ has the HCP property?

Arithmetic and Geometry of Algebraic Surfaces (MS - ID 45)

Algebraic and Complex Geometry

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Generalized Kummer surfaces and a configuration of conics in the plane

Mrs. Alessandra Sarti, University of Poitiers

Abstract

A generalized Kummer surface is the minimal resolution of the quotient of an abelian surface by an automorphism of order three. The quotient surface contains nine cusps and this is a special example of a singular K3 surface. The converse is also true : the minimal resolution of a K3 surface containing nine cusps is a generalized Kummer surface. In this paper we study several configurations of nine cusps on the same K3 surface, which is the double cover of the plane ramified on a sextic with nine cusps. We get the new configurations by studying conics through the singular points of the sextic. The aim of the construction is to investigate the following question: is it possible that the K3 surface is the generalized Kummer surface associated to two non-isomorphic abelian surfaces ? This is a joint work with D. Kohl and X. Roulleau.

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Irregular covers of K3 surfaces

Mrs. Alice Garbagnati, Universita' Statale di Milano

Abstract

The K3 surfaces are regular surfaces. They can be covered by irregular surfaces in several ways and the easiest example is given by the abelian surfaces: these often appear as the minimal model of Galois covers of K3 surfaces for several Galois group (e.g. all the Abelian surfaces are birational to the double cover of their Kummer surfaces). Nevertheless is considerably more difficult to obtain an irregular cover of a K3 surface which is a surface of general type. During the talk we discuss the construction of irregular covers of K3 surfaces, obtaining both surfaces with Kodaira dimension 1 and 2. In particular we focus our attention to the covers of order 2 and 3 (the latter not necessarly Galois).

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Infinitesimal Torelli for elliptic surfaces revisited

Prof. Remke Kloosterman, University of Padova

Abstract

In this talk we consider the infinitesimal Torelli problem for elliptic surfaces without multiple fibers. We give a new proof for the case of elliptic surfaces without multiple fibers with Euler number at least 24 and nonconstant j-invariant and with Euler number at least 72 and constant j-invariant. For all of the remaining cases we will indicate whether infinitesimal Torelli holds, does not hold or our methods are insufficient to decide. This solve an issue raised by Atsushi Ikeda in a recent paper, in connection with his construction a counterexample to infinitesimal Torelli with $p_g=q=1$.

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Z/2-Godeaux surfaces

Prof. Carlos Rito, University of Porto - CMUP

Abstract

I will report on work in progress in collaboration with Eduardo Dias towards the classification of Z/2-Godeaux surfaces.

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New families of surfaces with canonical map of high degree

Prof. Roberto Pignatelli, University of Trento

Abstract

By a celebrated result of A. Beauville, if the dimension of the image of the canonical map of a complex surface is two, then the image is either of genus zero or a canonical surface. In both cases the degree of the map is uniformely bounded from above. Until a few years ago, very few examples of surfaces with canonical map of high degree were known. Thanks to the work of some collegues, new examples have been obtained in recent years, opening up new research questions and perspectives. In this seminar we will present a new method for constructing surfaces with canonical map of high degree and some new examples produced with this method. Finally, we will discuss the contribution our examples make to the aforementioned issues.

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On a family of surfaces with $p_g=q=2$ and $K^2=7$

Prof. Matteo Penegini, Università di Genova

Abstract

In this talk we shall study a family of surfaces of general type with $p_g=q=2$ and $K^2=7$, originally constructed by C. Rito. We provide an alternative construction of these surfaces, that allows us to describe their Albanese map and the corresponding locus $\mathcal{M}$ in the moduli space of the surfaces of general type. In particular we prove that $\mathcal{M} $ is an irreducible component, two dimensional and generically smooth.

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Subbundles of the Hodge bundle, fibred surfaces and the Coleman-Oort conjecture

Dr. Victor Alonso, Humboldt-Universität zu Berlin

Abstract

The Hodge bundle of a semistable family of complex projective curves has two nested subbundles: the flat unitary subbundle (spanned by flat sections with respect to the Gauss-Manin connection), and the kernel of the Higgs field. The latter contains the flat subbundle, but it was not clear how strict the inclusion could be. In this talk I will present some techniques to estimate the ranks of both subbundles and to construct families where they are arbitrarily different. I will also discuss some implications of this result for the classification of fibred surfaces, as well as some connections with the Coleman-Oort conjecture on the (non-)existence of totally geodesic subvarieties in the Torelli locus of principally polarized abelian varieties. This is joint work with Sara Torelli.

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The Mumford–Tate conjecture for surfaces — state of the art

Dr. Johan Commelin, Universität Freiburg

Abstract

The Mumford–Tate conjecture (MTC) provides a bridge between the Hodge structure on the singular cohomology of an algebraic variety (over the complex numbers) and the Galois representation on the étale cohomology of a model of that same variety over a number field. Recently, there have been several new results on MTC for algebraic surfaces. In this talk I will give an overview of what is currently known, and explain some of the techniques used in the new results.

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Holomorphic one forms on projective surfaces and applications

Ms. Sara Torelli, University of Hannover

Abstract

In the talk I present a result, recently proven in collaboration with F.Favale and G.P.Pirola, that aims to recover holomorphic $1$-forms on smooth projective surfaces from divisors contracted to points by big and semiample line bundles. I'll then discuss its application to investigate holomorphic forms on higher dimensional varieties (e.g. the moduli space of pointed curves $\mathcal{M}_{g,n}$) and related open problems.

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Rigid complex manifolds via product quotients

Dr. Christian Gleissner, University of Bayreuth

Abstract

Product quotients give rise to a large class of varieties equipped with a lot of symmetry. They are particularly useful as a tool to provide examples of complex manifolds with special properties. In this talk I will explain how to construct in each dimension dimension n (at least three) an example of a rigid n-manifold of Kodaira dimension one, as a resolution of a singular product quotient. Their existence was conjectured by I. Bauer and F. Catanese and constructed in a joint project with I. Bauer.

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Moduli space of stable (1,2)-surfaces

Dr. Stephen Coughlan, Universitaet Bayreuth

Abstract

A (1,2)-surface is a surface of general type with geometric genus 2 and canonical degree 1. I describe ongoing work with several collaborators, aiming to understand the so-called KSBA compactification of the moduli space of (1,2)-surfaces. In particular, we are able to describe some irreducible components of the moduli space and some boundary strata too.

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Diagonal double Kodaira fibrations with minimal signature

Prof. Francesco Polizzi, Università della Calabria

Abstract

We study some special systems of generators on finite groups, introduced in previous work by the first author and called \emph{diagonal double Kodaira structures}, in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$, where $\Sigma_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last part, as a geometrical application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having signature $16$. This is joint work with P. Sabatino

CA18232: Variational Methods and Equations on Graphs (MS - ID 40)

Partial Differential Equations and Applications

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Nonlinear models of kinetic type: On the Cauchy problem and Banach space regularity for Boltzmann flows of monatomic gas mixtures

Dr. Milana Pavić-Čolić, Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad

Abstract

This talk will focus on the analysis of kinetic models for multi-component mixtures of monatomic gases with different masses. The model corresponds to a Boltzmann system for the evolution of vector valued distribution function. The collision or interaction law, as much as the modelling of the transition probability rates for pairwise interactions, are crucial components in the dynamics. We will present some recent rigorous results for the full non-linear space homogeneous Boltzmann system of equations describing multi-component monatomic gas mixtures for binary interactions. More precisely, we will show existence and uniqueness of the vector value solution in the case of hard potentials and integrable angular scattering kernels associated to each pair of interacting species, by means of an existence theorem for ODE systems in Banach spaces. In addition, we will present several properties for such a solution, including integrability properties of the multispecies collision operator. These properties together with a control by below imply propagation of the polynomially and exponentially weighted $L^p$ norms, $1\leq p \leq \infty$, associated to the system solution. Additionally, for $p=1$ we have generation of such moments.

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Hidden symmetries in non-self-adjoint graphs

Mr. Amru Hussein, TU Kaiserslautern

Abstract

On finite metric graphs Laplace operators subject to general non-self-adjoint matching conditions imposed at graph vertices are considered. A regularity criterium related to the Cayley transform of boundary conditions is discussed and spectral properties of such regular operators are investigated, in particular similarity transforms to self-adjoint operators and generation of $C_0$-semigroups. Concrete examples are discussed exhibiting that non-self-adjoint boundary conditions can yield to unexpected spectral features. The talk is based on joint works with David Krej\v{c}i\v{r}\'{i}k (Czech Technical University in Prague), Petr Siegl (Queen's University Belfast) and Delio Mugnolo (FernUniversit\"at Hagen).

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Gibbs Evolution Families

Prof. András Bátkai, Pädagogische Hochschule Vorarlberg

Abstract

We extend results of Zagrebnov and coauthors on Gibbs semigroups to the nonautonomous case. Sufficient conditions will be given to evolution equations in the parabolic case so that the generated evolution family belogs to appropriate operator ideals. Applications to Schrödinger operators and the trace class ideal will be also given. The results were achieved in part in a joint work with Bálint Farkas (Wuppertal). Support by the COST Action CA18232 - Mathematical models for interacting dynamics on networks is acknowledged.

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Spectral geometry of quantum graphs via surgery principles

Dr. James Kennedy, University of Lisbon

Abstract

``Surgery" on a (metric) graph means making a small, generally local, change to its structure: for example, joining two vertices, lengthening an edge, or maybe removing an edge and reinserting it somewhere else. We will introduce a number of sharp new surgery principles which allow one to control the eigenvalues of the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or delta). We will illustrate how these principles can be used to give new proofs and sharper versions of existing ``isoperimetric"-type eigenvalue estimates by sketching a result which interpolates between the theorems of Nicaise and Band--L\'evy for the first non-trivial eigenvalue of the Laplacian with natural vertex conditions. This is based on joint work with Gregory Berkolaiko, Pavel Kurasov and Delio Mugnolo.

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Uniqueness and non-uniqueness of prescribed mass NLS ground states on metric graphs

Dr. Simone Dovetta, Università degli Studi di Roma "La Sapienza"

Abstract

This talk addresses the problem of uniqueness of ground states of prescribed mass for the Nonlinear Schr\"odinger Energy with power nonlinearity on noncompact metric graphs with half--lines. We first show that, up to an at most countable set of masses, all ground states at given mass solve the same equation, that is the Lagrange multiplier appearing in the NLS equation is constant on the set of ground states of mass $\mu$. On the one hand, we apply this result to prove uniqueness of ground states on two specific families of noncompact graphs. On the other hand, we construct a graph that admits at least two ground states with the same mass having different Lagrange multipliers. This shows that the result for Lagrange multipliers is sharp in general, in the sense that one cannot get rid of the at most countable set of masses where it may fail without further assumptions. Our proofs are based on careful variational arguments and rearrangement techniques, and hold both for the subcritical regime $p\in (2,6)$ and in the critical case $p=6$. This is a joint work with Enrico Serra and Paolo Tilli.

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Convergence to equilibrium of stochastic semigroups and an application to buffered networks flows

Jochen Glück, University of Passau

Abstract

The long-time behaviour of flows on finite metric graphs is known to depend heavily on the network topology. Depending on this topology, the lengths of the edges and the flow velocities, the flow might converge or behave asymptotically periodic as $t \to \infty$. In this talk we show that the situation changes if we introduce a mass buffer in at least one of the vertices. Such a buffer has a smoothing effect on the flow and thus enforces convergence as $t \to \infty$. As we consider finite graphs only, one would even expect that the convergence is uniform (i.e., with respect to the operator norm over the $L^1$-space over the graph). In order to prove that this is indeed true we employ a novel characterisation of operator norm convergence to equilibrium for stochastic $C_0$-semigroups.

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Self-adjoint extensions of infinite quantum graphs

Mrs. Noema Nicolussi, University of Vienna

Abstract

In the last decades, quantum graphs (Laplacians on metric graphs) have become popular objects of study and the analysis of spectral properties relies on the self-adjointness of the Laplacian. Whereas on finite metric graphs the Kirchhoff Laplacian is always self-adjoint, much less is known about the self-adjointness problem for graphs having infinitely many edges and vertices. Intuitively the question is closely related to finding appropriate boundary notions for infinite graphs. \\ In this talk we study the connection between self-adjoint extensions and the notion of graph ends, a classical graph boundary introduced independently by Freudenthal and Halin. Our discussion includes a lower estimate on the deficiency indices of the minimal Kirchhoff Laplacian and a geometric characterization of self-ajointness of the Gaffney Laplacian.\\ Based on joint work with Aleksey Kostenko (Ljubljana \& Vienna) and Delio Mugnolo (Hagen).

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Stability and asymptotic properties of dissipative equations coupled with ordinary differential equations

Prof. Serge Nicaise, Université Polytechnique Hauts-de-France

Abstract

In this talk, we will present some stability results of a system corresponding to the coupling between a dissipative equation (set in an infinite dimensional space) and an ordinary differential equation. Namely we consider $U, P$ solution of the system \begin{equation}\label{1} \left\{\begin{array}{ll}U_t=\mathcal{A} U +M P, \hbox{ in } H,\\ P_t=BP+ N U, \hbox{ in } X,\\ U(0)=U_0,P(0)=P_0, \end{array}\right. \end{equation} where $\mathcal{A}$ is the generator of a $C_0$ semigroup in the Hilbert space $H$, $B$ is a bounded operator from another Hilbert space $X$, and $M$, $N$ are supposed to be bounded operators. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of {ODE}-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators $\mathcal{A}$, $B$, $M$ and $N$ that garantee stability properties of system (\ref{1}), i.e., strong stability, exponential stability or polynomial one. We will illustrate our general results by an example of generalized telegraph equations set on networks.

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Telegraph systems on networks and port-Hamiltonians

Prof. Jacek Banasiak, University of Pretoria

Abstract

In this talk we consider a system of linear hyperbolic differential equations on a network coupled through general transmission conditions of Kirchhoff’s type at the nodes. We discuss the reduction of such a problem to a system of 1-dimensional hyperbolic problems, also called port-Hamiltonian, for the associated Riemann invariants and provide a semigroup theoretic proof of its well-posedness in any $L_p$. In the second part of the talk we consider a reverse question, that is, we derive conditions under which such a port-Hamiltonian with general linear Kirchhoff’s boundary conditions can be written as a system of $2 \times 2$ hyperbolic equations on a metric graph $\Gamma$. This is achieved by interpreting the matrix of the boundary conditions as a potential map of vertex connections of $\Gamma$ and then showing that, under the derived assumptions, that matrix can be used to determine the adjacency matrix of $\Gamma$.

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Kinetic and macroscopic diffusion models for gas mixtures in the context of respiration

Dr. Bérénice GREC, University of Paris

Abstract

In this talk, I will first discuss shortly the context of respiration, and in particular the need to describe accurately the diffusion of respiratory gases in the lower part of the lung. At the macroscopic level, diffusion processes for mixtures are often modelled using cross-diffusion models. In order both to determine the regime in which such models are valid, and to compute the binary diffusion coefficients, it is of particular interest to derive these models from a description at the mesoscopic level by means of kinetic equations. More precisely, we consider the Boltzmann equations for mixtures with general cross-sections (i.e. for any kind of molecules interactions), and obtain the so-called Maxwell-Stefan equations by performing a Hilbert asymptotic expansion at low Knudsen and Mach numbers. This allows us to compute the values of the Maxwell-Stefan diffusion coefficients with explicit formulae with respect to the cross-sections. We also justify the specific ansatz we use thanks to the so-called moment method. This is a joint work with Laurent Boudin and Vincent Pavan.

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Stochastic completeness of graphs: curvature and volume growth

Radoslaw Wojciechowski, York College and the Graduate Center/City University of New York

Abstract

We will summarize some recent work concerning stochastic completeness of graphs. In particular, we will discuss curvature criteria, uniqueness class and volume growth results.

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On transmission conditions in modeling equations on graphs.

Prof. Adam Bobrowski, Lublin University of Technology

Abstract

I will discuss the role of transmission conditions in modeling equations on graph-like spaces, by considering two recent examples. The first of these is a theorem on thin layer approximation, the other is a simple kinetic model with interface.

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Flows in infinite networks

Prof. Marjeta Kramar Fijavž, University of Ljubljana

Abstract

We consider linear transport processes in infinite metric graphs in the $L^{\infty}$-setting. We apply the theory of bi-continuous operator semigroups to obtain well-posedness of the problem under different assumptions on the velocities and for general stochastic matrices appearing in the boundary conditions.

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Trace formulas for general Hermitian matrices: a scattering approach on their associated graphs

Dr. Sven Gnutzmann, University of Nottingham

Abstract

Two trace formulas for the spectra of arbitrary Hermitian matrices are presented. In either case the one associates a unitary scattering matrix to the given Hermitian matrix H such that the unitary matrix depends on the spectral parameter. In the first type the unitary matrix is obtained by exponentiation. The new feature in this case is that the spectral parameter appears in the final form as an argument of Eulerian polynomials—thus connecting the periodic orbits to combinatorial objects in a novel way. To obtain the second type, one expresses the input in terms of a unitary scattering matrix in a larger Hilbert space. One of the surprising features here is that the locations and radii of the spectral discs of Gershgorin's theorem appear naturally as the pole parameters of the scattering matrix. Both formulas are discussed and possible applications are outlined.

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Semilinear evolution problems in fractal domains

Prof. Maria Rosaria Lancia, Sapienza

Abstract

We present some results on semilinear evolutions equations, possibly non autonomous , in fractal domains. Local existence, uniqueness and regularity results for the mild solution are proved.

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Semigroups for flows on limits of graphs

Dr. Christian Budde, North-West University

Abstract

Transport of goods is nowadays of extreme importance and indispensable considering what mankind needs for daily life. Now imagine a start-up company shipping special goods all over the world. Of course, the company starts with a small network of customers. However, assuming the company grows and retains the already existing routes and customers, the ship network grows and grows. It might come to the point in the development of the company, that one actually lost the view on all specific routes but only knows how the network works since it becomes too big. However, one still wants to know how the transport is going on the whole network. That a network is growing, through adding vertices and edges, means that one has a sequence of graphs and each graph is a subgraph of the subsequent graph in the sequence, describing the above mentioned situation of growing networks. We will describe the transition from finite to infinite graphs by means of direct limits in a certain category. The approximation of the transport process on the direct limit graph, is done by a version of the (first) Trotter--Kato approximation theorem, which is originally due to T. Kato \cite[Chapter IX, Thnm.~3.6]{K1976} and H.F. Trotter \cite[Thm. 5.2 \& 5.3]{T1958} and modified by a version by K. Ito and F. Kappel \cite[Thm. 2.1]{TK1998}. We will also present an extension the work of Ito and Kappel which is related to the well-known second Trotter–Kato theorem. \begin{thebibliography}{10} \bibitem{TK1998} K.~Ito and F.~Kappel. \newblock The {T}rotter-{K}ato theorem and approximation of {PDE}s. \newblock {\em Math. Comp.}, 67(221):21--44, 1998. \bibitem{K1976} T.~Kato. \newblock {\em Perturbation theory for linear operators}. \newblock Springer-Verlag, Berlin-New York, second edition, 1976. \newblock Grundlehren der Mathematischen Wissenschaften, Band 132. \bibitem{T1958} H.~F. Trotter. \newblock Approximation of semi-groups of operators. \newblock {\em Pacific J. Math.}, 8:887--919, 1958. \end{thebibliography}

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Mathematical modeling of traffic flow

Prof. Helge Holden, Norwegian University of Science and Technology

Abstract

Traditionally, there are two types of mathematical models for vehicular traffic, namely the Follow-the-Leader (FtL) models and the continuum models, using variants of the classical Lighthill--Whitham--Richards (LWR) models. In the FtL models individual vehicles are tracked, and this leads to a system of ordinary differential equations. On the other hands, in LWR models, traffic is represented by the density of vehicles, and the resulting equation is a first order hyperbolic conservation law. We will study the continuum limit of the FtL model when traffic becomes dense. We will also mention the problem of modeling traffic flow on networks. In the second part of the talk, we will discuss a novel model for multi-lane traffic within the LWR framework. The talk is based on joint work with Nils Henrik Risebro (University of Oslo).

Combinatorial Designs (MS - ID 16)

Combinatorics and Discrete Mathematics

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Testing Arrays for Fault Localization

Prof. Charles Colbourn, Arizona State University

Abstract

A {\sl separating hash family} ${\sf SHF}_\lambda(N; k,v,\{w_1,\dots,w_s\})$ is an $N \times k$ array on $v$ symbols, with the property that no matter how we choose disjoint sets $C_1, \dots, C_s$ of columns with $|C_i| = w_i$, there are at least $\lambda$ rows in which, for every $1 \leq i < j \leq s$, no entry in a column of $C_i$ equals that in a column of $C_j$. (That is, there are $\lambda$ rows in which sets $\{C_1,\dots,C_s\}$ are {\sl separated}.) Separating hash families have numerous applications in combinatorial cryptography and in the construction of various combinatorial arrays; typically, one only considers whether two symbols are the same or different. We instead employ symbols that have algebraic significance. We consider an ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ whose symbols are column vectors from ${\mathbb F}_q^s$. The entry in row $r$ and column $c$ of the ${\sf SHF}$ is denoted by ${\bf v}_{r,c}$. Suppose that $C_1, \dots, C_s$ is a set of disjoint sets of columns. Row $r$ is {\sl covering} for $\{C_1, \dots, C_s\}$ if, whenever we choose $s$ columns $\{ \gamma_i \in C_i : 1 \leq i \leq s\}$, the $s \times s$ matrix $[ {\bf v}_{r,\gamma_1} \cdots {\bf v}_{r,\gamma_s} ]$ is nonsingular over ${\mathbb F}_q$. Then the ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ is {\sl covering} if, for every way to choose $\{C_1, \dots, C_s\}$, there are at least $\lambda$ covering rows. We establish that covering separating hash families of type $1^t d^1$ give an effective construction for detecting arrays, which are useful in screening complex systems to find interactions among $t$ or fewer factors without being masked by $d$ or fewer other interactions. This connection easily accommodates outlier and missing responses in the screening. We explore asymptotic existence results and explicit constructions using finite geometries for covering separating hash families. We develop randomized and derandomized construction algorithms and discuss consequences for detecting arrays. This is joint work with Violet R. Syrotiuk (ASU).

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A reduction of the spectrum problem for sun systems

Prof. Anita Pasotti, University of Brescia

Abstract

A $k$-cycle with a pendant edge attached to each vertex is called a $k$-sun. When we approached the existence problem for $k$-sun systems of order $v$, complete solutions were known only for $k=3,4,5,6,8,10,14$ and for $k=2^t$. Here, we reduce this problem to the orders $v$ in the range $2k< v < 6k$ satisfying the obvious necessary conditions. Thanks to this result, we provide a complete solution whenever $k$ is an odd prime, and some partial results whenever $k$ is twice a prime. \begin{itemize} \item[1)] M. Buratti, A. Pasotti, T. Traetta, \emph{A reduction of the spectrum problem for odd sun systems and the prime case}, J. Combin. Des. \textbf{29} (2021), 5--37. \item[2)] A. Pasotti, T. Traetta, \emph{Even sun systems of the complete graph}, in preparation. \end{itemize}

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Heffter Arrays and Biembbedings of Cycle Systems on Orientable Surfaces

Prof. E. Şule Yazıcı , Koç University

Abstract

In this talk we will review the recent developments on square Heffter arrays, $H(n;k)$, and their applications on face $2$-colourable embeddings of the complete graph $K_{2nk+1}$ on an orientable surfaces. Square Heffter arrays, $H(n;k)$, are $n\times n$ arrays such that each row and each column contains $k$ filled cells, each row and column sum is divisible by $2nk+1$ and either $x$ or $-x$ appears in the array for each integer $1\leq x\leq nk$. Archdeacon noted that a Heffter array, satisfying two additional conditions, yields a face $2$-colourable embedding of the complete graph $K_{2nk+1}$ on an orientable surface, where for each colour, the faces give a $k$-cycle system. These necessary conditions pertain to cyclic orderings of the entries in each row and each column of the Heffter array and are: (1) for each row and each column the sequential partial sums determined by the cyclic ordering must be distinct modulo $2nk+1$; (2) the composition of the cyclic orderings of the rows and columns is equivalent to a single cycle permutation on the entries in the array. We construct Heffter arrays that satisfy condition (1) whenever (a) $k\equiv 0\ mod\ 4$; or (b) $n\equiv 1\ mod\ 4$ and $k\equiv 3\ mod\ 4$; or (c) $n\equiv 0\ mod\ 4$, $k\equiv 3\ mod\ 4$ and $n\gg k$. As a corollary to the above we obtain pairs of orthogonal $k$-cycle decompositions of $K_{2nk+1}$. Furthermore we study when these arrays satisfy condition (2). We show the existence of face $2$-colourable embeddings of cycle decompositions of the complete graph when $n\equiv 1\ mod\ 4$ and $k\equiv 3\ mod\ 4$, $n\gg k\geq 7$ (provided that when $n\equiv 0\ mod\ 3$ then $k\equiv 7\ mod\ 12$).

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Bipartite 2-factorizations of complete multigraphs via layering

Dr. Mateja Sajna, University of Ottawa

Abstract

Layering is in principle a simple method that allows us to obtain a type-specific 2-factorization of a complete multigraph (or complete multigraph minus a 1-factor) from existing 2-factorizations of complete multigraphs and complete multigraphs minus a 1-factor. This technique is particularly effective when constructing bipartite 2-factorizations; that is, 2-factorizations with all cycles of even length. In this talk, we shall give a thorough introduction to layering, and then describe new bipartite 2-factorizations of complete multigraphs obtained by layering. In particular, for complete multigraphs and bipartite 2-factors with no 2-cycles, we obtain a complete solution to the Oberwolfach Problem and an almost complete solution to the Hamilton-Waterloo Problem. This is joint work with Amin Bahmanian.

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Strictly additive 2-designs

Dr. Anamari Nakic, Faculty of Electrical Engineering and Computing, University of Zagreb

Abstract

This work draws inspiration from an interesting theory developed in [1]. A design $(V,{\cal B})$ is said to be {\it additive} if $V$ is a subset of an abelian group $G$ and the elements of any block $B\in{\cal B}$ sum up to zero. We propose to speak of a {\it strictly additive} design when $V$ coincides with $G$. Up to last year, apart from the obvious examples of the $2-(q^n,q,1)$ designs associated with the affine geometry AG$(n,q)$, all known strictly additive 2-designs had a quite ``big" $\lambda$. Very recently, a strictly additive $2-(81,6,2)$ design has been found in [3]. This design, besides being {\it simple} (the only design with these parameters previously known [2] has sixteen pairs of repeated blocks), has the property that every block is union of two parallel lines of AG$(4,3)$. In the attempt of getting other strictly additive designs with this property we found some infinite series of 2-designs whose parameter-sets are probably new. In this talk, besides presenting the above series, I will try to outline a proof that for every odd $k$, there are infinitely many values of $v$ for which a strictly additive $2-(v,k,1)$ design exists. \bigskip \noindent [1] A. Caggegi, G. Falcone, M. Pavone,\textit{ On the additivity of block designs}, J. Algebr. Comb. 45, 271--294 (2017). \noindent [2] H. Hanani, {\it Balanced incomplete block designs and related designs}, Discrete Math. 11 (1975), 255--369. \noindent [3] A. Nakic, \textit{The first example of a simple $2-(81,6,2)$ design}, Examples and Counterexamples 1 (2021).

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Mutually orthogonal cycle systems

Dr. Andrea Burgess, University of New Brunswick

Abstract

An $\ell$-cycle system of order $n$ is a set of $\ell$-cycles whose edges partition the edge set of $K_n$. We say that two cycle systems, say $\mathcal{C}$ and $\mathcal{C}'$, are {\em orthogonal} if any cycle of $\mathcal{C}$ and any cycle of $\mathcal{C}'$ share at most one edge. Orthogonal cycle systems arise naturally from simple Heffter arrays and biembeddings of cycle decompositions. A collection of cycle systems is {\em mutually orthogonal} if any two of the systems are orthogonal. In this talk, we give bounds on the number of mutually orthogonal $\ell$-cycle systems of order $n$, and provide constructions for sets of mutually orthogonal cyclic cycle systems.

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Testing isomorphism of circulant objects in polynomial time

Prof. Mikhael Muzychuk, Ben-Gurion University of the Negev

Abstract

We show that isomorphism testing of two cyclic combinatorial objects may be done in a polynomial time of their sizes provided that both objects share the same regular cyclic group of automorphisms given in advance.

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Sequences in $\mathbb{Z}_n$ with Distinct Partial Sums

Dr. Matt Ollis, Marlboro College

Abstract

Let~$\mathbb{Z}_n = \{0, 1, \ldots, n-1 \}$ be the integers modulo~$n$. For a sequence of elements $a_1, \ldots, a_k$ of~$\mathbb{Z}_n$, define its {\em partial sums} $b_0, \ldots, b_k$ by $b_0 = 0$ and $b_i = a_1 + \cdots + a_i$ for $1 \leq i \leq k$. For which subsets~$S \subseteq \mathbb{Z}_n \setminus \{ 0 \}$ is it possible to order the elements of~$S$ so that the partial sums are distinct? When the sum of the elements of~$S$ is~0, there can be no such ordering. Alspach conjectures that this is the only obstacle; that is, every subset~$S$ whose sum is nonzero has an ordering with distinct partial sums. We show how to translate the problem into one of finding monomials with non-zero coefficients in particular polynomials over $\mathbb{Z}_p$, where~$p$ is a prime divisor of~$n$, using Alon's Combinatorial Nullstellensatz. The approach offers hope for a full proof of the conjecture, and can be used in conjunction with a computational approach in cases where $n = pt$ with $p$ prime and $t$ and $|S|$ small.

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Merging Combinatorial Design and Optimization: the Oberwolfach Problem

Fabio Salassa, Politecnico di Torino

Abstract

Combinatorial optimization is a subset of mathematical optimization that is related to operations research, algorithm theory, and computational complexity theory. It consists of finding an (optimal) object from a finite set of objects and in many such problems, exhaustive search is not tractable. Combinatorial design theory is the part of combinatorial mathematics that deals with the existence, construction and properties of systems of finite sets whose arrangements satisfy generalized concepts of balance and/or symmetry. Combinatorial Design and Combinatorial Optimization, though apparently different research fields, share common problems, such as for example sudokus, covering arrays, tournament design and more in general problems that can be represented on graphs. Aim of the talk is to present intersections and possible contributions to Combinatorial Design given by the application of Combinatorial Optimization techniques and solution methods. This is accomplished by the presentation of results on the Oberwolfach Problem (OP) where interaction of methods from both domains enabled us to solve large OP instances in limited computational time and at the same time to derive a theoretical result for general classes of instances.

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A few new triplanes

Prof. Sanja Rukavina, University of Rijeka

Abstract

An incidence structure ${\cal D}=({\cal P},{\cal B},I)$, with point set ${\cal P}$, block set ${\cal B}$ and incidence $I$ is a $t$-$(v,k,\lambda)$ design, if $|{\cal P}|=v$, every block $B \in {\cal B}$ is incident with precisely $k$ points, and every $t$ distinct points are together incident with precisely $\lambda$ blocks. We consider triplanes, i.e. symmetric block designs with $\lambda = 3$. Triplanes of order $12$, i.e. symmetric (71,15,3) designs, have the greatest number of points among all known triplanes and it is not known if a triplane $(v,k,3)$ exists for $v>71$. \\ In this talk, in addition to reviewing previously known results, we give the first example of a triplane of order 12 that doesn't admit an automorphism of order 3. \begin{thebibliography}{12} \bibitem{CDDDR} D. Crnkovi\'{c}, D. Dumi\v{c}i\'{c} Danilovi\'{c}, S. Rukavina, Enumeration of symmetric (45,12,3) designs with nontrivial automorphisms, J. Algebra Comb. Discrete Struct. Appl. 3 (2016), 145--154. \bibitem{geocr} D. Crnkovi\'{c}, S. Rukavina, L. Sim\v{c}i\'{c}, On triplanes of order twelve admitting an automorphism of order six and their binary and ternary codes, Util. Math. 103 (2017), 23--40. \end{thebibliography}

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Universal sequences and Euler tours in hypergraphs

Prof. Deryk Osthus, University of Birmingham

Abstract

We show that a quasirandom $k$-uniform hypergraph $G$ has a tight Euler tour subject to the necessary condition that $k$ divides all vertex degrees. The case when $G$ is complete confirms a \$100 conjecture of Chung, Diaconis and Graham from 1989 on the existence of universal cycles for the $k$-subsets of an $n$-set. Our proof is based on random walks and the proof of the existence of $H$-designs (by Glock, K\"uhn, Lo and Osthus), i.e. decompositions of a complete hypergraph into copies of an arbitrary hypergraph $H$ (subject to divisibility conditions).

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Packings of Partial Difference Sets

Dr. Shuxing Li, Simon Fraser University

Abstract

As the underlying configuration behind many elegant finite structures, partial difference sets have been intensively studied in design theory, finite geometry, coding theory, and graph theory. Over the past three decades, there have been numerous constructions of partial difference sets in abelian groups with high exponent, accompanied by numerous very different and delicate techniques. Surprisingly, we manage to unify and extend a great many previous constructions in a common framework, using only elementary methods. The key insight is that, instead of focusing on one single partial difference set, we consider a packing of partial difference sets, namely, a collection of disjoint partial difference sets in a finite abelian group. Although the packing of partial difference sets has been implicitly studied in various contexts, we recognize that a particular subgroup reveals crucial structural information about the packing. Identifying this subgroup allows us to formulate a recursive lifting construction of packings in abelian groups of increasing exponent. This is joint work with Jonathan Jedwab.

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A lower bound on permutation codes of distance $n-1$

Dr. Peter Dukes, University of Victoria

Abstract

A construction for mutually orthogonal latin squares (MOLS) inspired by pairwise balanced designs is shown to hold more generally for a class of permutation codes of length $n$ and minimum distance $n-1$. Using ingredients when $n$ equals a prime or a prime plus one, and applying a number sieve, we obtain a general lower bound $M(n,n-1) \ge n^{1.0797}$ on the size of such codes for large $n$. This represents a small improvement on the guarantee given from MOLS.

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Regular $1$-factorizations of complete graphs with orthogonal spanning trees

Prof. Gloria Rinaldi, University of Modena and Reggio Emilia

Abstract

A $1$-factorization ${\cal F}$ of a complete graph $K_{2n}$ is said to be $G$-regular if $G$ is an automorphism group of ${\cal F}$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (1985) which solved the problem when $G$ is a cyclic group. It is also well known that this problem simplifies somewhat when $n$ is odd: $G$ must be the semi-direct product of $Z_2$ with its normal complement and $G$ always realizes a $1-$factorization of $K_{2n}$ upon which it acts sharply transitively on vertices. When $n$ is even the problem is still open, even though several classes of groups were tested in the recent past. An attempt to obtain a fairly precise description of groups and $1$-factorizations satisfying this symmetry constraint could be done by imposing further conditions. For example some non existence results were achieved by assuming the existence of a $1$-factor fixed by the action of the group, further results were obtained when the number of fixed $1-$factors is as large as possible. In this talk we focus our attention on the possibility of constructing $G$-regular $1$-factorizations of $K_{2n}$ together with a complete set of isomorphic spanning trees orthogonal to the $1-$factorization. Here orthogonal tree means that the tree shares exactly one edge with each $1-$factor. We see how to realize such a situation when $n$ is odd and examine some classes of groups in the case $n$ even.

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Balancedly splittable orthogonal designs

Prof. Hadi Kharaghani, University of Lethbridge

Abstract

As an extension of balancedly splittable Hadamard matrices, the concept of balancedly splittable orthogonal designs is introduced along with a recursive construction. Among the other results, a class of equiangular tight frames over the real, complex, and quaternions meeting the Delsarte-Goethals-Seidel upper bound is obtained.

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Factorizations of infinite graphs

Dr. Simone Costa, University of Brescia

Abstract

Let $\mathcal{F}=\{F_{\alpha}: \alpha\in \mathcal{A}\}$ be a family of infinite graphs. The Factorization Problem $FP(\mathcal{F}, \Lambda)$ asks whether $\mathcal{F}$ can be realized as a factorization of a given infinite graph $\Lambda$, namely, whether there is a factorization $\mathcal{G}=\{\Gamma_{\alpha}: \alpha\in \mathcal{A}\}$ of $\Lambda$ such that each $\Gamma_{\alpha}$ is a copy of $F_{\alpha}$. Inspired by the results on regular $1$-factorizations of infinite complete graphs $[1]$ and on the resolvability of infinite designs $[4]$, we study this problem when $\Lambda$ is either the Rado graph $R$ or the complete graph $K_\aleph$ of infinite order $\aleph$. When $\mathcal{F}$ is a countable family, we show that $FP(\mathcal{F}, R)$ is solvable if and only if each graph in $\mathcal{F}$ has no finite dominating set. Generalizing the existence result of $[2]$, we also prove that $FP(\mathcal{F}, K_\aleph)$ admits a solution whenever the cardinality $\mathcal{F}$ coincides with the order and the domination numbers of its graphs. Finally, in the case of countable complete graphs, we show some non-existence results when the domination numbers of the graphs in $\mathcal{F}$ are finite. \vskip0.4cm \begin{large} \textbf{References} \end{large} \begin{enumerate}[{[1]}] \item S. Bonvicini and G. Mazzuoccolo. Abelian $1$-factorizations in infinite graphs, \textit{European J. Combin.}, 31 (2010), 1847--1852. \item S. Costa. A complete solution to the infinite Oberwolfach problem, \textit{J. Combin. Des.}, 28 (2020), 366--383. \item S. Costa and T. Traetta. Factorizing the Rado graph and infinite complete graphs, \textit{Submitted (arXiv:2103.11992)}. \item P.~Danziger, D.~Horsley and B.~S.~Webb. Resolvability of infinite designs, \textit{J. Combin. Theory Ser. A}, 123 (2014), 73--85. \end{enumerate}

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On the Oberwolfach Problem for single-flip 2-factors via graceful labelings

Dr. Tommaso Traetta, Università degli Studi di Brescia

Abstract

The Oberwolfach Problem, posed by Ringel in 1967 and still open, asks for each odd integer $v>1$ and each $2$-regular graph $F$ of order $v$ to determine whether there is a decomposition of the complete graph $K_v$ into copies of $F$. We construct solutions whenever $F$ has a sufficiently large odd cycle meeting a specified lower bound and, in addition, $F$ has a single-flip automorphism (i.e. an involutory automorphism acting as a reflection on exactly one cycle). For even orders $v$, we give analogous results for the maximum packing and minimum covering variants of the problem. % for single-flip $2$-factors with a sufficiently large even cycle. We also show a similar result when the edges of $K_v$ have multiplicity 2, but in this case %we do not require that $F$ be single-flip. we only require that $F$ has a sufficiently large cycle. Our methods build on the techniques used in [2] and involve a doubling construction defined in [1] which we apply to graceful labelings of 2-regular graphs with a vertex removed, allowing us to explicitly construct solutions to the Oberwolfach Problem with well-behaved automorphisms. This is joint work with Andrea Burgess and Peter Danziger. %\\ [.5cm] \begin{enumerate} \item[{[1]}] M.\ Buratti, T.\ Traetta. 2-starters, graceful labelings, and a doubling construction for the Oberwolfach Problem. {J.\ Combin.\ Des.} 20 (2012), 483--503. \item[{[2]}] T.\ Traetta. A complete solution to the two-table Oberwolfach Problems. J.\ Combin.\ Theory Ser.\ A {120} (2013), 984--997. \end{enumerate}

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On the mini-symposium problem

Prof. Peter Danziger, Ryerson University

Abstract

\begin{center} Joint work with E.\ Mendelsohn, B.\ Stevens, T.\ Traetta. \end{center} The Oberwolfach problem was originally stated as a seating problem: \begin{quote} Given $v$ attendees at a conference with $t$ circular tables each of which {seats $a_i$} people $\left({\sum_{i=1}^t a_i = v}\right)$. Find a seating arrangement so that every person sits next to each other person around a table exactly once over the $r$ days of the conference. \end{quote} The Oberwolfach problem thus asks for a decomposition of $K_n$ ($K_n-I$ when $n$ is even) into 2-factors consisting of cycles with lengths $a_1,\ldots, a_t$. In this talk we introduce the related {\em mini-symposium problem,} which asks for solutions to the Oberwolfach problem on $v$ points which contains a subsystem on $m$ points. In the seating context above, the larger conference contains a mini-symposium of $m$ participants, and we also require these $m$ participants to be seated together for $\left\lfloor\frac{m-1}{2}\right\rfloor$ of the days. We obtain a complete solution when the cycle sizes are as large as possible, i.e.\ $m$ and $v-m$. In addition, we provide extensive results in the case where all cycle lengths are equal, of size $k$ say, completely solving all cases when $m\mid v$, except possibly when $k$ is odd and $v$ is even. In particular, we completely solve the case when all cycles are of length $m$ ($k=m$).

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An Evans-style result for block designs

Dr. Daniel Horsley, Monash University

Abstract

A now-proven conjecture of Evans states that any partial latin square with at most $n-1$ filled cells can be completed to a latin square. This is sharp: there are uncompletable partial latin squares with $n$ filled cells. This talk will discuss the analogous problem for block designs. An \textit{$(n,k,1)$-design} is a collection of $k$-subsets (\textit{blocks}) of a set of $n$ \textit{points} such that each pair of points occur together in exactly one block. If this restriction is relaxed to require only that each pair of points occur together in at most one block we instead have a \textit{partial $(n,k,1)$-design}. I will outline a proof that any partial $(n,k,1)$-design with at most $\frac{n-1}{k-1}-k+1$ blocks is completable to a $(n,k,1)$-design provided that $n$ is sufficiently large and obeys the obvious necessary conditions for an $(n,k,1)$-design to exist. This result is sharp for all $k$. I will also mention some related results concerning edge decompositions of almost complete graphs into copies of $K_k$.

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The classification of $2$-$(v,k,\lambda )$ designs, with $\lambda >1$ and $(r,\lambda )=1$, admitting a flag-transitive automorphism group.

Prof. Alessandro Montinaro, University of Salento

Abstract

A classical subject in Theory of Designs is the study of $2$-designs admitting an automorphism group fulfilling prescribed properties. Within this research area, it is of great interest the study of $2$-$(v,k,\lambda )$ designs $\mathcal{D}$ admitting a flag-transitive automorphism group $G$. Since they have been classified for $\lambda =1$ and $G\nleq A\Gamma L_{1}(q)$ by Buekenhout et al. (1990), a special attention is devoted to the general case $\lambda >1$. In this setting, a first natural generalization of the case $\lambda =1$ is represented by $\lambda >1$ and $\gcd (r,\lambda )=1$, where $r$ is the replication number of $\mathcal{D}$. Then $G$ acts point-primitively on $\mathcal{D}$ by a result of Dembowski (1968), and $Soc(G)$, the socle of $G$, is either an elementary abelian $p$-group for some prime $p$, or a non abelian simple group by a result of Zeischang (1988). Starting from these two results, such $2$-designs have been recently classified for $G\nleq A\Gamma L_{1}(q)$ by Biliotti et al. and by Alavi, Zhou et al. according to whether $Soc(G)$ is an elementary abelian $p$-group or a non abelian simple group, respectively. The aim of the talk is to survey the classification\ of $2$-$(v,k,\lambda )$ designs $\mathcal{D}$, with $\lambda >1$ and $(r,\lambda )=1$, admitting a flag-transitive automorphism group $G$, mostly focusing on the constructions of the various examples contained in it.

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Characterizing isomorphism classes of Latin squares by fractal dimensions of image patterns

Dr. Raul M. Falcon, Universidad de Sevilla

Abstract

Based on the construction of pseudo-random sequences arisen from a given Latin square, Dimitrova and Markovski [1] described in 2007 a graphical representation of quasigroups by means of fractal image patterns. The recognition and analysis of such patterns have recently arisen [2,3] as an efficient new approach for classifying Latin squares into isomorphism classes. This talk delves into this topic by focusing on the use of the differential box-counting method for determining the mean fractal dimension of the homogenized standard sets associated to these fractal image patterns. It constitutes a new Latin square isomorphism invariant which is analyzed in this talk for characterizing isomorphism classes of non-idempotent Latin squares in an efficient computational way. \medskip {\bf References}: \begin{enumerate} \item[1)] V. Dimitrova, S. Markovski, {\em Classification of quasigroups by image patterns}. In: Proceedings of the Fifth International Conference for Informatics and Information Technology, Bitola, Macedonia, 2007; 152--160. \item[2)] R. M. Falc\'on, {\em Recognition and analysis of image patterns based on Latin squares by means of Computational Algebraic Geometry}, Mathematics {\bf 9} (2021), paper 666, 26 pp. \item[3)] R. M. Falc\'on, V. \'Alvarez, F. Gudiel, {\em A Computational Algebraic Geometry approach to analyze pseudo-random sequences based on Latin squares}, Adv. Comput. Math. {\bf 45} (2019), 1769--1792. \end{enumerate}

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On the existence of large set of partitioned incomplete Latin squares

Ms. Cong Shen, Nanjing Normal University

Abstract

In this talk, we survey the existence of large sets of partitioned incomplete Latin squares (LSPILS). Algebraic and combinatorial methods are employed to construct the large sets of partitioned incomplete Latin squares of type $g^nu^1$. Furthermore, we prove that there exists a pair of orthogonal LSPILS$(1^pu^1)$s for any odd $u$ and some even values of $u$, where $p$ is a prime. Lastly, we propose some problems for further research.

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Equitably $2$-colourable even cycle systems

Dr. Francesca Merola, Università Roma Tre

Abstract

An $\ell$-cycle decomposition of a graph $G$ is said to be \emph{equitably $c$-colourable} if there is a $c$-vertex-colouring of $G$ such that each colour is represented (approximately) an equal number of times on each cycle: more precisely, we ask that in each cycle $C$ of the decomposition, each colour appears on $\lfloor \ell/c \rfloor$ or $\lceil \ell/c \rceil$ of the vertices of $C$. In this talk, we consider the case $c=2$ and present some new results on the existence of $2$-colourable even $\ell$-cycle systems of the cocktail party graph $K_v-I$. In particular, we determine a complete existence result for equitably $2$-colourable $\ell$-cycle decompositions of $K_v-I$, $\ell$ even, in the cases that $v\equiv 0,2 \pmod{\ell},$ or $\ell$ is a power of 2, or $\ell\in \{2q, 4q\}$ for $q$ an odd prime power, or $\ell \leq 30$. We will also discuss some work in progress on analogous problems for cycles of odd length.

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Nov\'{a}k's conjecture on cyclic Steiner triple systems and its generalization

Prof. Tao Feng, Beijing Jiaotong University

Abstract

Nov\'{a}k conjectured in 1974 that for any cyclic Steiner triple systems of order $v$ with $v\equiv 1\pmod{6}$, it is always possible to choose one block from each block orbit so that the chosen blocks are pairwise disjoint. In this talk, we shall consider the generalization of this conjecture to cyclic $(v,k,\lambda)$-designs with $1 \leq \lambda \leq k-1$. Superimposing multiple copies of a cyclic symmetric design shows that the generalization cannot hold for all $v$, but we conjecture that it holds whenever $v$ is sufficiently large compared to $k$. We confirm that the generalization of the conjecture holds when $v$ is prime and $\lambda=1$ and also when $\lambda \leq (k-1)/2$ and $v$ is sufficiently large compared to $k$. As a corollary, we show that for any $k \geq 3$, with the possible exception of finitely many composite orders $v$, every cyclic $(v,k,1)$-design without short orbits is generated by a $(v,k,1)$-disjoint difference family.

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Kirkman triple systems with many symmetries

Prof. Marco Buratti, Università di Perugia

Abstract

A {\it Steiner triple system} of order $v$, STS$(v)$ for short, is a set $V$ with $v$ {\it points} together with a collection ${\cal B}$ of 3-subsets of $V$ ({\it blocks}) such that that any pair of distinct points is contained in exactly one block. A {\it Kirkman triple system} of order $v$, KTS$(v)$ for short, is a STS$(v)$ whose blocks are partitioned in {\it parallel classes} each of which is a partition of the point-set $V$. Steiner and Kirkman triple systems are among the most popular objects in combinatorics and their existence has been established a long time ago. Yet, very little is known about the automorphism groups of KTSs of an arbitrary order. In particular, from the very beginning of my research in design theory, I found surprising that there was no known pair $(r,n)$ for which whenever $v\equiv r$ (mod $n$) one may claim that there exists a KTS$(v)$ with a number of automorphisms at least close to $v$. After pursuing the target of getting such a pair $(r,n)$ for more than twenty years, we recently managed to find the following: $(39,72)$ and $(4^e48 + 3,4^e96)$ for any $e\geq0$. Indeed, for $v\equiv r$ (mod $n$) with $(r,n)$ as above, we are able to exhibit, concretely, a KTS$(v)$ with an automorphism group of order at least equal to $v-3$. The proof is very long and elaborated; so I will try to speak about the main ideas which led us to this result as for instance the invention of some variants of well known combinatorial objects which could be also used in the search of other combinatorial designs.

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On the classification of unitals on 28 points of low rank

Prof. Alfred Wassermann, University of Bayreuth

Abstract

Unitals are combinatorial $2$-$(q^3 + 1, q + 1, 1)$ designs. In 1981, Brouwer constructed $138$ nonisomorphic unitals for $q=3$, i.e.~$2$-$(28, 4, 1)$ designs. He observed that the $2$-rank of the constructed unitals is at least $19$, where the $p$-rank of a design is defined as the rank of the incidence matrix between points and blocks of the design over the finite field GF($p$). In 1998, McGuire, Tonchev and Ward proved that indeed the $2$-rank of a unital on $28$ points is between $19$ and $27$ and that there is a unique $2$-$(28, 4, 1)$ design, the Ree unital $R(3)$, with $2$-rank 19. In the same year, Jaffe and Tonchev showed that there is no unital on $28$ points of $2$-rank $20$ and there are exactly $4$ isomorphism classes of unitals of rank $21$. Here, we present the complete classification by computer of unitals of $2$-rank $22$, $23$ and $24$. There are $12$ isomorphism classes of unitals of $2$-rank $22$, $78$ isomorphism classes of unitals of $2$-rank $23$, and $298$ isomorphism classes of unitals of $2$-rank $24$.

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20 years of Legendre Pairs

Prof. ilias kotsireas, WLU

Abstract

Legendre pairs were introduced in 2001 by Seberry and her students, as a means to construct Hadamard matrices via a two-circulant core construction. A Legendre pair consists of two sequences of odd length $\ell$, with elements from $\{-1,+1\}$, such that their respective autocorrelation coefficients sum to $-2$, or (equivalently) their respective power spectral density coefficients sum to $2\ell +2$. Legendre pairs of every odd prime length $\ell$ exist, via a simple construction using the Legendre symbol. We will review known constructions for Legendre pairs. We will discuss various results on Legendre pairs during the past 20 years, including the concept of compression, introduced in a joint paper with Djokovic, as well as the computational state-of-the-art of the search for Legendre pairs. In particular, we recently contributed the only known Legendre pair of length $\ell = 77$ in a joint paper with Turner/Bulutoglu/Geyer. In addition, we recently contributed in a joint paper with Koutschan, several Legendre pairs of new lenghts $\ell \equiv 0 \, (\mod 3)$, as well as an algorithm that allows one to determine the full spectrum of values for the $\frac{\ell}{3}$-th power spectral density value. The importance of Legendre pairs lies in the fact that they constitute a promising avenue to the Hadamard conjectrure.

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Block designs constructed from orbit matrices using a modified genetic algorithm

Mr. Tin Zrinski, Department of Mathematics, University of Rijeka

Abstract

Genetic algorithms (GA) are search and optimization heuristic population-based methods which are inspired by the natural evolution process. In this talk, we will present a method of constructing incidence matrices of block designs combining the method of construction with orbit matrices and a modified genetic algorithm. With this method we managed to find some new non-isomorphic S(2,5,45) designs.

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On some periodic Golay pairs and pairwise balanced designs

Dr. Andrea Svob, University of Rijeka

Abstract

\renewcommand*\labelenumi{[\theenumi]} In this talk we will show how a relationship between certain pairwise balanced designs with $v$ points and periodic Golay pairs of length $v$ can be useful to construct periodic Golay pairs. The talk is based on recent work [1] where we construct pairwise balanced designs with $v$ points under specific block conditions having an assumed cyclic automorphism group, and using isomorph rejection which is compatible with equivalence of corresponding periodic Golay pairs, we complete a classification of periodic Golay pairs of length less than 40, up to equivalence. Further, we will show how we use similar tools to construct new periodic Golay pairs of lengths greater than 40 where classifications remain incomplete, and demonstrate that under some extra conditions on its automorphism group, a periodic Golay pair of length 90 do not exist. \bigskip \begin{enumerate} \item D. Crnkovi\' c, D. Dumi\v ci\' c Danilovi\' c, R. Egan, A. \v Svob, Periodic Golay pairs and pairwise balanced designs, J. Algebraic Combin., to appear. \end{enumerate}

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Dual incidences and $t$-designs in elementary abelian groups

Dr. Kristijan Tabak, Rochester Institute of Technology - Croatia

Abstract

Let $q$ be a prime and $E_{q^n}$ is an elementary abelian group of order $q^n.$ Let $\mathcal{H}$ be a collection of some subgroups of $E_{q^n}$ of order $q^k$. A pair $(E_{q^n}, \mathcal{H})$ is a $t-(n,k,\lambda)_q$ design if every subgroup of $E_{q^n}$ of order $q^t$ is contained in exactly $\lambda$ groups from $\mathcal{H}.$ This definition corresponds to the classical definition of a $q$-analog design. We introduce two incidence structures denoted by $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ with $\mathcal{H}$ as set of points. The blocks of $\mathcal{D}_{max}$ are labeled by maximal subgroups of $E_{q^n}$, while the blocks of $\mathcal{D}_{min}$ are labeled by groups of order $q.$ We fully describe a duality between $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ by proving some identities over group rings. The proven results are used to provide a full description of incidence matrices of $\mathcal{D}_{max}$ and $\mathcal{D}_{min}$ and their mutual dependance.

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Balanced Equi-$n$-squares

Dr. Trent Marbach, Ryerson University

Abstract

We define a $d$-balanced equi-$n$-square $L=(l_{ij})$, for some divisor $d$ of $n$, as an $n \times n$ matrix containing symbols from $\mathbb{Z}_n$ in which any symbol that occurs in a row or column, occurs exactly $d$ times in that row or column. We show how to construct a $d$-balanced equi-$n$-square from a partition of a Latin square of order $n$ into $d \times (n/d)$ subrectangles. In design theory, $L$ is equivalent to a decomposition of $K_{n,n}$ into $d$-regular spanning subgraphs of $K_{n/d,n/d}$. We also study when $L$ is diagonally cyclic, defined as when $l_{(i+1)(j+1)}=l_{ij}+1$ for all $i,j \in \mathbb{Z}_n$, which corresponds to cyclic such decompositions of $K_{n,n}$ (and thus $\alpha$-labellings). We identify necessary conditions for the existence of (a) $d$-balanced equi-$n$-squares, (b) diagonally cyclic $d$-balanced equi-$n$-squares, and (c) Latin squares of order $n$ which partition into $d \times (n/d)$ subrectangles. We prove the necessary conditions are sufficient for arbitrary fixed $d \geq 1$ when $n$ is sufficiently large, and we resolve the existence problem completely when $d \in \{1,2,3\}$. Along the way, we identify a bijection between $\alpha$-labellings of $d$-regular bipartite graphs and, what we call, $d$-starters: matrices with exactly one filled cell in each top-left-to-bottom-right unbroken diagonal, and either $d$ or $0$ filled cells in each row and column. We use $d$-starters to construct diagonally cyclic $d$-balanced equi-$n$-squares, but this also gives new constructions of $\alpha$-labellings.

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Resolving sets and identifying codes in finite geometries

Prof. György Kiss, Eötvös Loránd University, Budapest & University of Primorska, Koper

Abstract

Let $\Gamma=(V,E)$ be a finite, simple, undirected graph. A vertex $v \in V$ is resolved by $S=\{v_1,\ldots,v_n\}\subset V$ if the list of distances $(d(v,v_1),d(v,v_2),\ldots,$ $d(v,v_n))$ is unique. $S$ is a \emph{resolving set} for $\Gamma$ if it resolves all the elements of $V$. A subset $D\subset V$ is a \emph{dominating set} if each vertex is either in $D$ or adjacent to a vertex in $D.$ A vertex $s$ \emph{separates} $u$ and $v$ if exactly one of $u$ and $v$ is in $N[s].$ A subset $S\subset V$ is a \emph{separating set} if it separates every pair of vertices of $G.$ Finally, a subset $C\subset V$ is an \emph{identifying code} for $V$ if it is both a dominating and separating set. \smallskip In this talk resolving sets and identifying codes for graphs arising from finite geometries (e.g. Levi graphs of projective and affine planes and spaces, generalized quadrangles) are considered. We present several constructions and give estimates on the sizes of these objects.

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Weakly self-orthogonal designs and related linear codes

Prof. Vedrana Mikulić Crnković, University of Rijeka

Abstract

A 1-design is weakly self-orthogonal if all the block intersection numbers have the same parity. If both $k$ and the block intersection numbers are even then 1-design is called self-orthogonal and its incidence matrix generates a self-orthogonal code. We analyze extensions, of the incidence matrix and an orbit matrix of a weakly self-orthogonal 1-design, that generates a self-orthogonal code over the finite field. Additionally, we develop methods for constructing LCD codes by extending the incidence matrix and an orbit matrix of a weakly self-orthogonal 1-design.

Complex Analysis and Geometry (MS - ID 17)

Algebraic and Complex Geometry

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A criterion of Nakano positivity

Prof. Xiangyu Zhou, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences

Abstract

How to characterize Nakano positivity of holomorphic vector bundles is a difficult problem in complex geometry. In this talk, we'll give a criterion of Nakano positivity in terms of $L^2$ extensions. This is joint work with Fusheng Deng, Jiafu Ning, Zhiwei Wang.

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Metric geometry of domains in complex Euclidean spaces

Mr. Hervé Gaussier, Université Grenoble Alpes

Abstract

The boundary geometry of a domain strongly constrains the asymptotic behaviour of invariant metrics and of their curvatures, as shown in important contributions. I will present recent results that, conversely, give metric characterizations of the boundary geometry of some domains.

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Seiberg-Witten equations and pseudoholomorphic curves

Prof. Armen Sergeev, Steklov Mathematical Institute

Abstract

%\documentclass{article} %\begin{document} \begin{center}SEIBERG--WITTEN EQUATIONS AND PSEUDOHOLOMORPHIC CURVES\\ Armen SERGEEV\end{center} %\address{Steklov Mathematical Institute, Moscow} \bigskip % \maketitle Seiberg--Witten equations (SW-equations for short) were proposed in order to produce a new kind of invariant for smooth 4-dimensional manifolds. These equations, opposite to the conformally invariant Yang--Mills equations, are not invariant under scale transformations. So to draw a useful information from these equations one should plug the scale parameter $\lambda$ into them and take the limit $\lambda\to\infty$. If we consider such limit in the case of 4-dimensional symplectic manifolds solutions of SW-equations will concentrate in a neighborhood of some pseudoholomorphic curve (more precisely, pseudoholomorphic divisor) while SW-equations reduce to some vortex equations in normal planes of the curve. The vortex equations are in fact static Ginzburg--Landau equations known in the superconductivity theory. So solutions of the limiting adiabatic SW-equations are given by families of vortices in the complex plane parameterized by the point z running along the limiting pseudoholomorphic curve. This parameter plays the role of complex time while the adiabatic SW-equations coincide with a nonlinear $\bar\partial$-equation with respect to this parameter.

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Angular Derivatives and Boundary Values of H(\MakeLowercase{b}) Spaces of Unit Ball of $\mathbb{C}^n$

Dr. Sibel Sahin, Mimar Sinan Fine Arts University

Abstract

In this talk we will consider a special subclass of the Hardy-Hilbert space $H^2(\mathbb{B}^n)$, namely deBranges-Rovnyak spaces $H(b)$, in the setting of the unit ball of $\mathbb{C}^n$. One of the main problems in the study of $H(b)$ functions is their representation and in the first part of this talk we will see how we can represent these classes through the Clark measure on $S^n$ associated with $b$. In the second part we will give a characterization of admissible boundary limits in relation with finite angular derivatives and we will see the interplay between Clark measures and angular derivatives showing that Clark measure associated with $b$ has an atom at a boundary point if and only if $b$ has finite angular derivative at the same point. More detailed analysis of the concepts mentioned in this talk can be found in the following study: \\ \d{S}ahin,S. Angular Derivatives and Boundary Values of H(\MakeLowercase{b}) Spaces of Unit Ball of $\mathbb{C}^n$, Complex Varaibles and Elliptic Equations, \\ DOI: 10.1080/17476933.2020.1715373, (2020).

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The Density Property for Calogero--Moser Spaces

Prof. Rafael Andrist, American University of Beirut

Abstract

\noindent A Calogero--Moser space describes the (completed) phase space of a system of finitely many particles in classical physics. Since the past two decades, these spaces are also an object of ongoing study in pure mathematics. In particular, a Calogero--Moser space of $n$ particles is known to be a smooth complex-affine variety, and to be diffeomorphic to the Hilbert scheme of $n$ points in the affine plane. We establish the density property for the Calogero--Moser spaces and describe their group of holomorphic automorphisms.

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The CR Ahlfors derivative and a new invariant for spherically equivalent CR maps

Dr. Ngoc Son Duong, Universität Wien

Abstract

In this talk, I will discuss a notion of the Ahlfors derivative for CR maps. This notion possesses several important properties similar to those of the conformal counterpart and provides a new invariant for spherical equivalent CR maps from strictly pseudoconvex CR manifolds into a sphere. The invariant can be computed easily and distinguishes many well-known equivalent classes of CR maps between spheres. In particular, it vanishes precisely when the map is spherical equivalent to the linear embedding. This is a joint work with Bernhard Lamel.

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Weighted-$L^2$ polynomial approximation in $\mathbb{C}$

Dr. Jujie Wu, Sun Yat-Sen University

Abstract

We study the density of polynomials in $H^2(\Omega,e^{-\varphi})$, the space of square integrable holomorphic functions in a bounded domain $\Omega$ in $\mathbb{C}$, where $\varphi$ is a subharmonic function. In particular, we prove that the density holds in Carath\'{e}odory domains for any subharmonic function $\varphi$ in a neighborhood of $\overline{\Omega}$. In non-Carath\'{e}odory domains, we prove that the density depends on the weight function, giving examples. This is joint with S\'everine Biard and John Erik Forn\ae ss.

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Images of CR manifolds

Prof. Jiri Lebl, Oklahoma State University

Abstract

We study CR singular submanifolds that have a removable CR singularity, that is, manifolds that are singular, but whose CR structure extends through the singularity. Bishop surfaces are trivial examples, but in higher CR dimension, generically the CR singularity is not removable. In particular, we study real codimension 2 submanifolds in ${\mathbb{C}}^3$. This is joint work with Alan Noell and Sivaguru Ravisankar.

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Equivalence of neighborhoods of embedded compact complex manifolds and higher codimension foliations

Mr. Laurent Stolovitch, CNRS-Université Côte d'Azur

Abstract

We consider an embedded $n$-dimensional compact complex manifold in $n+d$ dimensional complex manifolds. We are interested in the holomorphic classification of neighborhoods as part of Grauert's formal principle program. We will give conditions ensuring that a neighborhood of $C_n$ in $M_{n+d}$ is biholomorphic to a neighborhood of the zero section of its normal bundle. This extends Arnold's result about neighborhoods of a complex torus in a surface. We also prove the existence of a holomorphic foliation in $M_{n+d}$ having $C_n$ as a compact leaf, extending Ueda's theory to the high codimension case. Both problems appear as a kind linearization problem involving {\it small divisors condition} arising from solutions to their cohomological equations.

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Parabolic Fatou components with holes

Mr. Josias Reppekus, Università di Roma "Tor Vergata"

Abstract

Many examples of invariant non-recurrent Fatou components of automorphisms $F$ of $\mathbb{C}^{2}$ arise from local projections to one variable in which the dynamics are approximately parabolic. After constructing a local basin $B$, a critical step is to show that the associated completely invariant global basin $\Omega$ of all orbits ending up in $B$ is not just a proper subset of a Fatou component. I will present two types of examples in which the initial projection is non-linear and the resulting Fatou component $\Omega$ is biholomorphic to $\mathbb{C}\times\mathbb{C}^{*}$, i.e.~``has a hole''. The constructions are based on the dynamics near a fixed point $p$ on the boundary of $U$ such that the eigenvalues of $F$ at $p$ are complex conjugate irrational rotations. For the first type, all orbits in $\Omega$ converge to $p$, whereas for the second type the orbits in $\Omega$ converge precisely to the points of an entire curve minus $p$.

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Fekete Configurations, Products of Vandermonde Determinants and Canonical Point Processes

Dr. Jakob Hultgren, University of Maryland

Abstract

The asymptotic behavior of Fekete configurations is a classical topic in complex analysis. By definition, Fekete configurations are arrays of points that maximize Vandermonde determinants. From a complex geometric perspective, any Hermitian ample line bundle over a compact Kähler manifold defines a sequence of Vandermonde determinants of increasing dimension and thus a notion of Fekete configurations. In this talk we will consider a collection of Hermitian ample line bundles over a fixed compact Kähler manifold. I will present two results. One regarding the product of the associated Vandermonde determinants and the asymptotic behavior of its maximizers and one regarding existence of sequences of point configurations which are asymptotically Fekete (a slightly weaker concept than Fekete) with respect to all Hermitian ample line bundles simultaneously. If time permits I will also outline a conjectural picture involving a related class of point processes and its connection to canonical metrics in complex geometry.

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``Levi core", Diederich-Fornaess index, and D'Angelo forms

Mr. Gian Maria Dall'Ara, Istituto Nazionale di Alta Matematica "F. Severi"

Abstract

We introduce the notion of ``Levi core" of a CR manifold $M$ of hypersurface type, and prove that, whenever $M$ is the boundary of a smoothly bounded pseudoconvex domain $\Omega$, the Diederich-Fornaess index of $\Omega$ equals a numerical invariant of the Levi core. This line of research is connected and extends recent results of B. Liu, M. Adachi, and J. Yum.

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Strict density inequalities for interpolation in weighted spaces of holomorphic functions in several complex variables.

Joaquim Ortega Cerdà, Universitat de Barcelona

Abstract

I will present a joint work with Karlheinz Gröchenig, Antti Haimi and José Luis Romero were we solve a problem posed by Lindholm and prove strict density inequalities for sampling and interpolation in Fock spaces of entire functions in several complex variables defined by a plurisubharmonic weight. In particular, these spaces do not admit a set that is simultaneously sampling and interpolating. To prove optimality of the density conditions, we construct sampling sets with a density arbitrarily close to the critical density.

Computational aspects of commutative and noncommutative positive polynomials (MS - ID 77)

Algebra

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Non-commutative polynomial in quantum physics

Prof. Antonio Acin, ICFO

Abstract

Quantum physics is a natural source of problems involving the optimization of polynomials over non-commuting variables. The talk first introduces some of the most relevant problems, explains how they can be tackled using semi-definite programming hierarchies and concludes with some of the open problems. References: [1] S. Pironio, M. Navascues, A. Acin, Convergent relaxations of polynomial optimization problems with non-commuting variables, SIAM J. Optim. 20 (5), 2157 (2010). [2] E. Wolfe, A. Pozas-Kerstjens, M. Grinberg, D. Rosset, A. Acín, M. Navascues, Quantum Inflation: A General Approach to Quantum Causal Compatibility, to appear in Phys. Rev. X. [3] A. Pozas-Kerstjens, R. Rabelo, L. Rudnicki, R. Chaves, D. Cavalcanti, M. Navascues, A. Acín, Bounding the sets of classical and quantum correlations in networks, Phys. Rev. Lett. 123, 140503 (2019). [4] F. Baccari, C. Gogolin, P. Wittek, A. Acín, Verifying the output of quantum optimizers with ground-state energy lower bounds, Phys. Rev. Research 2, 043163 (2020).

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Tensor Decompositions on Simplicial Complexes: Existence and Applications

Mr. Tim Netzer, University of Innsbruck

Abstract

Inspired by the tensor network approach from theoretical quantum physics, we develop a framework to define and analyze invariant decompositions of elements of tensor product spaces. We define an invariant decomposition with indices arranged on a simplicial complex, which is explicitly invariant under a given group action. Several versions of such decompositions also allow to cover positivity of the involved objects. We prove that these decompositions exists for all invariant/positive tensors, after possibly enriching the structure of the simplicial complex. The approach cannot only be applied to tensor products of matrices (as done in quantum physics), but to multivariate polynomials as well. This yields new types of decompositions and complexity measures for polynomials, containing certificates for positivity and symmetries.

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Hilbert's 17th problem for noncommutative rational functions

Dr. Jurij Volčič, Texas A&M University

Abstract

One of the problems on Hilbert's 1900 list asked whether every positive rational function can be written as a sum of squares of rational functions. Its affirmative resolution by Artin in 1927 was a breakthrough for real algebraic geometry. This talk addresses the analog of this problem for noncommutative rational functions. More generally, a rational Positivstellensatz on matricial sets given by linear matrix inequalities is presented. The crucial intermediate step is an extension theorem on invertible evaluations of linear matrix pencils. A consequence of the Positivstellensatz is an algorithm for eigenvalue optimization of noncommutative rational functions. Lastly, the talk discusses some contrast between polynomial and rational Positivstellens\"atze.

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Quantum de Finetti theorems and Reznick’s Positivstellensatz

Ion Nechita, CNRS, Toulouse University

Abstract

We present a proof of Reznick’s quantitative Positivstellensatz using ideas from Quantum Information Theory. This result gives tractable conditions for a positive polynomial to be written as a sum of squares. We relate such results to de Finetti theorems in Quantum Information.

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Optimizing conditional entropies for quantum correlations

Dr. Omar Fawzi, Inria

Abstract

The rates of quantum cryptographic protocols are usually expressed in terms of a conditional entropy minimized over a certain set of quantum states. In the so-called device-independent setting, the minimization is over all the quantum states of arbitrary dimension jointly held by the adversary and the parties that are consistent with the statistics that are seen by the parties. We introduce new quantum divergences and use techniques from noncommutative polynomial optimization to approximate such entropic quantities. Based on https://arxiv.org/abs/2007.12575

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Dual nonnegativity certificates and efficient algorithms for rational sum-of-squares decompositions

Dr. David Papp, North Carolina State University

Abstract

We study the problem of computing rational weighted sum-of-squares (WSOS) certificates for positive polynomials over a compact semialgebraic set. In the first part of the talk, we introduce the concept of dual cone certificates, which allows us to interpret vectors from the dual of the WSOS cone as rigorous nonnegativity certificates. Every polynomial in the interior of the WSOS cone admits a full-dimensional cone of dual certificates; as a result, rational WSOS certificates can be constructed from numerically computed dual certificates at little additional cost. In the second part of the talk, we use this theory to develop an almost entirely numerical hybrid algorithm for computing the optimal WSOS lower bound of a given polynomial along with a rational dual certificate, with a polynomial-time computational cost per iteration and linear rate of convergence. As a special case, we obtain a new polynomial-time algorithm for certifying the nonnegativity of strictly positive polynomials over an interval.

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Optimization over trace polynomials

Dr. Victor Magron, LAAS CNRS

Abstract

Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascu\'es and Ac\'in scheme [New J. Phys., 2008] for optimization of noncommutative polynomials. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.

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Fast Semidefinite Optimization with Latent Basis Learning

Dr. Georgina Hall, INSEAD

Abstract

When faced with a semidefinite program (SDP), it is often the case that we do not need to solve just one specific SDP, but rather a family of very similar problems with varying data, for example, when solving matrix completion problems to obtain movie recommendations for users with similar preferences. In this talk, I will present Fast Semidefinite Optimization (FSDO), a data-driven method to quickly solve SDPs coming from the same family. Our method learns a shared latent basis representation across the family, which is then used as input to a second-order cone program, whose solution constitutes an approximate solution to the original SDP. The learning is done using neural networks and leverages recent advances in differentiable convex optimization.

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Efficient noncommutative polynomial optimization by exploiting sparsity

Dr. Jie Wang, LAAS-CNRS

Abstract

Many problems arising from quantum information can be modelled as noncommutative polynomial optimization problems. The moment-SOHS hierarchy approximates the optimum of noncommutative polynomial optimization problems by solving a sequence of semidefinite programming relaxations with increasing sizes. In this talk, I will show how to exploit various sparsity patterns encoded in the problem data to improve scalability of the moment-SOHS hierarchy for eigenvalue and trace optimization problems.

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Quantum polynomial optimisation problems for dimension $d$ variables, with symmetries

Dr. Marc-Olivier Renou, ICFO - The Institute of Photonic Sciences

Abstract

Quantum Information Theory (QIT) involves quantum states and measurements, mathematically represented as non-commutative positive operators. A typical problem in QIT is to find the minimum of a polynomial expression in these operators. For instance, finding the maximum violation of a Bell inequality, or the ground state energy of a Hamiltonian are polynomial optimisation problems in non-commuting variables. The NPA hierarchy [New J. Phys. 10, 073013 (2008)], which can be viewed as the “eigenvalue” version of Lasserre’s hierarchy, provides a converging hierarchy of SDP relaxation of a non-commutative polynomial optimisation problem involving variables of unbounded dimension. This hierarchy converges, it is one of QIT main technical tool. Importantly, some QIT problems concern operators of bounded dimension $d$. The NPA hierarchy was extended into the NV hierarchy [Phys. Rev. Lett.115, 020501(2015)] to tackle this case. In this method, one first sample at random many dimension $d$ operators satisfying the constraint, and compute the associated moment matrix. This first step discovers the moment matrix vector space, over which the relaxed SDP problem is solved in a second step. In this talk, we will first review this method. Then, based on [Phys. Rev. Lett. 122, 070501], we will show how one can reduce the computational requirements by several orders of magnitude, exploiting the eventual symmetries present in the optimization problem.

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Sum-of-Squares proofs of logarithmic Sobolev inequalities on finite Markov chains

Dr. Hamza Fawzi, University of Cambridge

Abstract

Logarithmic Sobolev inequalities play an important role in understanding the mixing times of Markov chains on finite state spaces. It is typically not easy to determine, or indeed approximate, the optimal constant for which such inequalities hold. In this paper, we describe a semidefinite programming relaxation for the logarithmic Sobolev constant of a finite Markov chain. This relaxation gives certified lower bounds on the logarithmic Sobolev constant. Numerical experiments show that the solution to this relaxation is often very close to the true constant. Finally, we use this relaxation to obtain a sum-of-squares proof that the logarithmic Sobolev constant is equal to half the Poincaré constant for the specific case of a simple random walk on the odd n-cycle, with n in {5,7,…,21}. Previously this was known only for n=5 and even n.

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Quantum Isomorphism of Graphs: an Overview

Dr. Laura Mančinska, University of Copenhagen

Abstract

In this talk I will introduce a quantum version of graph isomorphism. Our point of departure will be an interactive protocol (nonlocal game) where two provers try to convince a verifier that two graphs are isomorphic. Allowing provers to take advantage of shared quantum resources will then allow us to define quantum isomorphism as the ability of quantum players to win the corresponding game with certainty. We will see that quantum isomorphism can be naturally reformulated in the languages of quantum groups and counting complexity. In particular, we will see that two graphs are quantum isomorphic if and only if they have the same homomorphism counts from all planar graphs.

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The tracial moment problem on quadratic varieties

Dr. Aljaž Zalar, University of Ljubljana

Abstract

The truncated moment problem asks to characterize finite sequences of real numbers that are the moments of a positive Borel measure on $\mathbb{R}^n$. Its tracial analog is obtained by integrating traces of symmetric matrices. The solution of the bivariate quartic tracial moment problem with a nonsingular $7\times 7$ moment matrix $\mathcal M_2$ whose columns are indexed by words of degree 2 was established by Burgdorf and Klep. In the talk we will present the solution of the singular bivariate tracial moment problem on quadratic varieties. In case the moment matrix $\mathcal M_n$ satisfies two quadratic column relations the existence of the measure can be completely characterized by certain numerical conditions, while in the presence of one quadratic column relation the existence of a representing measure is equivalent to the feasibility of certain linear matrix inequalities. The atoms in the representing measure are always pairs of $2\times 2$ matrices. This is joint work with Abhishek Bhardwaj. \vskip3mm \noindent{\bf References:} \begin{itemize} \item A. Bhardwaj, A. Zalar, \textit{The singular bivariate quartic tracial moment problem}. Complex Anal.\ Oper.\ Theory \textbf{12:4} (2018), 1057--1142. \item A. Bhardwaj, A. Zalar, \textit{The tracial moment problem on quadratic varieties}. J. Math. Anal. Appl. \textbf{498} (2021). \end{itemize}

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Positive maps and trace polynomials from the symmetric group

Dr. Felix Huber, Jagiellonian University

Abstract

With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{\alpha_1} \cdots X_{\alpha_r}$ and their traces $\operatorname{tr}(X_{\alpha_1} \cdots X_{\alpha_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, characterize the set of multilinear equivariant positive maps in terms of Werner state witnesses, and construct permutation polynomials and tensor polynomial identities on tensor product spaces. We give connections to concepts in quantum information theory and invariant theory.

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Semidefinite relaxations in non-convex spaces

Dr. Alex Pozas-Kerstjens, Universidad Complutense de Madrid

Abstract

In analogy with Lassere's and Parillo's hierarchies of semidefinite relaxations for polynomial optimization problems, a semidefinite hierarchy exisits for non-commutative polynomial optimization problems of the form \begin{equation*} \begin{split} p^* = &\underset{(\mathcal{H},\phi,X)}{\text{min}}\quad \langle \phi,p(X) \phi\rangle \\ &\,\,\,\,\text{s.t.}\quad\quad q_i(X)\succeq 0 \end{split}, \end{equation*} where $\phi$ is a normalized vector in the Hilbert space $\mathcal{H}$, $X=(X_1,\dots,X_n)$ are the non-commuting variables in the problem, and $p(X)$ and $q_i(X)$ are polynomials in the variables $X$. This hierarchy, known as the Navascu\'es-Pironio-Ac\'in hierarchy, has encountered important applications in physics, where it has become a central tool in quantum information theory and in the certification of quantum phenomena. Its success has motivated its application, within quantum information theory, in more complex scenarios where the relevant search spaces are not convex, and thus the characterizing constraints are in conflict with semidefinite formulations. The paradigmatic example is optimizing over probability distributions obtained by parties performing measurements on quantum systems where not all parties receive a share from every system available. The most illustrating consequence is that, in certain situations, marginalization over a selected number of parties makes the resulting probability distribution to factorize. In this talk we address the problem of non-commutative polynomial optimization in non-convex search spaces and present two means of providing monotonically increasing lower bounds on its solution, that retain the characteristic that the newly formulated problems remain being semidefinite programs. The first directly addresses factorization-type constraints by adding auxiliary commuting variables that allow to encode (relaxations of) polynomial trace constraints, while the second considers multiple copies of the problem variables and constrains them by requiring the satisfaction of invariance under suitable permutations. In addition to presenting the methods and exemplifying their applicability in simple situations, we highlight the fundamental questions that remain open, mostly regarding to their convergence.

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SOS relaxations for detecting quamtum entanglement

Dr. Abhishek Bhardwaj, CNRS

Abstract

Positive maps which are not completely positive (PNCP maps) are of importance in quantum information theory, in particular they can be used to identify entanglement in quantum states. PNCP maps can naturally be associated to non-negative polynomials which are not sums of squares (SOS). An algorithm for constructing such maps is given in ``There are many more positive maps than completely positive maps" by Klep et al. In this talk we will present a summary of their construction and discuss theoterical and numerical issues that arise in practice.

Configurations (MS - ID 81)

Combinatorics and Discrete Mathematics

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Connected $(n_{k})$ configurations exist for almost all $n$

Dr. Leah Berman, University of Alaska Fairbanks

Abstract

A geometric $(n_{k})$ configuration is a collection of points and straight lines, typically in the Euclidean plane, so that each line passes through $k$ of the points and each of the points lies on $k$ of the lines. In a series of papers, Branko Gr\"unbaum showed that geometric $(n_{4})$ configurations exist for all $n \geq 24$, using a series of geometric constructions later called the ``Gr\"unbaum Calculus''. In this talk, we will show that for each $k > 4$, there exists an integer $N_{k}$ so that for \emph{all} $n \geq N_{k}$, there exists at least one $(n_{k})$ configuration, by generalizing the Gr\"unbaum Calculus operations to produce more highly incident configurations. This is joint work with G\'abor G\'evay and Toma\v{z} Pisanski.

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Barycentric configurations in real space

Prof. Hendrik Van Maldeghem, Ghent University

Abstract

Barycentric configurations are configurations with three points per line in real projective space such that (homogeneous) coordinates exist with the property that, for each line, the sum of the coordinate tuples of the three points on that line is the zero tuple. Such configurations turn up naturally and we develop some theory about them. In particular there exist universal barycentric embeddings and a general construction method if the geometry is self-polar. We apply these results to the Biggs-Smith geometry on 102 points, providing a (new) geometric construction of the Biggs-Smith graph making the full automorphism group apparent. We also mention a connection with ovoids.

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Combinatorial Configurations and Dessins d'Enfants

Prof. Milagros Izquierdo, Linköping University

Abstract

In this talk we give an overview, with many examples, of how dessins d'enfants (maps and hypermaps) on Riemann surfaces produce point-circle realisations on higher genus of combinatorial configurations. The symmetry of the hypermap is transfered to the symmetry of the configuration.

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Transitions between configurations

Prof. Gábor Gévay, University of Szeged

Abstract

In this talk we briefly review (without aiming at completeness) various procedures by means of which new configurations can be obtained from old configurations. Among them, there are binary operations, like e.g.\ the \emph{Cartesian product} and the \emph{incidence sum}. Several other operations are collectively called the \emph{Gr\"unbaum calculus}. The \emph{incidence switch} operation can be defined on the level of incidence graphs (also called Levi graphs) of configurations, and in some cases it is in close connection with realization problems of configurations as well as with incidence theorems. Considering point-line, point-circle and point-conic configurations, there are interesting ad hoc constructions by which from a configuration of one of these geometric types another one can be derived, thus realizing \emph{transitions} between configurations in a very general sense. We also present interesting examples and applications. The most recent results mentioned in this talk are based on joint work with Toma\v{z} Pisanski and Leah Wrenn Berman.

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Splittability of cubic bicirculants and their related configurations

Dr. Nino Bašić, University of Primorska

Abstract

Recently, it was shown that there exist infinitely many splittable and also infinitely many unsplittable cyclic $(n_3)$ configurations. This was achieved by studying splittability of trivalent cyclic Haar graphs. We extend this study to include cubic bicirculant and their related configurations.

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Taxonomy of Three-Qubit Doilies

Dr. Metod Saniga, Slovak Academy of Sciences

Abstract

We study doilies (i.\,e., $W(3,2)$'s) living in $W(5,2)$, when the points of the latter space are parametrized by canonical three-fold products of Pauli matrices and the associated identity matrix (i.\,e., by three-qubit observables). Key characteristics of such a doily are: the number of its negative lines, distribution of types of observables, character of the geometric hyperplane the doily shares with the distinguished (non-singular) quadric of $W(5,2)$ and the structure of its Veldkamp space. $W(5,2)$ is endowed with 90 negative lines of two types and its 1344 doilies fall into 13 types. 279 out of 480 doilies with three negative lines are composite, i.\,e. they all originate from the two-qubit doily by selecting in the latter a geometric hyperplane and formally adding to each two-qubit observable, at the same position, the identity matrix if an observable lies on the hyperplane and the same Pauli matrix for any other observable. Further, given a doily and any of its geometric hyperplanes, there are other three doilies possessing the same hyperplane. There is also a particular type of doilies a representative of which features a point each line through which is negative.

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Hirzebruch-type inequalities and extreme point-line configurations

Prof. Piotr Pokora, Pedagogical University of Cracow

Abstract

In my talk I would like to report on very recent developments devoted to extreme point-line configurations  from an algebraic perspective of Hirzebruch-type inequalities. We recall some inequalities, especially the most powerful variant based on orbifolds, and then I will report on extreme combinatorial problems for which Hirzebruch-type inequalities played a decisive role. Time permitting, we will present a short proof of the Weak Dirac Conjecture.

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Strongly regular configurations

Prof. Vedran Krčadinac, University of Zagreb Faculty of Science

Abstract

We report on the recent study~[1] of combinatorial configurations with the associated point and line graphs being strongly regular. A prominent family of such configurations are the partial geometries, introduced by R.~C.~Bose in~1963. We focus on such configurations that are not partial geometries nor known generalisations such as semipartial geometries and strongly regular $(\alpha,\beta)$-geometries, neither are they elliptic semiplanes of P.~Dembowski. Several families are constructed, necessary existence conditions are proved, and a table of feasible parameters with at most $200$ points is presented. \begin{enumerate}[{[1]}] \item M.~Abreu, M.~Funk, V.~Kr\v{c}adinac, D.~Labbate, \emph{Strongly regular configurations}, preprint, 2021. \verb|https://arxiv.org/abs/2104.04880| \end{enumerate}

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Configurations from strong deficient difference sets

Dr. Marién Abreu, Università degli Studi della Basilicata

Abstract

In [1] we have studied combinatorial configurations with the associated point and line graphs being strongly regular, which we call \emph{strongly regular configurations}. In the talk ``Strongly regular configurations'' of this minisymposium, Vedran Kr\v{c}adinac will present existing known families of strongly regular configurations; constructions of several other families; necessary existence conditions and a table of feasible parameters with at most $200$ points. Let $G$ be a group of order $v$. A subset $D \subset G$ of size $k$ is a \emph{deficient difference set} the left differences $d^{-1}_1d_2$ are all distinct. Considering the elements of $G$ as points and the\emph{ development} $devD = \{gD | g \in G\}$ as lines a symmetric $(v_k)$ configuration is obtained and it has $G$ as its automorphism group acting regularly on the points and lines. Let $\Delta(D) = \{d^{-1}_1d_2 | d_1, d_2 \in D, d_1 \ne d_2\}$ be the set of left differences of $D$. For a group element $x \in G \setminus \{1\}$, denote by $n(x) = |\Delta(D) \cap x\Delta(D)|$. If $n(x) = \lambda$ for every $x \in \Delta(D)$, and $n(x) = \mu$ for every $x \notin \Delta(D)$, $D$ is said to be a \emph{strong deficient difference set} ($SDDS$) for $(v_k; \lambda, \mu)$. Here, we present one of the new families of strongly regular configurations constructed in [1], with parameters different from semipartial geometries and arising from strong deficient difference sets, as well as two examples arising from Hall's plane and its dual. Moreover, from the exhaustive search performed in groups of order $v \le 200$ further four examples corresponding to strong deficient difference sets, but not in the previous families, are obtained. \medskip \noindent [1] M.~Abreu, M.~Funk, V.~Kr\v{c}adinac, D.~Labbate, Strongly regular configurations, preprint, 2021. https://arxiv.org/abs/2104.04880

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Highly symmetric configurations

Dr. Jurij Kovič, UP FAMNIT Koper and IMFM Ljubljana

Abstract

In the talk some general methods and special techniques for the construction of highly symmetric configurations will be presented. As an application of these theoretical principles, some examples of highly symmetric spatial geometric configurations of points and lines with the symmetry of Platonic solids will be given.

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4-lateral matroids induced by $n_3$-configurations

Dr. Michael Raney, Georgetown University

Abstract

A \emph{4-lateral matroid} induced by an $n_3$-configuration is a rank-4 matroid whose ground set consists of the blocks (lines) of the configuration, and for which any 4 elements of the ground set are independent if and only if they do not determine a 4-lateral within the configuration. We characterize the $n_3$-configurations which induce 4-lateral matroids, and provide examples of small $n_3$-configurations which do so.

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Jordan schemes

Dr. Sven Reichard, Dresden International University

Abstract

Association schemes (also called homogeneous coherent configurations) first appeared in Statistics in relation to the design of experiments, due to the efforts of Bose and his collaborators. These systems of binary relations defined on the same set, color graphs in other terms, give rise to certain matrix algebras, not necessarily commutative. In this way a bridge is formed between combinatorics and algebra, in particular with permutation groups. In statistial applications we want the relations to be symmetric, while the product of symmetric matrices is not symmetric in general. Therefore Bailey and Cameron, following Shah (from the same school as Bose) suggested to replace the matrix product \(AB\) by the Jordan product \(A*B = (AB+BA)/2\), which is commutative but not necessarily associative. The resulting structures are called Jordan schemes. Given any association scheme, its symmetrization is a Jordan scheme. This led Peter Cameron to the following question: Do all Jordan schemes arise in this way, or do there exist ``proper'' Jordan schemes? We gave a positive answer to this question by constructing a first proper Jordan scheme on 15 points using so-called Siamese color graphs, investigated earlier by our group. First elements of the theory of these structures were established; a few infinite classes of proper Jordan schemes were discovered. Moreover an efficient computational criterion for recognizing proper Jordan schemes is given, based on the classical Weisfeiler-Leman stabilization. The current talk mainly focuses on the computer search for proper Jordan schemes, which is based on algorithmic ideas of Hanaki and Miyamoto, who enumerated all small association schemes. It turns out that the initial example is indeed the smallest. Besides it, up to isomorphism three more examples on less than 20 points were discovered. Each such small example of orders 15, 16 and 18 can be constructed from a suitable association scheme by a certain switching operation. For more on the background of Jordan schemes see Cameron's blog:\\ {https://cameroncounts.wordpress.com/2019/06/28/proper-jordan-schemes-exist/} This is part of a joint project with Misha Klin and Misha Muzychuk from Ben-Gurion University, Beer Sheva.

Convex bodies - approximation and sections (MS - ID 61)

Analysis and its Applications

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On the complex hypothesis of Banach

Prof. Luis Montejano, Instituto de Matematicas

Abstract

The following is known as the geometric hypothesis of Banach: let $V$ be an m-dimensional Banach space (over the real or the complex numbers) with unit ball $B$ and suppose all $n$-dimensional subspaces of $V$ are isometric (all the n-sections of B are affinely equivalent). In 1932, Banach conjectured that under this hypothesis $V$ is a Hilbert space (the boundary of $B$ is an ellipsoid). Gromow proved in 1967 that the conjecture is true for $n$=even and Dvoretzky and V. Milman derived the same conclusion under the hypothesis $n$=infinity. We prove this conjecture for $n=4k+1$, with the possible exception of $V$ a real Banach space and $n=133$. [G.Bor, L.Hernandez-Lamoneda, V. Jim\'enez and L. Montejano. To appear Geometry $\&$ Topology] for the real case and [J. Bracho, L. Montejano, submitted to J. of Convex Analysis] for the complex case. The ingredients of the proof are classical homotopic theory, irreducible representations of the orthogonal group and convex geometry. For the complex case, suppose B is a convex body contained in complex space $C^{n+1}$, with the property that all its complex n-sections through the origin are complex affinity equivalent to a fixed complex n-dimensional body $K$. Studying the topology of the complex fibre bundle $SU(n) -> SU(n-1)-> S^{2n+1}$, it is possible to prove that if $n$=even, then $K$ must be a ball and using homotopical properties of the irreducible representations we prove that if $n=4+1$ then $K$ must be a body of revolution. Finally, we prove, using convex geometry and topology that, if this is the case, then there must be a section of $B$ which is an complex ellipsoid and consequently $B$ must be also a complex ellipsoid.

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The $L_p$ Minkowski problem and polytopal approximation

Prof. Karoly Boroczky, Alfred Renyi Institute of Mathematics

Abstract

The $L_p$ Minkowski problem , a Monge-Ampere type equation on the sphere, is a recent version of the classical Minkowski problem. I will review cases when one can reduce the equation to properties of polytopes, and then use polytopal approximation to solve the PDE in general.

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On bodies floating in equalibrium in every direction

Prof. Dmitry Ryabogin, Kent State University

Abstract

We give a negative answer to Ulam's Problem 19 from the Scottish Book asking {\it is a solid of uniform density which will float in water in every position a sphere? } Assuming that the density of water is 1, we show that there exists a strictly convex body of revolution $K\subset {\mathbb R^3}$ of uniform density $\frac{1}{2}$ , which is not a Euclidean ball, yet floats in equilibrium in every direction. We prove an analogous result in all dimensions $d\ge 3$.

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Strengthened inequalities for the mean width

Mr. Ferenc Fodor, University of Szeged

Abstract

According to a result of Barthe the regular simplex maximizes the mean width of convex bodies whose John ellipsoid is the Euclidean unit ball. The reverse statement that the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball is also true as proved by Schmuckenschl\"ager. In this talk we prove strengthened stability versions of these theorems and also some related stability statements for the convex hull of the support of centered isotropic measures on the sphere. This is joint work with K\'aroly J. B\"or\"oczky (Budapest, Hungary) and Daniel Hug (Karlsruhe, Germany).

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A new look at the Blaschke-Leichtweiss theorem

Prof. Karoly Bezdek, University of Calgary

Abstract

The Blaschke-Leichtweiss theorem (Abh. Math. Sem. Univ. Hamburg 75: 257--284, 2005) states that the smallest area convex domain of constant width $w$ in the $2$-dimensional spherical space ${\mathbb S}^2$ is the spherical Reuleaux triangle for all $0<w\leq\frac{\pi}{2}$. In this paper we extend this result to the family of wide $r$-disk domains of ${\mathbb S}^2$, where $0<r\leq\frac{\pi}{2}$. Here a wide $r$-disk domain is an intersection of spherical disks of radius $r$ with centers contained in their intersection. This gives a new and short proof for the Blaschke-Leichtweiss theorem. Furthermore, we investigate the higher dimensional analogue of wide $r$-disk domains called wide $r$-ball bodies. In particular, we determine their minimum spherical width (resp., inradius) in the spherical $d$-space ${\mathbb S}^d$ for all $d\geq 2$. Also, it is shown that any minimum volume wide $r$-ball body is of constant width $r$ in ${\mathbb S}^d$, $d\geq 2$.

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Pea Bodies of Constant Width

Dr. Deborah Oliveros, Universidad Nacional Autónoma de México (UNAM)

Abstract

Besides the two Meissner solids, the obvious constant width bodies of revolution, and the Meissner polyhedra, there are few concrete examples of bodies of constant width or a concrete finite procedure to construct them. In this talk, we will describe an infinite family of 3-dimensional bodies of constant width obtained from the Reuleaux Tetrahedron by replacing a small neighborhood of all six edges with sections of an envelope of spheres using the classical notion of confocal quadrics. This family includes Meissner solids as well as one with tetrahedral symmetry. (Joint work with I. Arelio and L. Montejano).

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The Golden ratio and high dimensional mean inequalities

Dr. Bernardo González Merino, Universidad de Murcia

Abstract

The classical inequalities between means state that \begin{equation}\label{eq:mean_inequalities} \min\{a,b\}\leq\left(\frac{a^{-1}+b^{-1}}{2}\right)^{-1}\leq \sqrt{ab}\leq\frac{a+b}{2}\leq\max\{a,b\}, \end{equation} for any $a,b>0$, with equality if and only if $a=b$. One can naturally extend \eqref{eq:mean_inequalities} considering means of $n$-dimensional compact, convex sets. Notice that means of $K$ and $-K$ are commonly known as symmetrizations of $K$. In this context, we will show that in even dimensions, if $K$ has a large Minkowski asymmetry, then the corresponding inequalities between the symmetrizations of $K$ can no longer be optimal. Especially in the planar case, we compute that the range of asymmetries of $K$ for which the inequalities between the symmetrizations of $K$ can be optimal is $[1,\phi]$, where $\phi$ is the Golden ratio. Indeed, we introduce the Golden House, (up to linear transformations) the only Minkowski centered set with asymmetry $\phi$ such that the symmetrizations of $K$ are successively optimally contained in each other.

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Colorful Helly-type Theorems for Ellipsoids

Mrs. Viktória Földvári, Alfréd Rényi Institute of Mathematics

Abstract

Helly-type theorems have been widely studied and applied in discrete geometry. In this talk, I am going to give a brief overview of quantitative Helly-type theorems and introduce our joint result with Gábor Damásdi and Márton Naszódi, a colorful Helly-type theorem for ellipsoids.

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A cap covering theorem

Mr. Alexandr Polyanskii, MIPT

Abstract

A \textit{cap} of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set of points within spherical distance $\alpha$ from a given point on the sphere. Let $\mathcal F$ be a finite set of caps lying on $S$. We prove that if no hyperplane through the center of \( S \) divides $\mathcal F$ into two non-empty subsets without intersecting any cap in $\mathcal F$, then there is a cap of radius equal to the sum of radii of all caps in $\mathcal F$ covering all caps of $\mathcal F$ provided that the sum of radii is less $\pi/2$. This is the spherical analog of the so-called circle covering theorem by Goodman and Goodman and the strengthening of Fejes T\'oth's zone conjecture proved by Jiang and the author.

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A solution to some problems of Conway and Guy on monostable polyhedra

Dr. Zsolt Langi, Budapest University of Technology

Abstract

A convex polyhedron is called monostable if it can rest in stable position only on one of its faces. In this talk we investigate three questions of Conway, regarding monostable polyhedra, from the open problem book of Croft, Falconer and Guy (Unsolved Problems in Geometry, Springer, New York, 1991), which first appeared in the literature in a 1969 paper. In this talk we answer two of these problems. The main tool of our proof is a general theorem describing approximations of smooth convex bodies by convex polyhedra in terms of their static equilibrium points.

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Covering techniques in Integer \& Lattice Programming

Mr. Moritz Venzin, École polytechnique fédérale de Lausanne

Abstract

I will present two ideas based on high-dimensional coverings that yield some of the currently best approximation algorithms for Integer \& Lattice Programming. These problems are as follows: Given some convex body $K \subseteq \mathbb{R}^n$, is there an integer point in $K$? This problem is central in the algorithmic geometry of numbers and has found applications in Integer Programming and plays a key role in proposed schemes for post-quantum Cryptography. The following two natural geometric considerations have led to improvements in the current state-of-the-art algorithms for Lattice Programming: \begin{itemize} \item Given some convex body $K\subseteq \mathbb{R}^n$, how many convex bodies $Q_i$ are required to cover $K$ so that, when scaled around their respective centroids by a factor of $2$, these convex bodies are contained inside $(1+\epsilon) \cdot K$? This question was first considered by Eisenbrand et al. in the context of Integer Programming for $K = B_{\infty}^n$. In that case, a simple dyadic decomposition along the facets reveals that $O(1+\log(1/\epsilon))^n$ convex bodies suffice. For general $\ell_p$ norm balls, exploiting the modulus of smoothness, $O(1+1/\epsilon)^{n/{\min(p,2)}}$ convex bodies suffice. This is tight for the Euclidean ball, but it is wide open whether for general $K$ (even for the cross-polytope) there is an improvement over $O(1+1/\epsilon)^n$. \item Let $K$ be some convex body and let $\mathcal{E}$ be some ellipsoid enclosing $K$. We consider the translative covering number $N(\mathcal{E},K)$ but with a twist: For $\epsilon > 0$, is there some constant $C_{\epsilon}>0$ so that $N(\mathcal{E},C_{\epsilon}\cdot K) < 2^{\epsilon n}$? When $K = B_{\infty}^n$ and the ellipsoid is $\sqrt{n}B_2^n$, the answer is yes, i.e. for any $\epsilon > 0$, we can scale the cube by some constant and cover $\sqrt{n}B_2^n$ by fewer than $2^{\epsilon n}$ translates. Similarly, when $K$ is the $\ell_p$ norm ball for $p \geq 2$ and the ellipsoid is an adequately scaled ball, the answer is yes, while for $p < 2$, the answer is no (compare the volume). \end{itemize} For both problems I will describe the techniques to obtain these results, briefly sketch their relevance to Lattice Programming and present some open questions. The talk is based on joint works with Márton Naszódi and Fritz Eisenbrand respectively.

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Rigidity of compact Fuchsian manifolds with convex boundary

Mr. Roman Prosanov, TU Wien

Abstract

It is known that convex bodies in the Euclidean 3-space are globally rigid, i.e., their shape is determined by the intrinsic geometry of the boundary. This story was developed separately in smooth and in polyhedral settings until in 50s it was unified by Pogorelov who proved the rigidity of general convex bodies without any assumptions on their boundaries except convexity. Later another approach was proposed by Volkov with the help of polyhedral approximation.  On the other hand, in 70s Thurston revolutionized the field of 3-dimensional topology by formulating his geometrization program culminated in the famous works of Perelman. In particular Thurston highlighted the abundance and the ubiquity of hyperbolic 3-manifolds. In the scope of this framework some amount of attention was directed towards hyperbolic 3-manifolds with convex boundary. It is conjectured that their shape is determined by the topology and the intrinsic geometry of the boundary. So far this was established only for smooth strictly convex boundaries by Schlenker.  In my recent work I obtained the general rigidity for a toy family of hyperbolic 3-manifolds with convex boundary. This was achieved by reviving the approach of Volkov. 

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Coverings by homothets of a convex body

Dr. Nora Frankl, Carnegie Mellon University

Abstract

Rogers proved that for any convex body $K$, we can cover $R^d$ by translates of $K$ of density roughly $d \log d$. We discuss several related results. First, we extend Roger's result by showing that, if we are given a family of positive homothets of $K$ of infinite total volume, then we can find appropriate translation vectors for each given homothet to cover $R^d$ with the same density. Second, we consider an extension to multiple coverings of space by translates of a convex body. Finally, we also prove a lower bound on the total volume of a family $\mathcal{F}$ of homothets of $K$ that guarantees the existence of a covering of $K$ by members of $\mathcal{F}$.

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Functional John and L\"owner Ellipsoids

Dr. Grigory Ivanov, Institute of Science and Technology Austria (IST Austria)

Abstract

In this talk, we will speak about functional analogs of the John and L\"owner ellipsoids for log-concave functions. We will discuss the existence and uniqueness results for these objects, their general properties such as volume ratio, containment, John's condition, etc. We will note the difference between the behavior of convex sets and log-concave functions concerning our problems, and highlight a number of open problems.

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Extremal sections and local optimization

Dr. Gergely Ambrus, Alfréd Rényi Institute of Mathematics

Abstract

We will demonstrate how to apply local optimization methods in order to find, or, at least characterize, maximal or minimal central sections of convex bodies. We are mainly interested in the special cases when the convex body is the $d$-dimensional cube, or the $d$-dimensional regular simplex. Maximal sections of the former were determined by K. Ball in 1986, while monotinicity of the volume of diagonal central sections for $d \geq 3$ was proven by F. Bartha, F. Fodor and B. Gonz\'{a}lez Merino in 2021. On the other hand, finding minimal central sections of the regular simplex is still an open question. Among other results, we provide a geometric characterization for central sections of locally maximal volume of the $d$-dimensional cube.

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Cells in the box and a hyperplane

Prof. Imre Bárány, Alfréd Rényi Institute of Mathematics

Abstract

It is well known that a line can intersect at most $2n-1$ cells of the $n \times n$ chessboard. We consider the high dimensional version: how many cells of the $d$-dimensional $n\times \ldots \times n$ box can a hyperplane intersect? We also prove the lattice analogue of the following well-known fact. If $K,L$ are convex bodies in $\mathbb{R}^d$ and $K\subset L$, then the surface area of $K$ is smaller than that of $L$. Joint work with Peter Frankl.

Current topics in Complex Analysis (MS - ID 32)

Analysis and its Applications

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Geometry of planar domains and their applications in study of conformal and harmonic mappings

Prof. Stanisława Kanas, University of Rzeszow

Abstract

The famous Riemann Mapping Theorem states that for every simply connected domain $\Omega\neq \mathbb{C}$ containing a point $w_0$ there exists a essentially unique univalent function $f$ such that $f(0) = w_0$ and $f_z(0) > 0$, that maps the unit disk $\mathbb{D}$ onto $\Omega$. Open sense-preserving quasiconformal mappings of $\mathbb{D}$ arised as a solutions of linear elliptic partial differential equations of the form $$\overline{f}_{\bar{z}}(z)=\omega(z)f_z(z),\quad z\in \mathbb{D},$$ where $\omega$ is an analytic function from $\mathbb{D}$ into itself, known as a dilatation of $f$ and such that $|\omega(z)|< k < 1$. A natural generalization of the classical class of normalized univalent functions on $\mathbb{D}$ is the class of sense-preserving univalent harmonic mappings on $\mathbb{D}$ of the form $f=h+\overline{g}$ normalized by $h(0)=g(0) = h'(0)-1=0$. In the context of univalent, quasiconformal and planar harmonic mappings a problem of convexity, linear convexity, starlikeness, etc. have been intensively studied in the past decades. Additional properties of a planar domains exhibits a very rich geometric and analytic properties. We discuss behavior of the function $f$ for which some functional are limited to the Both leminiscates, Pascal snail, hyperbola and conchoid of the Sluze. Some appropriate examples are demonstrated.

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Almost periodic functions revisited

Dr. Juan Matías Sepulcre, University of Alicante

Abstract

In the beginnings of the twentieth century, Harald Bohr (1887-1951) gave important steps in the understanding of Dirichlet series and their regions of convergence, uniform convergence and absolute convergence. As a result of his investigations concerning those functions which could be represented by a Dirichlet series, he developed in its main features the theory of almost periodic functions (both for functions of a real variable and for the case of a complex variable). Based on a new equivalence relation on these classes of functions, in this talk we will refine Bochner's result that characterizes the property of almost periodicity in the Bohr's sense. Furthermore, we will present a thorough extension of Bohr's equivalence theorem which states that two equivalent almost periodic functions take the same set of values on every open half-plane or open vertical strip included in their common region of almost periodicity.

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Harmonic quasiconformal mappings and hyperbolic type metrics

Prof. Vesna Todorcevic, FON and MI SASA

Abstract

We are going to present results about connections between quasiconformal and harmonic quasiconformal mappings and some metrics of hyperbolic type.

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Extremal decomposition of the complex plane

Mrs. Iryna Denega, Institute of mathematics of NAS of Ukraine

Abstract

The talk is devoted to a few well-known an extremal problems in geometric function theory of a complex variable associated with estimates of the functionals defined on the systems of mutually non-overlapping domains [1--9]. An improved method is proposed for solving problems on extremal decomposition of the complex plane. And effective upper estimates for maximum of the products of inner radii of mutually non-overlapping domains for any fixed systems of points of the complex plane at all possible values of some parameter $\gamma$ are obtained [6--9]. Also we established conditions under which the structure of points and domains is not important. In particular, we obtained full solution of an open problem about extremal decomposition of the complex plane with two free poles located on the unit circle [8]. \textbf{Problem.} [1] Consider the product $$r^\gamma\left(B_0,0\right)\prod\limits_{k=1}^n r\left(B_k,a_k\right),$$ where $B_{0}$,..., $B_{n}$, $n\geqslant 2$, are pairwise non-overlapping domains in $\overline{\mathbb{C}},$ $a_0=0,$ $|a_{k}|=1$, $k=\overline{1,n}$ and $\gamma\in (0, n]$ ($r(B,a)$ be an inner radius of the domain $B\subset\overline{\mathbb{C}}$ relative to a point $a\in B$). For all values of the parameter $\gamma\in (0, n]$ to show that it attains its maximum at a configuration of domains $B_{k}$ and points $a_{k}$ possessing rotational $n$-symmetry. The proof is due to Dubinin for $\gamma=1$ [1] and to Kuz'mina [2] for $0 <\gamma< 1$. Subsequently, Kovalev [3] solved this problem under the additional assumption that the angles between neighbouring line segments $[0, a_{k}]$ do not exceed $2\pi / \sqrt{\gamma}$. Let $$I_{n}^{0}(\gamma)=\left(\frac{4}{n}\right)^n \frac{\left(\frac{4\gamma}{n^2}\right)^\frac{\gamma}{n}} {\left(1-\frac{\gamma}{n^2}\right)^{n+\frac{\gamma}{n}}} \left(\frac{1-\frac{\sqrt{\gamma}}{n}} {1+\frac{\sqrt{\gamma}}{n}}\right)^{2\sqrt{\gamma}}.$$ \textbf{Theorem 1.} [8] Let $\gamma\in (1,\,2]$. Then, for any different points $a_{1}$ and $a_{2}$ of the unit circle and any mutually non-overlapping domains $B_0$, $B_1$, $B_2$, $a_1\in B_1\subset\overline{\mathbb{C}}$, $a_2\in B_2\subset\overline{\mathbb{C}}$, $a_{0}=0\in B_0\subset\overline{\mathbb{C}}$, the inequality $$r^\gamma\left(B_0,0\right)r\left(B_1,a_1\right) r\left(B_2,a_2\right)\leqslant I_{2}^{0}(\gamma)\left(\frac{1}{2}\,|a_{1}-a_{2}|\right)^{2-\gamma}$$ is true. The sign of equality in this inequality is attained, when the points $a_0$, $a_1$, $a_2$ and the domains $B_0$, $B_1$, $B_2$ are, respectively, the poles and circular domains of the quadratic differential $$Q(w)dw^{2} =-\frac{(4-\gamma)w^{2}+\gamma}{w^{2}(w^{2}-1)^{2}}dw^{2}.$$ \textbf{Theorem 2.} [8] Let $n\in\mathbb{N}$, $n\geqslant 3$, $\gamma\in(1, n]$. Then, for any system of different points $\{a_k\}_{k=1}^n$ of the unit circle and for any collection of mutually non-overlapping domains $B_0$, $B_k$, $a_{0}=0\in B_0\subset\overline{\mathbb{C}}$, $a_k\in B_k\subset\overline{\mathbb{C}}$, $k=\overline{1,n}$, the following inequality holds $$r^\gamma(B_0,0)\prod\limits_{k=1}^n r(B_k, a_k)\leqslant \left(\sin \frac{\pi}{n}\right)^{n-\gamma}\left(I_{2}^{0}\left(\frac{2\gamma}{n}\right)\right)^{\frac{n}{2}}.$$ \begin{center} \textbf{References} \end{center} 1. Dubinin V.N. Condenser capacities and symmetrization in geometric function theory. Birkh\"{a}user/Springer, Basel, 2014. 2. Kuz’mina G.V. The method of extremal metric in extremal decomposition problems with free parameters // J. Math. Sci., 2005, V. 129, No. 3, pp. 3843--3851. 3. Kovalev L.V. On the problem of extremal decomposition with free poles on a circle // Dal’nevostochnyi Mat. Sb., 1996, No. 2, pp. 96--98. (in Russian) 4. Bakhtin A.K., Bakhtina G.P., Zelinskii Yu.B. Topological-algebraic structures and geometric methods in complex analysis. Zb. prats of the Inst. of Math. of NASU, 2008. (in Russian) 5. Bakhtin A.K. Separating transformation and extremal problems on nonoverlapping simply connected domains. J. Math. Sci., 2018, V. 234, No. 1, pp. 1--13. 6. Bakhtin A.K., Denega I.V. Weakened problem on extremal decomposition of the complex plane // Matematychni Studii, 2019, V. 51, No. 1, pp. 35--40. 7. Denega I. Estimates of the inner radii of non-overlapping domains. J. Math. Sci., 2019, V. 242, No. 6, pp. 787--795. 8. Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles // J. Math. Sci., 2020, V. 246, No. 1, pp. 1--17. 9. Bakhtin A.K., Denega I.V. Extremal decomposition of the complex plane with free poles II // J. Math. Sci., 2020, V. 246, No. 5, pp. 602--616.

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Convex sums of biholomorphic mappings and Extension operators in $\mathbb{C}^n$

Mr. Eduard Stefan Grigoriciuc, Babes-Bolyai University of Cluj-Napoca

Abstract

Let $\mathbb{B}^n$ be the Euclidean unit ball in $\mathbb{C}^n$ and let $U$ be the unit disc in $\mathbb{C}$. The aim of this work is to study convex combinations of biholomorphic mappings on $\mathbb{B}^n$ starting from a result proved by Chichra and Singh in the case of one complex variable. They obtained the conditions in which a convex combination of the form $(1-\lambda)f+\lambda g$ is starlike on $U$, when $f$ and $g$ are starlike on the unit disc $U$ and $\lambda\in[0,1]$. Using this ideea, we can construct a similar result for the case of several complex variables. Then, we use this result to characterize convex sums of biholomorphic starlike mappings on the Euclidean unit ball $\mathbb{B}^n$. Moreover, we obtain some remarks on convex sums of extension operators defined for locally univalent functions (for example, the Graham-Kohr extension operator).

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Asymptotic estimates for one class of homeomorphisms

Bogdan Klishchuk, Institute of Mathematics of National Academy of Sciences of Ukraine

Abstract

Let $\Gamma$ be a family of curves $\gamma$ in $\Bbb R^{n}$, $n\geqslant2$. A Borel measurable function $\rho:{\Bbb R^{n}}\to[0,\infty]$ is called {\it admissible} for $\Gamma$, (abbr. $\rho\in{\rm adm}\,\Gamma$), if $$ \int\limits_{\gamma}\rho(x)\,ds\ \geqslant\ 1 $$ for any curve $ \gamma\in\Gamma$. Let $p\in (1,\infty)$. The quantity $$ M_p(\Gamma)\ =\ \inf_{\rho\in\mathrm{adm}\,\Gamma}\int\limits_{{\Bbb R^{n}}}\rho^p(x)\,dm(x)\, $$ is called {\it $p$--modulus} of the family $\Gamma$. For arbitrary sets $E$, $F$ and $G$ of $\Bbb R^{n}$ we denote by $\Delta(E, F, G)$ a set of all continuous curves $\gamma: [a, b] \rightarrow \Bbb R^{n}$ that connect $E$ and $F$ in $G$, i.å., such that $\gamma(a) \in E$, $\gamma(b) \in F$ and $\gamma(t) \in G$ for $a < t < b$. Let $D$ be a domain in $\Bbb R^{n}$, $n\geqslant2$, $x_0\in D$ and $d_0 = {\rm dist}(x_{0}, \partial D)$. Set $$ \mathbb{A}(x_{0}, r_{1}, r_{2}) = \{ x \in \Bbb R^{n}: r_{1} < |x- x_{0}| < r_{2}\} \,, $$ $$ S_{i}=S(x_0,r_{i})=\{x\in \Bbb R^{n}: \, |x-x_0| = r_{i}\}\,, \quad i = 1,\,2\,. $$ Let a function $Q: D\rightarrow [0,\infty]$ be Lebesgue measurable. We say that a homeomorphism $f: D \rightarrow \Bbb R^{n}$ is ring $Q$-homeomorphism with respect to $p$-modulus at $x_0 \in D$ if the relation $$ M_p(\Delta(fS_{1}, fS_{2}, fD))\ \leqslant\ \int\limits_{{\mathbb{A}}}Q(x)\,\eta^{p}(|x-x_{0}|)\,dm(x)\, $$ holds for any ring $\mathbb{A} = \mathbb{A}(x_{0}, r_{1}, r_{2})$\,, $0< r_{1} < r_{2} < d_0$, $d_0 = {\rm dist}(x_{0}, \partial D)$ and for any measurable function $\eta: (r_{1}, r_{2}) \rightarrow [0,\infty]$ such that $$ \int\limits_{r_{1}}^{r_{2}} \eta(r)\, dr = 1\,. $$ Let $$ L(x_0, f, R) = \sup\limits_{|x - x_{0}|\leqslant R}\, |f(x) - f(x_{0})|\,. $$ {\bf Theorem.} {\it Suppose that $f:\Bbb R^{n} \rightarrow \Bbb R^{n}$ is a ring $Q$-homeomorphism with respect to $p$-modulus at a point $x_0$ with $p>n$ where $x_0$ is some point in $\Bbb R^{n}$ and for some numbers $c >0$, $\kappa \leqslant p$, $r_0 >0$ the condition $$ \int\limits_{{\mathbb{A}(x_{0}, r_{0}, R)}}Q(x)\,\psi^{p}(|x-x_{0}|)\,dm(x)\ \leqslant\ c\, I^{\kappa}(r_{0}, R) \, \quad \forall\, R > r_{0}\,, $$ holds, where $\psi(t)$ is a nonnegative measurable function on $(0, +\infty)$ such that $$ 0< I(r_{0}, R) = \int\limits_{r_0}^{R} \psi(t) dt < \infty \quad \forall\, R > r_{0}\,, $$ then $$ \varliminf\limits_{R \rightarrow \infty}\, L(x_0, f, R)\, I^{\frac{\kappa-p}{p-n}}(r_{0}, R) \geqslant \omega_{n-1}^{\frac{1}{n-p}}\left(\frac{p-n}{p-1}\right)^{\frac{p-1}{p-n}}\, c^{\frac{1}{n-p}}\,, $$ where $\omega_{n-1}$ is an area of the unit sphere $\mathbb{S}^{n-1} = \{x\in \Bbb R^{n}: \, |x| = 1\}$ in $\Bbb R^{n}$. %\end{equation} }

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Generalizations of Hardy type inequalities via new Green functions

Dr. Kristina Krulić Himmelreich, University of Zagreb, Faculty of Textile Technology

Abstract

This talk deals with Hardy inequality and its famous generalizations, extensions and refinements, i.e. with Hardy-type inequalities. The classical Hardy inequality reads: \begin{equation} \label{1} \int\limits_0^{\infty} \left( \frac{1}{x} \int\limits_0^x f(t) \,dt \right)^p dx \leq \left ( \frac{p}{p-1} \right )^p \int\limits_0^{\infty} f^p(x) \,dx, \, p>1, \end{equation} where $f$ is non negative function such that $f \in L^p(\mathbb{R}_{+})$ and $\mathbb{R}_{+} =(0,\infty)$. The constant $\left(\frac{p}{p-1}\right)^p$ is sharp. This inequality has been generalized and extended in several directions. In this talk the Hardy inequality is generalized by using new Green functions.

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Recent studies on the image domain of starlike functions

Ms. Sahsene Altinkaya, Beykent University

Abstract

The geometric properties of image domain of starlike functions is of prominent significance to present a comprehensive study on starlike functions. For this purpose, by using a relation of subordination, we investigate a family of starlike functions in the open unit disc \begin{eqnarray*} \mathbf{U}=\left\{ z\in \mathbb{C}:\left\vert z\right\vert <1\right\} . \end{eqnarray*} Subsequently, we discuss some interesting geometric properties, radius problems, general coefficients for this class. Further, we point out several recent studies related to image domain of starlike functions.

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Sharp estimates of the Hankel determinant and of the coefficients for some classes of univalent functions

Prof. Nikola Tuneski, Ss. Cyril and Methodius University in Skopje

Abstract

In recent times, the problem of finding upper bound, preferably sharp, of the Hankel determinant for classes of univalent functions, is being rediscovered and attracts significant attention among the mathematicians working in the field. In that direction, this presentation will provide some new and improvements of existing results for various classes of univalent functions. Most of the results are sharp.

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Some properties of functions from families of even holomorphic functions of several complex variables

Dr. Edyta Trybucka, University of Rzeszów

Abstract

In the presentation we will consider some properties of two families of holomorphic functions defined on bounded complete n-circular domain $G$ of $\mathbb{C}^n$ by the evenness (see [3], [5]). For instance, we present some relationships between considered families and another families investigated by Bavrin (see [1]), the growth and distortion type theorems for functions belonging to this families and the estimates of $G$-balances of homogeneous polynomials. \medskip \noindent {\bf References} \begin{itemize}{} \itemsep=0pt \item[] \hspace*{-20pt}{[1]} I. I. Bavrin, \emph{Classes of regular functions in the case of several complex variables and extremal problems in that class}, Moskov Obl. Ped. Inst., Moscov (1976), 1-99, (in Russian). \item[] \hspace*{-20pt}{[2]} R. Długosz, P. Liczberski \emph{Some results of Fekete-Szeg\"{o} type. Results for some holomorphic functions of several complex variables}, Symmetry 2020, {\bf 12}, 1707. \item[] \hspace*{-20pt}{[3]} E. Le\'{s}-Bomba, P. Liczberski, \emph{On some family of holomorphic functions of several complex variables}, Sci. Bull. Che{\l}m, Sec. Math. and Comput. Sci. \textbf{2} (2007), 7-16. \item[] \hspace*{-20pt}{[4]} A. A. Temljakov, \emph{Integral representation of functions of two complex variables}, Izv. Akad. Nauk SSSR. Ser. Mat. \textbf{21} (1957), 89-92, (in Russian). \item[] \hspace*{-20pt}{[5]} E. Trybucka, \emph{Extremal Problems of Some Family of Holomorphic Functions of Several Complex Variables}, Symmetry – Basel, Vol. \textbf{11}, Article number: 1304, 2019. \end{itemize}

Differential Geometry: Old and New (MS - ID 15)

Differential Geometry and Applications

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Non-holonomic equations for sub-Riemannian extremals and metrizable parabolic geometries

Prof. Jan Slovak, Masaryk University

Abstract

I will report several recent attempts linking sub-Riemannian geometries to the rich geometry of filtered manifolds, particularly the parabolic ones. After touching on some relation between the canonical Cartan geometries, I shall present an approach to sub-Riemannian extremals motivated by the tractor calculus. Finally, I will explore some implications of the BGG machinery to the (sub-Riemannian) metrizability of parabolic geometries. All that will be based on joined work with D. Alekseevsky, A. Medvedev, R. Gover, D. Calderbank, V. Soucek.

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Differential geometry of submanifolds in flag varieties via differential equations

Mr. Boris Doubrov, Belarusian State University

Abstract

We give a unified method for the general equivalence problem osculating embeddings \[ \varphi\colon (M,\mathfrak f) \to \operatorname{Flag}(V,\phi) \] from a filtered manifold $(M,\mathfrak f)$ to a flag variety $\operatorname{Flag}(V, \phi)$. We establish an algorithm to obtain the complete systems of invariants for the osculating maps which satisfy the reasonable regularity condition of constant symbol of type $(\mathfrak g_-, \operatorname{gr} V)$. We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weightedly) involutive systems of linear differential equations of finite type. Therefore we also obtain a complete system of invariants for a general involutive systems of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group $H^1_+(\mathfrak g_-, \mathfrak{gl}(V) / \operatorname{Prol}(\mathfrak g_-))$, which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries.

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Contact CR submanifolds in odd-dimensional spheres: new examples

Prof. Marian Ioan MUNTEANU, University Alexandru Ioan Cuza of Iasi

Abstract

The notion of $CR$-submanifold in K\"{a}hler manifolds was introduced by A. Bejancu in 70's, with the aim of unifying two existing notions, namely complex and totally real submanifolds in K\"{a}hler manifolds. Since then, the topic was rapidly developed, mainly in two directions: $\bullet$ Study $CR$-submanifolds in other almost Hermitian manifolds. $\bullet$ Find the odd dimensional analogue of $CR$-submanifolds. Thus, the notion of semi-invariant submanifold in Sasakian manifolds was introduced. Later on, the name was changed to contact CR-submanifolds. A huge interest in the last 20 years was focused on the study of $CR$-submanifolds of the nearly K\"{a}hler six dimensional unit sphere. Interesting and important properties of such submanifolds were discovered, for example, by M. Antic, M. Djoric, F. Dillen, L. Verstraelen, L. Vrancken. As the odd dimensional counterpart, contact $CR$-submanifolds in odd dimensional spheres were, recently, intensively studied. In this talk we focus on those proper contact $CR$-submanifolds, which are as closed as possible to totally geodesic ones in the seven dimensional spheres endowed with its canonical structure of a Sasakian space form. We give a complete classification for such a submanifold having dimension $4$ and describe the techniques of the study. We present also some very recent developments concerning dimension $5$ and $6$ and propose further problems in this direction. This presentation is based on some papers in collaboration with M. Djoric and L. Vrancken. {\bf Keywords:} (contact) CR-submanifold, Sasakian manifolds, minimal submanifolds, mixed and nearly totally geodesic CR-submanifolds {\bf References:} [1] M. Djoric, M.I. Munteanu, L. Vrancken: Four-dimensional contact $CR$-submanifolds in ${\mathbb{S}}^7(1)$, Math. Nachr. {\bf 290} (2017) 16, 2585--2596. \smallskip [2] M. Djoric, M.I. Munteanu, On certain contact $CR$-submanifolds in ${\mathbb{S}}^7$, Contemporary Mathematics, {\it Geometry of submanifolds}, Eds. (J. van der Veken et al.) {\bf 756} (2020) 111-120. \smallskip [3] M. Djoric, M.I. Munteanu, Five-dimensional contact $CR$-submanifolds in ${\mathbb{S}}^7(1)$, Mathematics, Special Issue {\it Riemannian Geometry of Submanifolds}, Guest Editor: Luc Vrancken, {\bf 8} (2020) 8, art. 1278.

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Cauchy-Riemann geometry of Legendrian curves in the 3-dimensional Sphere.

Prof. Emilio Musso, Politecnico di Torino

Abstract

Let $S^3$ be the unit 3-sphere with its standard Cauchy–Riemann (CR) structure. We consider the CR geometry of Legendrian curves in $S^3$, thought of as a 3-dimensional homogeneous CR manifold. We introduce the two main local invariants : a line element (the cr-infinitesimal strain) and the cr-bending. Integrating the invariant line element we get the simplest cr-invariant variational problem for Legendrian curves in $S^3$ . We discuss Liouville integrability and the existence of closed critical curves.

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Similarlity geometry revisited: Differential Geometry and CAGD

Prof. Jun-ichi Inoguchi, University of Tsukuba

Abstract

Similarity geometry is a Klein geometry whose transformation group is the similarity transformation group. The similarity transformation group is generated by Euclidean isometries and scalings. One can develop differential geometry of plane curves under similarity transformation group. In particular we obtain similarity curvature, similarity Frenet formula and fundamental theorem of plane curves in similarity geometry. On the other hand, in industrial design (CAGD), log-aesthetic curves are studied extensively. In this talk, we give a similarity geometric reformulation of log-aesthetic curves and discuss relations to curve flows derived form Burgers flows.

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On the Bishop frame of a partially null curve in Minkowski spacetime

Prof. Emilija Nešović, University of Kragujevac Faculty of Science

Abstract

The Bishop frame $\{T,N_1,N_2\}$ (relatively parallel adapted frame) of a regular curve in Euclidean space ${E}^{3}$ contains the tangent vector field $T$ of the curve and two relatively parallel vector fields $N_1$ and $N_2$ whose derivatives in arc length parameter $s$ make minimal rotations along the curve. In Minkowski spaces $E^3_1$ and $E^4_1$, the Bishop frame of a non-null curve and a null Cartan curve has analogous property. In this talk, we present a method for obtaining the Bishop frame (rotation minimizing frame) of a partially null curve $\alpha$ lying in the lightlike hyperplane of Minkowski spacetime. We show that $\alpha$ has two possible Bishop frames, one of which coincides with its Frenet frame. By using spacetime geometric algebra, we derive the Darboux bivectors of Frenet and Bishop frame and give geometric interpretation of the Frenet and the Bishop curvatures in terms of areas obtained by projecting the Darboux bivector onto a spacelike or a lightlike plane.

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Null scrolls, B-scrolls and associated evolute sets in Lorentz-Minkowski 3-space

Dr. Ljiljana Primorac Gajcic, University of Osijek

Abstract

In classical differential geometry in Euclidean space, the Bonnet’s theorem states that there are two surfaces of constant mean curvature parallel a surface of constant positive Gaussian curvature. These two constant mean curvature surfaces are so-called harmonic evolutes of each other. In this short presentation, we present results of the analogous investigation in Lorentz-Minkowski 3-space, however, restricted to the case of surfaces that have no Euclidean counterpart, the quasi-umbilical surfaces, \cite{Clelland}--\cite{Inoguchi}. These surfaces are characterized by the property that their shape operator is not diagonalizable, and they can be parametrized as null scrolls or B-scrolls, \cite{Milin}. In \cite{Lopez} we have shown that they are the only surfaces whose evolute set degenerates to a curve. The curve is of either null or spacelike causal character, and we analyse them respectively. \begin{thebibliography}{} \bibitem{Clelland} J. N. Clelland, Totally quasi-umbilical timelike surfaces in $\mathbf{R}^{1,2}$, Asian J. Math, 16(2012), 189-208 \bibitem{Duggal} K. L. Duggal and A. Bejancu, Lightlike Submanifolds of semi-Riemannian Manifolds and Applications, Mathematics and its Applications, Kluwer Academic Publishers, Dordrecht, 1996. \bibitem{Duggal2}K. L. Duggal and B. Sahin, Differential geometry of lightlike submanifolds, Birkh\"auser, Basel, 2010. \bibitem{Inoguchi} J.-I. Inoguchi and S. Lee, Null curves in Minkowski 3-space, Int. Electron. J.Geom. 1(2008), 40-83. \bibitem{Lopez}R. Lopez, \v{Z}. Milin \v{S}ipu\v{s}, Lj. Primorac Gaj\v{c}i\'{c} and I. Protrka, Harmonic Evolutes of B-Scrolls with Constant Mean Curvature in Lorentz-Minkowski Space, Int. J. Geom. Methods Mod. Phys. 2019, https://doi.org/10.1142/S0219887819500762 \bibitem{Milin} \v{Z}. Milin \v{S}ipu\v{s}, Lj. Primorac Gaj\v{c}i\'{c} and I. Protrka, Null scrolls as B-scrolls in Lorentz-Minkowski Space, Turk J Math, 43(6) (2019): 2908-2920, doi: 10.3906/mat-1904-14. \end{thebibliography}

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Simple closed geodesics on regular tetrahedra in spaces of constant curvature

Ms. Darya Sukhorebska, B.Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine (B. Verkin ILTPE of NASU)

Abstract

Since regular triangles form a regular tiling of Euclidean plane it easily follows the full classification of closed geodesics on a regular tetrahedron in Euclidean space. We described all simple (without self-intersections) closed geodesics on regular tetrahedra in three-dimensional hyperbolic and spherical spaces. In these spaces the tetrahedron's curvature is concentrated not only into its vertices but also into its faces. The value $\alpha$ of the faces’ angle for hyperbolic space’s tetrahedron satisfies $0< \alpha< \pi/3$ and for a tetrahedron in spherical space the faces’ angle is measured $ \alpha$ that $\pi/3 < \alpha < 2\pi/3$. The intrinsic geometry of such tetrahedra depends on the value of its faces’ angles. A simple closed geodesic on a tetrahedron has the type $(p,q)$ if it has $p$ points on each of two opposite edges of the tetrahedron, $q$ points on each of another two opposite edges, and there are $(p+q)$ points on each edges of the third pair of opposite one. We prove that on a regular tetrahedron in hyperbolic space for any coprime integers $(p, q)$, $0 \le p<q$, there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type $(p,q)$. These geodesics exhaust all simple closed geodesics on a regular tetrahedron in hyperbolic space. The number of simple closed geodesics of length bounded by $L$ is asymptotic to constant (depending on $\alpha$) times $L^2$, when $L$ tending to infinity \cite{BorSuh}. On a regular tetrahedron in spherical space there exists the finite number of simple closed geodesic. The length of all these geodesics is less than $2\pi$. For any coprime integers $(p, q)$ we presented the numbers $\alpha_1$ and $\alpha_2$ depending on $p$, $q$ and satisfying the inequalities $\pi/3< \alpha_1 < \alpha_2 < 2\pi/3$ such that on a regular tetrahedron in spherical space with the faces’ angle of value $\alpha \in \left( \pi/3, \alpha_1 \right)$ there exists unique, up to the rigid motion of the tetrahedron, simple closed geodesic of type $(p,q)$ and on a regular tetrahedron with the faces’ angle of value $\alpha \in \left( \alpha_2, 2\pi/3 \right)$ there is no simple closed geodesic of type $(p,q)$. \begin{thebibliography}{99} \bibitem{BorSuh} A A Borisenko, D D Sukhorebska, "Simple closed geodesics on regular tetrahedra in hyperbolic space", Mat. Sb., 2020, 211(5), p.3-30. (in Russian). {\it English translation}: SB MATH, 2020, 211(5), DOI:10.1070/SM9212 \end{thebibliography}

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New results in the study of magnetic curves in quasi-Sasakian manifolds of product type

Dr. Ana Irina Nistor, "Gheorghe Asachi" Technical University of Iasi

Abstract

This presentation is based on the joint paper with M.I. Munteanu entitled "Magnetic curves in quasi-Sasakian manifolds of product type" which was accepted for publication in "New Horizons in Differential Geometry and its Related Fields", Eds. T. Adachi and H. Hashimoto, 2021. The main result represents a positive answer to sustain our conjecture about the order of a magnetic curve in a quasi-Sasakian manifold. More precisely, we show that the magnetic curves in quasi-Sasakian manifolds, obtained as the product of a Sasakian and a K\"ahler manifold, have maximum order $5$. Next, we study the magnetic curves in $\mathbb{S}^3\times\mathbb{S}^2$. First, we find the explicit parametrizations of such curves. Then, we find a necessary and sufficient condition for a magnetic curve in $\mathbb{S}^3\times\mathbb{S}^2$ to be periodic. Finally, we conclude with some examples of magnetic curves in $\mathbb{S}^3\times\mathbb{S}^2$.

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$J$-trajectories in $\mathrm{Sol}_0^4$

Prof. Zlatko Erjavec, University of Zagreb

Abstract

$J$-trajectories are arc length parameterized curves in almost Hermitian manifold which satisfy the equation $\nabla_{\dot{\gamma}}\dot{\gamma}=q J \dot{\gamma}$. $J$-trajectories are 4-dim analogon of 3-dim magnetic trajectories, curves which satisfy the Lorentz equation $\nabla_{\dot{\gamma}}\dot{\gamma}=q \phi \dot{\gamma}.$ In this talk $J$-trajectories in the 4-dimensional solvable Lie group $\mathrm{Sol}_0^4$ are considered. Moreover, the first and the second curvature of a non-geodesic $J$-trajectory in an arbitrary 4-dimensional LCK manifold whose anti Lee field has constant length are examined. In particular, the curvatures of non-geodesic $J$-trajectories in $\mathrm{Sol}_0^4$ are characterized.

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On composition of geodesic and conformal mappings between generalized Riemannian spaces preserving certain tensors

Dr. Miloš Petrović, University of Niš, Serbia

Abstract

Recently, O. Chepurna, V. Kiosak and J. Mike\v s studied geodesic and conformal mappings betweeen two Riemannian spaces preserving the Einstein tensor and among other things proved that in that case Yano's tensor of concircular curvature is also invariant with respect to these mappings. On the other hand I. Hinterleitner and J. Mike\v s recently investigated composition of geodesic and conformal mappings between Riemannian spaces that is at the same time harmonic. In the present paper we connect these results and consider it in the settings of generalized Riemannian spaces in Eisenhart's sense.

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Characterization of manifolds of constant curvature by spherical curves and ruled surfaces

Dr. Luiz C. B. da Silva, Weizmann Institute of Science

Abstract

Space forms, i.e., Riemannian manifolds of constant sectional curvature, play a prominent role in geometry and an important problem consists of finding properties that characterize them. In this talk, we report results from [1], where we show that the validity of some theorems concerning curves and surfaces can be used for this purpose. For example, it is known that the so-called rotation minimizing (RM) frames allow for a characterization of geodesic spherical curves in Euclidean, hyperbolic, and spherical spaces through a linear equation involving the coefficients that dictate the RM frame motion [2]. Here, we shall prove the converse, i.e., if all geodesic spherical curves on a manifold are characterized by a certain linear equation, then all the geodesic spheres with a sufficiently small radius are totally umbilical, and consequently, the ambient manifold is a space form. (We also present an alternative proof, in terms of RM frames, for space forms as the only manifolds where all geodesic spheres are totally umbilical [3].) In addition, we furnish two other characterizations in terms of (i) an inequality involving the mean curvature of a geodesic sphere and the curvature function of their curves and (ii) the vanishing of the total torsion of closed spherical curves in the case of 3d manifolds. (These are the converse of previous results [4].) Finally, we introduce ruled surfaces and show that if all extrinsically flat surfaces in a 3d manifold are ruled, then the manifold is a space form. \newline \newline {[1] Da Silva, L.C.B. and Da Silva, J.D.: ``Characterization of manifolds of constant curvature by spherical curves". Annali di Matematica \textbf{199}, 217 (2020); Da Silva, L.C.B. and Da Silva, J.D.: ``Ruled and extrinsically flat surfaces in three-dimensional manifolds of constant curvature". Unpublished manuscript 2021. \newline [2] Bishop, R.L.: ``There is more than one way to frame a curve". Am. Math. Mon. \textbf{82}, 246 (1975); Da Silva, L.C.B. and Da Silva, J.D.: ``Characterization of curves that lie on a geodesic sphere or on a totally geodesic hypersurface in a hyperbolic space or in a sphere". Mediterr. J. Math. \textbf{15}, 70 (2018) \newline [3] Kulkarni, R.S.: ``A finite version of Schur’s theorem". Proc. Am. Math. Soc. \textbf{53}, 440 (1975); Vanhecke, L. and Willmore, T.J.: ``Jacobi fields and geodesic spheres". Proc. R. Soc. Edinb. A \textbf{82}, 233 (1979); Chen, B.Y. and Vanhecke, L.: ``Differential geometry of geodesic spheres". J. Reine Angew. Math. \textbf{325}, 28 (1981). \newline [4] Baek, J., Kim, D.S., and Kim, Y.H.: ``A characterization of the unit sphere". Am. Math. Mon. \textbf{110}, 830 (2003); Pansonato, C.C. and Costa, S.I.R., ``Total torsion of curves in three-dimensional manifolds". Geom. Dedicata \textbf{136}, 111 (2008).}

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Topologically Embedded Pseudospherical Surfaces

Prof. Lorenzo Nicolodi, Università di Parma

Abstract

It is known that the class of traveling wave solutions of the sine-Gordon equation is in 1-1 correspondence with the class of (necessarily singular) pseudospherical helicoids, i.e., pseudospherical surfaces in Euclidean space with screw-motion symmetry. We illustrate our solution to the problem of explicitly describing all pseudospherical helicoids posed by A. Popov in [Lobachevsky Geometry and Modern Nonlinear Problems, Birkh\"auser, Cham, 2014]. As an application, countably many continuous families of topologically embedded pseudospherical helicoids are constructed. This is joint work with Emilio Musso.

Differential equations, dynamical systems and applications (MS - ID 52)

Analysis and its Applications

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On the mixed Cauchy-Dirichlet problem for equations with fractional time derivative

Prof. Davide Guidetti, Università di Bologna

Abstract

We consider first maximal regularity results for linear equations. In order to be applicable to fully nonlinear equations, we need to work in spaces of little Hoelder continuous functions. We extend to spaces of this type a recent theorem of the author in spaces of Hoelder continuous functions. Next, we prove the existence of a unique “maximal solution” in a suitable sense for fully nonlinear equations.

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A boundary value problem for system of differential equations with piecewise-constant argument of generalized type

Prof. Anar Assanova, Institute of mathematics and mathematical modeling

Abstract

In present communication, at the interval $[0, T]$ we consider the two-point boundary value problem for the system of differential equations with piecewise-constant argument of generalized type $$ \frac{dx}{dt}= A(t)x(t) + D(t)x(\gamma(t)) + f(t), \eqno (1)$$ $$ Bx(0) + C x(T)=d, \qquad d \in R^n, \eqno (2)$$ where $x(t)=col(x_1(t),x_2(t),...,x_n(t))$ is unknown vector function, the $(n\times n)$ matrices $A(t)$, $D(t)$ and $n$ vector function $f(t)$ are continuous on $ [0,T]$; $\gamma(t)=\zeta_j\ $ if $\ t\in [\theta_j, \theta_{j+1})$, $\ j=\overline{0,N-1}$; $\ \theta_j \leq \zeta_j\leq \theta_{j+1}\ $ for all $\ j=0,1,...,N-1$; $ 0=\theta_0<\theta_1 <...<\theta_{N-1}<\theta_N=T$; $ B$, $C$ are constant matrices, and $d$ is constant vector. A solution to problem (1), (2) is a vector function $ x(t)$ is continuously differentiable on $ [0,T]$, it satisfies the system (1) and boundary condition (2). Differential equations with piecewise-constant argument of generalized type (DEPCAG) are more suitable for modeling and solving various application problems, including areas of neural networks, discontinuous dynamical systems, hybrid systems, etc.[1-2]. To date, the theory of DEPCAG on the entire axis has been developed and their applications have been implemented. However, the question of solvability of boundary value problems for systems of DEPCAG on a finite interval still remains open. We are investigated the questions of existence and uniqueness of the solution to problem (1), (2). The parametrization method [3] is applied for the solving to problem (1), (2) and based on the construction and solving system of linear algebraic equations in arbitrary vectors of new general solutions [4]. By introducing additional parameters we reduce the original problem (1), (2) to an equivalent boundary value problem for system of differential equations with parameters. The algorithm is offered of findings of approximate solution studying problem also it is proved its convergence. Conditions of unique solvability to problem (1), (2) are established in the terms of initial data. The results can be used in the numerical solving of application problems. \vskip 0.1cm {\bf Acknowledgement.} The work is partially supported by grant of the Ministry of Education and Science of the Republic of Kazakhstan No AP 08855726. \vskip 0.1cm %\noindent {\centerline {\bf References}} 1. {\it Akhmet M.~U.} On the reduction principle for differential equations with piecewise-constant argument of generalized type, {\it J. Math. Anal. Appl.}, \textbf{336} (2007), 646-663. 2. {\it Akhmet M.~U.} Integral manifolds of differential equations with piecewise-constant argument of generalized type, {\it Nonlinear Analysis}, \textbf{66} (2007), 367-383. 3. {\it Dzhumabayev D.~S.} Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation, {\it U.S.S.R. Comput. Math. and Math. Phys.}, \textbf{29} (1989), 34-46. 4. {\it Dzhumabaev D.~S.} New general solutions to linear Fredholm integro-differential equations and their applications on solving the boundary value problems. {\it J. Comput. Appl. Math.}, \textbf{336} (2018), 79-108.

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Generalized concepts of almost periodicity

Dr. Daniel Velinov, Ss. Cyril and Methodius University,

Abstract

The main subject of this talk are semi-Bloch $(p,k)$-periodic functions and sequence and weighted almost $(\omega,c)$-pseudo periodic functions as a generalization of most of the all previously developed concepts in theory of almost periodic functions. A various types of weighted almost $(\omega,c)$-pseudo periodic composition results are given. Also, a qualitative analysis in this context of a semilinear (fractional) Cauchy inclusions is provided.

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Decay estimates for parabolic equations with inhomogeneous density in manifolds

Prof. Daniele Andreucci, Sapienza Università di Roma

Abstract

We consider the problem \begin{alignat}{2} \label{eq:pde}%1.1 \rho(x)u_{t} &= \Delta_{p,m}(u) \,, &\qquad& x\in M\,, t>0\,, \\ \label{eq:init}%1.2 u(x,0)&=u_{0}(x)\ge0 \,, &\qquad& x\in M\,. \end{alignat} in a suitable Riemannian manifold $M$, where $\Delta_{p,m}$ is the suitable analogue of the doubly nonlinear elliptic operator and $\rho$ is a density function vanishing at infinity. We obtain the asymptotic decay rate for positive solutions. We also discuss in this talk optimal Sobolev and Hardy inequalities in $M$ under assumptions on the isoperimetric profile of the manifold.

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On Muirhead and Schur inequalities

Prof. Abdulla Azamov, Institute ofMathematics of Uzbekistan Academy of Science

Abstract

It is considered three and four term generalizations of the well-known Muirhead inequality. Let positive integers $n$ and $m$ be given, $n \geqslant 2,\,\,m \geqslant 1.$ A vector $\boldsymbol{\alpha }= ({\alpha _1},\,{\alpha _2},\dots,\,{\alpha _n})$ with nonnegative integer components is called a power if ${\alpha _1} \geqslant \,{\alpha _2} \geqslant \dots \geqslant \,{\alpha _n}$ and ${\alpha _1} + \,{\alpha _2} + \dots+\,{\alpha _n} = m.$ The set of all the powers will be denoted by $A(n,m)$. Then $${\mu _{\boldsymbol{\alpha}} }\left(\boldsymbol{x}\right) = \frac{1}{{n!}}\sum\limits_{\sigma \in S_n {} } {x_{{\sigma _1}}^{{\alpha _1}}x_{{\sigma _2}}^{{\alpha _2}}\dots\,x_{{\sigma _n}}^{{\alpha _n}}} $$ is an elementary homogeneous symmetric polynomial of the vector $\boldsymbol{x} = ({x_1},\,{x_2},\dots,\,{x_n}) \in \mathbb{R}_ + ^n$ (where $S_n$ is a family of all permutations of the set $\{1,\,2,\dots,\,n\}$). The family $A(n,m)$ will be considered with the order $\boldsymbol{\alpha} \leqslant \boldsymbol{\beta },$ defined by the relations $$\begin{gathered} {\alpha _1} \leqslant {\beta _1}, \hfill \\ {\alpha _1} + {\alpha _2} \leqslant {\beta _1} + {\beta _2}, \hfill \\ ....................... \hfill \\ {\alpha _1} + {\alpha _2} + \dots + {\alpha _{n - 1}} \leqslant {\beta _1} + {\beta _2} + \dots + {\beta _{n - 1}}. \hfill \\ \end{gathered}$$ \noindent The following statements hold: \noindent a) $\left[ {\forall\boldsymbol{x} \in \mathbb{R}_ + ^n\,:\,\,{\mu _{\boldsymbol{\alpha}} }\left(\boldsymbol{x}\right) \leqslant {\mu _{\boldsymbol{\beta}} }\left(\boldsymbol{x}\right)} \right] \Leftrightarrow \boldsymbol{\alpha} \leqslant \boldsymbol{\beta} $ (Muirhead inequality) [1]; \noindent b) for $n = 3$ $\left( \alpha ,\,\beta \in {\mathbb{N}^{+} } \right)\,$ $${\mu _{(\alpha + 2\beta ,\,0,\,\,0)}}\left(\boldsymbol{x}\right) + {\mu _{(\alpha ,\,\beta ,\,\,\beta )}}\left(\boldsymbol{x}\right) \geqslant 2{\mu _{(\alpha + \beta ,\,\,\beta ,\,0)}}\left(\boldsymbol{x}\right)$$ (Schur inequality [2]). \textbf{Theorem.} Let $\alpha ,\,\beta ,\,\gamma \in {\mathbb{N} ^+ }.$ 1. If $\alpha + \gamma \geqslant 2\beta $ then $\,{\mu _\alpha }\left(\boldsymbol{x}\right) + {\mu _\gamma }\left(\boldsymbol{x}\right) \geqslant 2 {\mu _\beta }\left(\boldsymbol{x}\right).$ 2. For $n = 2$, $${\mu _{(m,\,\,m - \alpha )}}\left(\boldsymbol{x}\right) + {\mu _{(m,\,\,m - \gamma )}}\left(\boldsymbol{x}\right) \geqslant 2{\mu _{(m,\,\,m - \beta )}}\left(\boldsymbol{x}\right)$$ \noindent if and only if $\beta \left( {m - \beta } \right) \geqslant \left( {\beta - \gamma } \right)\left( {\beta + \gamma - m} \right).$ 3. For $n = 3$, $${\mu _{(\alpha + 2\beta + 2\gamma ,\,0,\,\,0)}}\left(\boldsymbol{x}\right) + {\mu _{(\alpha ,\,2\beta ,\,2\gamma )}}\left(\boldsymbol{x}\right) \geqslant 2{\mu _{(\alpha + \beta + \gamma ,\,\beta + \,\gamma ,\,0)}}\left(\boldsymbol{x}\right).$$ \begin{center} \textbf{References} \end{center} \noindent 1. \textbf{G.H.Hardy, J.E.Littlewood, G.P\'olya} (1952). Inequalities. Cambridge University Press. pp.324. \noindent 2. \textbf{Finta, B\'ela } (2015). A Schur Type Inequality for Five Variables. Procedia Technology. 19: 799-801. Doi:10.1016/j.protcy.2015.02.114.

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Invariants of group representations, dimension/degree duality and normal forms of vector fields

Prof. Henryk Zoladek, University of Warsaw

Abstract

We develop a constructive approach to the problem of polynomial first integrals for linear vector fi elds. As an application we obtain a new proof of the theorem of Wietzenbock about fi niteness of the number of generators of the ring of constants of a linear derivation in the polynomial ring. Moreover, we propose an alternative approach to the analyticity property of the normal form reduction of a germ of vector eld with nilpotent linear part in a case considered by Stolovich and Verstringe.

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Aspects of nonlinear differential equations

Dr. Galina Filipuk, University of Warsaw

Abstract

Painlev\'e equations appear in many applications of mathematics and mathematical physics. In this talk I shall give an overview how recurrence coeffi cients of semi-classical orthogonal polynomials are related to solutions of the Painlev\'e equations. I shall also show how discrete systems for the recurrence coefficients are related to B\"acklund transformations of the Painlev\'e equations.

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Modelling instability via a generalized Rulkov map: bursts, synchronization and chaos regularization in financial markets

Dr. Michele Bufalo, University of Bari

Abstract

The Rulkov map has been successfully employed for describing bursts, mutual synchronization and chaos regularization in biology. Then, its applications have been extended to physics, engineering, medicine, etc. In this work we find an application of a generalized Rulkov map to finance where the impact of COVID-19 can be clearly seen in the considered dataset. We examine the Financial Stress Index as well as a number of time series diversified by asset classes (swaps, equity and bonds), market (emerging vs developed), issuer (corporate vs government bond), maturity (short vs long). We find that a calibration of a single Rulkov map or of coupled Rulkov maps could describe the alternation between calm periods and financial turmoil (including a black swan) as well the synchronizing mechanism operating across markets. This result is compared to the so-called Naive and the ARIMA-GARCH model to determine how a chaotic deterministic model stands with respect first to a simple model and, second, to an advanced stochastic model expressly designed for handling moving average, autoregression, cointegration and heteroscedastic volatility.\\ Finally, we give some theoretical considerations about the stability of the nonlinear system described by our generalized Rulkov map.

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Non-linear convecting radiating fins: solutions, efficiency, entropy.

Prof. Federico Zullo, Università di Brescia

Abstract

The second order, nonlinear, ordinary differential equation describing the dissipation of heat in a convecting-radiating longitudinal fin is analyzed. The profile of the fin is arbitrary. With the introduction of an auxiliry variable, we obtain new solutions for the linear, purely convective, case and explicit solutions for the non-linear convecting-radiating case. The efficiency of the fin is discussed in both cases and it is compared with a novel introductiuon of the efficiency based on the entropy rates of the convecting-radiating processes.

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Ultracontractivity for p-fractional Robin-Venttsel' problems in extension domains

Dr. Simone Creo, Sapienza Università di Roma

Abstract

In this talk we consider a Robin-Venttsel' problem for the regional fractional p-Laplace operator on an extension domain $\Omega$; such domains can have a highly irregular boundary, for example of fractal type.\\ We first consider the parabolic problem and, by using nonlinear semigroup theory, we prove that it admits a unique weak solution. We then prove that the nonlinear semigroup associated to this problem is ultracontractive; this is achieved by means of a fractional logarithmic Sobolev inequality, adapted to the problem at hand.\\ We also consider the elliptic problem and we prove that it admits a unique bounded weak solution.\\ These results are obtained in collaboration with M. R. Lancia.

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Integrability and asymptotic behaviour of a matrix lattice equation

Dr. Pilar R Gordoa, Universidad Rey Juan Carlos

Abstract

In this talk we consider a matrix lattice equation, both in its autonomous and nonautonomous versions, and show integrability in both cases. Moreover, we explore the construction of Miura maps which relate these two lattices, though intermediate integrable lattice equations, to matrix analogues of autonomous and nonautonomous Volterra equations but in two matrix dependent variables which, in general, are subject to consistency conditions. For these last Volterra-type systems we consider certain special classes of matrices and derive coupled systems of integrable equations. We also consider asymptotic reductions to matrix partial differential equations. In addition, in the nonautonomous case, we construct a new nonisospectral matrix Volterra system together with its corresponding Lax pair. It is also interesting that in the nonautonomous case, the relation obtained between certain lattices even in the scalar case seems to be new.

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On classes of solutions of the matrix second Painlev\'e hierarchy

Dr. Andrew Pickering, Universidad Rey Juan Carlos

Abstract

We consider the iterative construction of solutions of members of the matrix second Painlev\'e hierarchy. We use compositions of auto-B\"acklund transformations applied to certain classes of initial solution. Results for other related matrix equations are also briefly discussed.

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Evolution problems with dynamical boundary conditions of Wentzell-type

Prof. Silvia Romanelli, Universita' degli Studi di Bari Aldo Moro (retired full professor)

Abstract

Different classes of evolution equations with dynamical boundary conditions will be considered. We will derive analyticity results for the operator semigroups generated by suitable uniformly elliptic operators with general Wentzell boundary conditions.

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Evolution problems for fractional operators in irregular domains

Prof. Maria Rosaria Lancia, Sapienza

Abstract

We consider nonlocal diffusion processes in non smooth domains of fractal type as well as in the corresponding smoother approximating domains. Existence, uniqueness and regularity issues will be discussed. The asymptotic behaviour of the smoother solutions, if any, will be discussed. These results are contained in joint papers with S.Creo and P. Vernole

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Thermoelastic Bresse system with dual-phase-lag model

Dr. Ivana Bochicchio, Università di Salerno

Abstract

The thermoelastic system modeling longitudinal, vertical and angular motion is the so-called Bresse system. It describes the behavior of a thin curved beam and it was introduced in 1856 by Bresse. The Bresse system generalizes the well-known Timoshenko model, obtained in the particular case where the longitudinal displacement is not considered and supposing zero initial curvature. The Bresse model becomes more interesting when also the thermal case is taken into account. It is commonly accepted that the classical heat conduction theory based on the Fourier law implies the fact that the thermal perturbations at any point of the body are felt instantly anywhere. So this work is devoted to analyze a non isothermal Bresse system where the dual-phase-lag heat conduction theory is used to model the heat transfer. In particular, setting the equations in an abstract framework, an existence and uniqueness result is obtained by using the theory of linear semi groups. Furthermore, the polynomial stability and the exponential energy decay are investigated.

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Materials with memory: an overview on admissible kernels in the integro-differential model equations

Prof. Sandra Carillo, Universita` "La Sapienza", Rome, Italy

Abstract

Materials with memory are modelled via integro-differential equations in which the integral term is introduced to take into account the past {\it history} of the material. The {\it classical} regularity requirements the kernel, which in viscoelasticity represents the relaxation function, is assumed to satisfy are considered. Aiming to model wider classes of materials, less restrictive assumptions are adopted. The results presented are part of a joint research program with M. Chipot, V. Valente and G. Vergara Caffarelli.

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Fractional Cauchy problem on random snowflakes

Prof. Raffaela Capitanelli, SAPIENZA-ROMA

Abstract

We consider random snowflakes which are domains whose boundary is constructed by mixtures of Koch curves with random scales. These domains are obtained as limit of domains with Lipschitz boundary, whereas for the limit object, the fractal given by the random Koch domain, the boundary has Hausdorff dimension between 1 and 2. We study time fractional Cauchy problems on the pre-fractal boundary and we prove asymptotic results for the corresponding fractional diffusions.

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Analytic properties of the complete formal normal form for the Bogdanov--Takens singularity

Dr. Ewa Stróżyna, Warsaw University of Technology

Abstract

%%%%%%%%%%%%%%%%%% ABSTRACT %%%%%%%%%%%%%%%%%%%%%%%%%% In our previous paper a complete formal normal forms for germs of 2--dimensional holomorphic vector fields with nilpotent singularity was obtained. That classification is quite nontrivial (7 cases), but it can be divided into general types like in the case of the elementary singularities. One could expect that also the analytic properties of the normal forms for the nilpotent singularities are analogous to the case of the elementary singularities. This is really true. In the cases analogous to the focus and the node the normal form is analytic. In the case analogous to the nonresonant saddle the normal form is often nonanalytic due to the small divisors phenomenon. In the cases analogous to the resonant saddles (including saddle--nodes) the normal form is nonanalytic due to properties of some homological operators associated with the first nontrivial term in the orbital normal form.

EU-MATHS-IN: mathematics for industry in Europe (MS - ID 66)

Numerical Analysis and Scientific Computing

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An overview of EU-MATHS-IN and its activities

Prof. Wil Schilders, TU Eindhoven

Abstract

EU-MATHS-IN, the European Service Network of Mathematics for Industry and Innovation, was established in November 2013 by the European Mathematical Society (EMS) and the European Consortium of Mathematics for Industry (ECMI). It is a unique network of national networks, currently with 21 countries on board. Since its foundation, EU-MATHS-IN has been active to spread best practices, organise events and issue reports. It has set up close contacts with the EU. In 2017, an Industrial Core Committee was established which has representatives of important industries in Europe. In this talk, we will go into more detail concerning the activities of EU-MATHS-IN, as well as on future plans.

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An overview of the transfer activity in Spain. Coordination with EU-MATHS-IN initiatives

Prof. Peregrina Quintela Estévez, Universidade de Santiago de Compostela / Spanish Network for Mathematics & Industry

Abstract

In this talk, the most relevant aspects of the transfer of Mathematics in Spain will be presented. In particular, the detail of the initiatives carried out to promote the transfer, some indicators on the impact of this activity in the Spanish productive landscape, as well as some relevant success stories of collaboration Academy - Industry will be introduced. Attention will also be paid to the coordination of math-in activities with those promoted by EU-MATHS-IN in order to multiply the potential of both structures facilitating transfer.

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An overview of the activities of the french network for mathematics and entreprises, France

Ms. Maume-Deschamps Véronique, AMIES

Abstract

From 2011, AMIES (French Agency for Mathematics in Interaction with Enterprises and the Society) acts to developp and make more visible the collaborations between mathematical research and companies in France. In the context of the digital transformation of companies and Industry 4.0' revolution, AMIES' provides financial support and environment to promote industrial mathematics. We have identified more than 600 French companies calling upon French mathematical research, of which nearly 300 have been supported by AMIES funding. During the talk, we will present the mechanisms put in place by AMIES and some examples of collaborations and mathematical services for companies.

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Quantitative Comparison of Deep Learning-Based Image Reconstruction Methods for Low-Dose and Sparse-Angle CT Applications

Mr. Johannes Leuschner, University of Bremen

Abstract

Over the last years, deep learning methods have significantly pushed the state-of-the-art results in applications like imaging, speech recognition and time series forecasting. This development also starts to apply to the field of computed tomography (CT). In medical CT, one of the main goals lies in the reduction of the potentially harmful radiation dose a patient is exposed to during the scan. Depending on the reduction strategy, such low-dose measurements can be more noisy or starkly under-sampled. Hence, achieving high quality reconstructions with classical methods like filtered back-projection (FBP) can be challenging, which motivated the investigation of a number of deep learning approaches for this task. With the purpose of comparing such approaches fairly, we evaluate their performance on fixed benchmark setups. In particular, two CT applications are considered, for both of which large datasets of 2D training images and corresponding simulated projection data are publicly available: a) reconstruction of human chest CT images from low-intensity data, and b) reconstruction of apple CT images from sparse-angle data. In order to include a large variety of methods in the comparison, we organized open challenges for either task. Our current study comprises results obtained with popular deep learning approaches from various categories, like post-processing, learned iterative schemes and fully learned inversion. The test covers image quality of the reconstructions, but also aspects such as the required data or model knowledge and generalizability to other setups are considered. For reproducibility, both source code and reconstructed images are made publicly available.

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MACSI and industrial mathematics in Ireland

Dr. Kevin Moroney, University of Limerick

Abstract

MACSI was established on foot of a Science Foundation Ireland grant so-called mathematics initiative grant in 2006. As a result, a fulltime business manager was employed, an industrial consultancy was established, and European study groups were brought to Ireland for the first time. Building on this, we have recently established the first Centre for Research Training in mathematics in the country (also supported by Science Foundation Ireland).

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Overview of HU-MATHS-IN, the Hungarian Service Network of Mathematics for Industry and Innovation

Prof. Zoltán Horváth, Széchenyi István University

Abstract

Inspired by the creation of EU-MATHS-IN, HU-MATHS-IN was established in 2013. Initially, the activities of the Hungarian network were based on the decades-long industrial collaborations of the national mathematical research groups. Since 2017 its activities are intensified by an EU and Hungary-supported project (EFOP 3.6.2-16-2017-00015) in which best practices of the EU-MATHS-IN national networks were analysed and some of them adapted. The main results of the project include, among others, the following ones:\\ \begin{itemize} \item more than 50 short-term aimed basic research projects with industry, \item two collaborative projects of the HU-MATHS-IN research groups, \item national one-stop-shop for services to industry with mathematical technologies. \end{itemize} HU-MATHS-IN has been participating in the EU-MATHS-IN activities strongly, e.g. with succesfull research and innovation actions projects within the H2020 framework program. In this talk an overview of the organization of the network and its activities will be given. In addition, some short-term industrial projects of HU-MATHS-IN will be overviewed from the fields of computational acoustics and model order reduction for compressible fluids, with mathematical details.

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Optimizing the routes of mobile agents in a network using a multiobjective travelling salesman model

Prof. José Luis Santos, University of Coimbra

Abstract

This work is a result of the consortium formed by the company Smartgeo Solutions, Lda., in the role of leading developer, and by the University of Coimbra. The main goal of the project was the development of an innovative Geographic Information System application, which operates on a Web platform. This application should incorporate a functionality that allows to optimise, according to multiple criteria, the routes of mobile agents in a network. The problem was modelled using a multicriteria version of the travelling salesman problem to obtain the efficient routes. Several new heuristics were proposed whose performances were evaluated and compared with classical approaches. It was also proposed a new dominance criteria relation which allows to speed up the algorithms to search efficient solutions. The proposed algorithms allowed us to explore new solutions that had not been found with classic ones.

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Hydrodynamic load on coupled “ship” – “breast dolphin” system using a conformal mapping approach

Dr. Aleksander Grm, Univesity of Ljubljana

Abstract

When a ship docks at a breast dolphin structure, such as a single pile flexible dolphin, one question remains unanswered: How do the hydrodynamic loads of the ships contribute to the deformation of the dolphins? One approach is the application of the potential flow theory and the section-by-section approximation of the ship geometry. In the literature, the method is usually referred to as "2.5D flow theory". The hydrodynamic loads of each section can now be calculated using the conformal mapping approach from circular to ship cross-section geometry. The proposed method is an improvement over Ursell's method, which uses a circular cylinder. We will show how a method is derived and applied to the case of a tanker berthing at the liquid jetty in the port of Koper. The results will discuss the dynamics of a coupled system.

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Simulation, optimal management and infrastructure planning of gas transmission networks

Dr. Ángel Manuel Rueda, University of A Coruña

Abstract

In this talk we plan to present the results of an ongoing collaboration with a Spanish company in the gas industry, for which we have developed a software, GANESO, that simulates and optimizes gas transmission networks. A gas transmission network consists basically of emission and consumption points, compressors and valves that are connected via pipes. The natural gas flows along the pipes, but the friction with the walls of the pipes decreases the pressure of the gas. At the demand points, the gas has to be delivered with a certain pressure, so it is necessary to counterbalance the pressure loss in the pipes using compressor stations. Yet, compressor stations operate consuming part of the gas that flows through the pipes, so it is important to manage the gas network in an efficient way to minimize such consumption. In order to tackle this problem in the steady-state setting, the methodology we have followed consists of implementing a slight variation of the standard sequential linear programming algorithms that can easily accommodate integer variables, combined with a control theory approach. Further, we have developed a simulator for the transient case, based on well-balanced finite volume methods for general flows and finite element methods for isothermal models. It is worth mentioning that we have also recently added two new features to GANESO in order to deal with energy coupling issues. On the one hand, GANESO was extended to be able to manage heterogeneous gas mixtures in the same network, including Hydrogen-rich mixtures. On the other hand, in order to integrate gas and electricity energy systems, a new simulation/optimization framework was developed for assessing the interdependency of both networks guaranteeing the security of supply of the whole system. In our collaboration we have also studied other related problems to gas networks, such as the allocation of gas losses, the infrastructure planning under uncertainty and the computation of tariffs for networks access according to the different methodologies proposed in EU directives. The developed software, along with the support and consulting analysis provided by the research group, has allowed the company to improve its strategic positioning in the gas sector.

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Suboptimal scheduling of a fleet of AGVs to serve online requests

Mr. Markó Horváth, Institute for Computer Science and Control

Abstract

In the talk we consider an online problem in which a fleet of vehicles must serve transportation requests arriving over time. We propose a dynamic scheduling strategy, which continuously updates the running schedule each time a new request arrives, or a vehicle completes a request. The vehicles move on the edges of a mixed graph modeling the transportation network of a workshop. Our strategy is complete in the sense that deadlock avoidence is guaranteed, and all requests get served. The primary objective is to minimize the total tardiness of serving the requests, and the secondary objective is to minimize the total distance traveled. We provide qualitative results and a comparison to another complete strategy from the literature. The main novelty of our approach is that we optimize the schedule to get better results, and we measure solution quality which is very rare in the AGV literature.

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Combining advanced mathematical methods, IoT and High-Performance Computing to optimize energy in existing buildings

Prof. Christophe Prud'homme, Université de Strasbourg

Abstract

At a time of mass awareness of the impacts of climate change, mainly due to the increase in carbon emissions, reducing and controlling energy consumption are major challenges for the future. Taking up this challenge will help to contain the runaway rise in climate change. This partnership between Cemosis, the platform for collaboration between mathematics and industry at University of Strasbourg, and the company Synapse-Concept will speed up the identification of sources of energy savings in existing buildings by means of clustered calculations and will enable the effectiveness of the technical solutions proposed by architects and engineers to be virtually tested before any improvement work is undertaken. In this talk, we present an overview of the mathematical and computational framework --- coupling modeling simulation data assimilation, reduced order modeling and high performance computing --- as well as some numerical results \vspace*{16pt} {\bf Acknowledgments}~ The authors are grateful for the support of the Region Grand Est and the Agency for Mathematics in Interaction with Enterprises and Society(AMIES).

Energy methods and their applications in material science (MS - ID 25)

Analysis and its Applications

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Fracture in random heterogeneous particle systems

Prof. Anja Schlömerkemper, University of Würzburg

Abstract

To understand the onset of fracture better, we study energy functionals for one-dimensional heterogenous particle systems with convex-concave interaction potentials that allow for fracture. We provide a notion of fracture in the discrete system and show that - in the continuum limit - this yields the same onset of fracture as corresponding energy functionals that are obtained by $\Gamma$-convergence methods.

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A derivation of Griffith functionals from discrete finite-difference models

Prof. Francesco Solombrino, Università di Napoli "Federico II"

Abstract

We analyse a finite difference approximation of an Ambrosio-Tortorelli-like functional in brittle fracture, in the discrete-to-continuum limit. In a suitable regime between the competing scales, namely if the discretization step is smaller than the ellipticity parameter, we show the $\Gamma$-convergence of the model to the Griffith functional. The limit analysis combines a blow-up procedure with suitable adaptions to the discrete setting of recent compactness and lower semicontinuity results in spaces of generalised functions of bounded deformation. In particular, we do not need to add any artificial $L^p$ fidelity term, which would be unnatural in fracture mechanics.

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Higher order far-field developments around lattice defects

Mr. Julian Braun, University of Warwick

Abstract

We examine the elastic deformation induced by lattice defects in crystalline materials. The deformation is given as a (local) minimizer of an atomistic energy. We give a novel far-field development of such minimizers based on continuum PDEs and multi-pol terms. This expansion is given up to arbitrary high-order. We also showcase how to use this expansion for very efficient atomistic simulations of defects.

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Variational Modeling of Paperboard Delamination under Bending

Prof. Patrick Dondl, Albert-Ludwigs-Universität Freiburg

Abstract

Paperboard is an engineering material consisting of a number of separate sheets of paper, that have been bonded together. Experimental evidence shows that paperboard undergoing bending develops phenomenologically plastic hinges. We consider a nonlinearly elastic mathematical model for paperboard, allowing debonding of the sheets at a given cost per unit area. Analysis of our model predicts a number of different regimes, including some where bending is concentrated in delaminated hinges, where the mid-plane of each individual layer may deform isometrically. This is joint work with Sergio Conti (Bonn) and Julia Orlik (Kaiserslautern).

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Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films

Dr. Marco Morandotti, Politecnico di Torino

Abstract

A one-dimensional evolution equation including a transport term is considered; it models a process of thin films deposition. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed-point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges to a purely diffusive Cahn-Hilliard equation. This is joint work with Marco Bonacini and Elisa Davoli.

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Geometric linearization of theories for incompressible elastic materials and applications

Prof. Bernd Schmidt, Universitaet Augsburg, Germany

Abstract

We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The resulting limiting theory is geometrically linearized in the sense that it acts on infinitesimal displacements rather than finite deformations, but will in general still have a limiting stored energy density that depends in a nonlinear way on the infinitesimal strains. Our results, in particular, establish a rigorous link of existing finite and infinitesimal theories for incompressible nematic elastomers.

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Asymptotic analysis of rigidity constraints modeling fiber-reinforced composites

Dr. Antonella Ritorto, Katholische Universität Eichstätt-Ingolstadt

Abstract

In this talk, we discuss recent analytical results describing the effective deformation behavior of composite materials. Our model involves a rigidity constraint acting on long fibers that are semi-periodically distributed in a material body. The main result characterizes the weak limits of sequences of Sobolev maps whose gradients on the fibers correspond to rotations. Such limits exhibit a restrictive anisotropic geometry in the sense that they locally preserve the length in the fiber direction. We illustrate the results with examples of attainable macroscopic deformations. Under additional regularization in terms of second-order derivatives in the directions orthogonal to the fibers, the model is less flexible, and only rigid body motions can occur. Joint work with Dominik Engl (Utrecht University) and Carolin Kreisbeck (KU Eichst\"att-Ingolstadt).

Extremal and Probabilistic Combinatorics (MS - ID 20)

Combinatorics and Discrete Mathematics

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Recent advances in Ramsey theory

Prof. Dhruv Mubayi, University of Illinois at Chicago

Abstract

For many decades, randomness has been the central idea to address the question of determining growth rates of graph Ramsey numbers. Recently, we proved a theorem which suggests that ``pseudo-randomness” and not complete randomness may in fact be a more important concept for this area. Consequently, we reduce one of the main open problems in graph Ramsey theory to the possibly simpler question of constructing certain pseudorandom graphs. This new connection widens the possibility to use tools from algebra, geometry, and number theory to address the fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.

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On Hadwiger's Conjecture

Dr. Luke Postle, University of Waterloo

Abstract

We discuss recent progress on Hadwiger's conjecture. In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. Recently, Norin, Song and I showed that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the $O(t\sqrt{\log t})$ bound. Here we show that every graph with no $K_t$ minor is $O(t (\log t)^{\beta})$-colorable for every $\beta > 0$; more specifically, they are $O(t (\log \log t)^{6})$-colorable.

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Progress towards Nash-Williams' Conjecture on Triangle Decompositions

Dr. Michelle Delcourt, Ryerson University

Abstract

Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on n vertices with minimum degree at least 0.75 n admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely one. We show that for any graph on n vertices with minimum degree at least 0.827327 n admits a fractional triangle decomposition. Combined with results of Barber, K\"{u}hn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on n vertices with minimum degree at least 0.82733 n admits a triangle decomposition. This is a significant improvement over the previous asymptotic result of Dross showing the existence of fractional triangle decompositions of sufficiently large graphs with minimum degree more than 0.9 n. This is joint work with Luke Postle.

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Erdos-Rademacher Problem

Prof. Oleg Pikhurko, University of Warwick

Abstract

We will give an overview of recent results on the Erdos-Rademacher Problem which asks for the minimum number of r-cliques that a graph with n vertices and m edges can have.

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On a problem of M. Talagrand

Dr. Jinyoung Park, Institute for Advanced Study

Abstract

We will discuss some special cases of a conjecture of M.~Talagrand relating two notions of ``threshold'' for an increasing family $\mathcal F$ of subsets of a finite set $X$. The full conjecture implies equivalence of the ``Fractional Expectation-Threshold Conjecture,'' due to Talagrand and recently proved by Frankston, Kahn, Narayanan, and myself, and the (stronger) ``Expectation-Threshold Conjecture'' of Kahn and Kalai. The conjecture under discussion here says there is a fixed $J$ such that if, for a given increasing family $\mathcal F$, $p\in [0,1]$ admits $\lambda: 2^X\rightarrow \mathbb R^+$ with \[\sum_{S\subseteq F} \lambda_S \ge 1 \quad \forall F\in \mathcal F\] and \[\sum_S \lambda_S p^{|S|} \le 1/2,\] then $p/J$ admits such a $\lambda$ taking values in $\{0,1\}$. Talagrand showed this when $\lambda$ is supported on singletons and suggested a couple of more challenging test cases. In the talk, I will give more detailed descriptions of this problem, and some proof ideas if time allows.

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A solution to Erd\H{o}s and Hajnal's odd cycle problem

Dr. Richard Montgomery, University of Birmingham

Abstract

This talk will address how to construct cycles of many different lengths in graphs, in particular answering the following two problems on odd and even cycles. Erd\H{o}s and Hajnal asked in 1981 whether the sum of the reciprocals of the odd cycle lengths in a graph diverges as the chromatic number increases, while, in 1984, Erd\H{o}s asked whether there is a constant $C$ such that every graph with average degree at least $C$ contains a cycle whose length is a power of 2. This is joint work with Hong Liu.

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On graph norms

Mr. Jan Hladky, Institute of Computer Science of the Czech Academy of Sciences

Abstract

The concept of graph norms, introduced by Hatami in 2010 (following suggestions from Lovász and Szegedy), is a convenient tool to work with subgraph densities. The definition is real-analytic, making use of the framework of graphons. I will describe several results obtained by in collaboration with Doležal-Grebík-Rocha-Rozhoň and Garbe-Lee. This in particular includes a characterization norming graphs (and alike) using the `step-Sidorenko' property and a characterization of disconnected norming graphs.

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Extremal density for sparse minors and subdivisions

Dr. Jaehoon Kim, KAIST

Abstract

We prove an asymptotically tight bound on the extremal density guaranteeing subdivisions of bounded-degree bipartite graphs with a mild separability condition. As corollaries, we prove that: $(1+o(1))t^2$ average degree is sufficient to force the $t\times t$ grid as a topological minor; $(3/2+o(1))t$ average degree forces every $t$-vertex planar graph as a minor, and the constant $3/2$ is optimal; a universal bound of $(2+o(1))t$ on average degree forcing every $t$-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible. This is joint work with John Haslegrave and Hong Liu.

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Counting transversals in group multiplication tables

Dr. Rudi Mrazović, University of Zagreb

Abstract

Hall and Paige conjectured in 1955 that the multiplication table of a finite group $G$ has a transversal (a set of $|G|$ cells in distinct rows and columns and having different symbols) if and only if $G$ satisfies a straightforward necessary condition. This was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. I will discuss joint work with Sean Eberhard and Freddie Manners in which we approach the problem in a more analytic way that enables us to asymptotically count transversals.

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Packing $D$-degenerate graphs

Ms. Diana Piguet, Institute of Computer Science of the Czech Academy of Sciences

Abstract

A family $\{H_1, \ldots, H_k\}$ of graphs packs in a host graph $G$, if there is a colouring of the edges of $G$ with colours $1,\ldots, k$ so that there is a copy of $H_i$ in the subgraph of $G$ induced by colour $i$. A conjecture of Gy\'arf\'as from 1976, now referred to as the Tree Packing Conjecture, says that if $T_i$ is an $i$-vertex tree for each $1\le i \le n$, then $\{T_1,\ldots, T_n\}$ packs in the complete graph $K_n$. We shall present a result on packing $D$-degenerate graphs which has direct implications on the Tree Packing Conjecture.

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Geometric constructions for Ramsey-Tur\'an theory

Dr. Hong Liu, University of Warwick

Abstract

Combining two classical notions in extremal combinatorics, the study of Ramsey-Tur\'an theory seeks to determine, for integers $m\le n$ and $p \leq q$, the number $\mathrm{RT}_p(n,K_q,m)$, which is the maximum size of an $n$-vertex $K_q$-free graph in which every set of at least $m$ vertices contains a $K_p$. Two major open problems in this area from the 80s ask: (1) whether the asymptotic extremal structure for the general case exhibits certain periodic behaviour, resembling that of the special case when $p=2$; (2) constructing analogues of Bollob\'as-Erd\H{o}s graphs with densities other than $1/2$. We refute the first conjecture by witnessing asymptotic extremal structures that are drastically different from the $p=2$ case, and address the second problem by constructing Bollob\'as-Erd\H{o}s-type graphs using high dimensional complex spheres with \emph{all rational} densities. Some matching upper bounds are also provided.

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Ramsey properties of randomly perturbed sets of integers and the asymmetric random Rado theorem

Prof. Yury Person, Technische Universität Ilmenau

Abstract

We discuss some results in randomly perturbed sets of integers, i.e.\ sets of the form $A\cup [n]_p$, and thresholds for asymmetric Rado properties in $[n]_p$, which guarantee existence of monochromatic solutions to certain systems of linear equations.

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New results for MaxCut in $H$-free graphs

Dr. Stefan Glock, ETH Zurich

Abstract

The MaxCut problem asks for the size ${\rm mc}(G)$ of a largest cut in a graph~$G$. It is well known that ${\rm mc}(G)\ge m/2$ for any $m$-edge graph $G$, and the difference ${\rm mc}(G)-m/2$ is called the \emph{surplus} of~$G$. The study of the surplus of $H$-free graphs was initiated by Erd\H{o}s and Lov\'asz in the 70s, who in particular asked what happens for triangle-free graphs. This was famously resolved by Alon, who showed that in the triangle-free case the surplus is $\Omega(m^{4/5})$, and found constructions matching this bound. We prove several new results in this area. First, we obtain an optimal bound when $H$ is an odd cycle, adding to the lacunary list of graphs for which such a result is known. Secondly, we extend the result of Alon in the sense that we prove optimal bounds on the surplus of general graphs in terms of the number of triangles they contain. Thirdly, we improve the currently best bounds for $K_r$-free graphs. Our proofs combine techniques from semidefinite programming, probabilistic reasoning, as well as combinatorial and spectral arguments.

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Progress on intersecting families

Mr. Andrey Kupavskii, Moscow Institute of Physics and Technology

Abstract

We say that a family of sets is intersecting, if any two of its sets intersect. The Erdos-Ko-Rado theorem characterizes the largest intersecting families of k-element subets of an n-element set, and a lot of studies have been devoted to the following vague question: what structure could a family have, given that it has size close to extremal? This question has many possible answers, depending on the structure we look for, and I am going to discuss several of them during my talk. These will include junta approximations, diversity and covering number. Such structural results have implications for other related problems, and I will cover at least one such application to the colorings of Kneser graphs. Recall that a Kneser graph is a graph on the collection of all k-element subsets of an n-element set, with two sets connected by an edge when they are disjoint. One of the famous results of Lovász is the topological proof that such graph has chromatic number $n-2k+2$. We discuss the following problem: for which values of $n=n(k)$ we can guarantee that one of the colors is trivial, i.e., it consists of sets containing a fixed element? Partially based on joint works with Peter Frankl and Sergei Kiselev.

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Universal phenomena for random constrained permutations

Mr. Jacopo Borga, University of Zurich

Abstract

How do local/global constraints affect the limiting shape of random permutations? This is a classical question that has received considerable attention in the last 15 years. In this talk we give an overview of some recent results on this topic, mainly focusing on random pattern-avoiding permutations. We first introduce a notion of scaling limit for permutations, called permutons. Then we present some recent results that highlight certain universal phenomena for permuton limits of various families of pattern-avoiding permutations. These results will lead us to the definition of three remarkable new limiting random permutons: the “biased Brownian separable permuton”, the “Baxter permuton” and the "skew Brownian permuton". We finally discuss some recent results that show how permuton limits are useful to investigate the behaviour of certain statistics on random pattern-avoiding permutations, such as the length of the longest increasing subsequence.

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How does the chromatic number of a random graph vary?

Dr. Annika Heckel, LMU Munich

Abstract

If we pick an $n$-vertex graph uniformly at random, how much does its chromatic number vary? In 1987 Shamir and Spencer proved that it is contained in some sequence of intervals of length about $n^{1/2}$. Alon improved this slightly to $n^{1/2} / \log n$. Until recently, however, no non-trivial lower bounds on the fluctuations of the chromatic number of a random graph were known, even though the question was raised by Bollobás many years ago. We will present the main ideas needed to prove that, at least for infinitely many values $n$, the chromatic number of a uniform $n$-vertex graph is not concentrated on fewer than $n^{1/2-o(1)}$ consecutive values. We will also discuss the Zigzag Conjecture, made recently by Bollobás, Heckel, Morris, Panagiotou, Riordan and Smith: this proposes that the correct concentration interval length 'zigzags' between $n^{1/4+o(1)}$ and $n^{1/2+o(1)}$, depending on $n$. Joint work with Oliver Riordan.

Geometric analysis and low-dimensional topology (MS - ID 59)

Differential Geometry and Applications

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Applications of Floer homology to clasp number vs genus.

Prof. Tomasz Mrowka, Massachusetts Institute of Technology

Abstract

A basic question in four dimensional topology is determining the smooth four-ball genus of a knot in the the three sphere. A closely related question is determining the minimal number of double points of a disk smoothly immersed bounding the knot. We'll survey recent advances on this problem using tools coming from gauge theory and related invariant. This circle of ideas is related to interesting questions in algebraic geometry.

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A Local Singularity Analysis for the Ricci Flow

Prof. Reto Buzano, Università degli studi di Torino

Abstract

The Ricci Flow is the most famous and most successful geometric flow, having led to resolutions of the Poincaré and Geometrisation Conjectures, as well as proofs of the Differentiable Sphere Theorem and the Generalised Smale Conjecture. For many of these applications, it is important to understand precisely how singularities form along the flow - which is a notoriously difficult task, in particular in dimensions strictly greater than three. In this talk, we develop a new and refined singularity analysis for the Ricci Flow by investigating curvature blow-up rates locally. We introduce general definitions of Type I and Type II singular points and show that these are indeed the only possible types of singular points in a Ricci Flow. In particular, near any singular point the Riemannian curvature tensor has to blow up at least at a Type I rate, generalising a result previously obtained with Enders and Topping under a global Type I assumption. We also prove analogous results for the Ricci tensor, as well as a localised version of Sesum’s result, namely that the Ricci curvature must blow up near every singular point of a Ricci flow, again at least at a Type I rate. If time permits, we will also see some applications of the theory to Ricci flows with bounded scalar curvature.

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Spaces of constrained positive scalar curvature metrics

Prof. Alessandro Carlotto, ETH

Abstract

In this lecture, I will present a collection of results concerning the interplay between the scalar curvature of a Riemannian manifold and the mean curvature of its boundary, with special emphasis on dimension-dependent phenomena. Our work is motivated by a network of far-reaching conjectures by Gromov on the one hand, and by the study of the space of admissible initial data sets for the Einstein field equation in general relativity on the other.

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Recent progress in Lagrangian mean curvature flow of surfaces

Prof. Jason Lotay, University of Oxford

Abstract

Lagrangian mean curvature flow is potentially a powerful tool for tackling several important open problems in symplectic topology. The first non-trivial and important case is the flow of Lagrangian surfaces in 4-manifolds. I will describe some recent progress in understanding the Lagrangian mean curvature flow of surfaces, including general results about ancient solutions and the study of the Thomas-Yau conjecture in explicit settings.

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The Witten Conjecture for homology $S^1 \times S^3$

Dr. Nikolai Saveliev, University of Miami

Abstract

The Witten conjecture (1994) poses that the Seiberg--Witten invariants contain all of the topological information of the Donaldson polynomials. The natural domain of this conjecture comprises closed simply connected oriented smooth 4-manifolds with $b_+ > 1$, where the Seiberg--Witten invariants are obtained by a straightforward count of irreducible solutions to the Seiberg--Witten equations. The Seiberg-Witten invariants have also been extended to manifolds with $b_+ = 1$ using wall-crossing formulas. In our work with Mrowka and Ruberman (2009) we defined the Seiberg--Witten invariant for a class of manifolds $X$ with $b_+ = 0$ having homology of $S^1 \times S^3$. The usual count of irreducible solutions in this case depends on metric and perturbation but we succeeded in countering this dependence by a correction term to obtain a diffeomorphism invariant of $X$. In the spirit of the Witten conjecture, we conjectured that the degree zero Donaldson polynomial of $X$ can be expressed in terms of this invariant. I will describe the special cases in which the conjecture has been verified, together with some applications. This is a joint project with Jianfeng Lin and Daniel Ruberman.

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Algebraic and geometric classification of Yang-Mills-Higgs flow lines

Dr. Graeme Wilkin, University of York

Abstract

The Yang-Mills-Higgs flow on a compact Riemann surface is modelled on a nonlinear heat equation, and therefore existence of the reverse flow is problematic in general. In this talk I will explain how the existence of a certain filtration (analogous to the Harder-Narasimhan filtration, but with the opposite inequality on the slopes) means that one can gauge the initial condition so that the reverse flow exists for all time. This leads to an algebraic classification of flow lines between pairs of critical points. A refinement of these ideas leads to a geometric classification of flow lines in terms of certain secant varieties, which can then be used to distinguish between broken and unbroken flow lines.

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Alternating links, rational balls, and tilings

Brendan Owens, University of Glasgow

Abstract

If an alternating knot is a slice of a knotted 2-sphere, then it follows from work of Greene and Jabuka, using Donaldson's diagonalisation combined with Heegaard Floer theory, that the flow lattice $L$ of its Tait graph admits an embedding in the integer lattice $\mathbb{Z}^n$ of the same rank which is ``cubiquitous”: every unit cube in $\mathbb{Z}^n$ contains an element of $L$. The same is true more generally for an alternating link whose double branched cover bounds a rational homology 4-ball. This results in an upper bound on the determinant of such links. In this talk I will describe recent joint work with Josh Greene in which we classify alternating links for which this upper bound on determinant is realised. This makes use of Minkowski’s conjecture, proved by Haj\'{o}s in 1941, which states that every lattice tiling of $\mathbb{R}^n$ by cubes has a pair of cubes which share a facet.

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Disoriented homology of surfaces and branched covers of the 4-ball

Sašo Strle, University of Ljubljana

Abstract

An often used construction in low-dimensional topology is to associate to a properly embedded surface $F \subset B^4$ the branched double cover $X$ of $B^4$ with branch set $F$. If the surface is obtained by pushing the interior of an embedded surface in $S^3$ into the interior of $B^4$, a classical result of Gordon and Litherland states that $H_2(X;\mathbb Z)$ is isomorphic to $H_1(F;\mathbb Z)$ and that the intersection pairing of $X$ may be described in terms of a pairing on $H_1(F;\mathbb Z)$ which is determined by the embedded surface in $S^3$. We generalize this result to any surface $F$ by defining a non-standard homology theory $DH_*(F)$ that depends on a description of $F$ in $S^3$. This homology captures the homological information of $X$ and may be equipped with a pairing on $DH_1(F)$ that corresponds to the intersection pairing of $X$. This construction also works for closed surfaces.

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A Brakke type regularity for the Allen-Cahn flow

Mr. Shengwen Wang, Queen Mary University of London

Abstract

We will talk about an analogue of the Brakke's local regularity theorem for the $\epsilon$ parabolic Allen-Cahn equation. In particular, we show uniform $C_{2,\alpha}$ regularity for the transition layers converging to smooth mean curvature flows as $\epsilon$ tend to 0 under the almost unit-density assumption. This can be viewed as a diffused version of the Brakke regularity for the limit mean curvature flow. This talk is based on joint work with Huy Nguyen.

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A Positive Mass Theorem for Fourth-Order Gravity

Prof. PAUL LAURAIN, Université de Paris

Abstract

A Positive Mass Theorem for Fourth-Order Gravity\\ In classical Einstein gravity, the metric is a critical point of the Einstein-Hilbert functional, i.e. $g\mapsto \int_M R_g \,dv$. It has been proved that there exists a conserved quantity along space-like hypersurface, called the ADM mass. In a celebrated work Schoen and Yau proved that if the scalar curvature of the hypersurface is none-negative then the mass is none-negative, with rigidity if the mass vanishes. After remembering some facts about this result, I will introduce a new mass associated to a fourth order functional, namely $g\mapsto \int_M \alpha R_g^2 + \beta \vert \mathrm{Ric}_g\vert^2\, dv_g$. Then I will explain how we obtain an analogue of the positive mass theorem for this new mass by replacing the none-negativity of the scalar curvature by the one of the $Q$-curvature.

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Free boundary minimal surfaces in the unit ball

Dr. Mario Schulz, Queen Mary University of London

Abstract

Free boundary minimal surfaces arise naturally in partitioning problems for convex bodies, in capillarity problems for fluids and in the study of extremal metrics for Steklov eigenvalues on manifolds with boundary. The theory has been developed in various interesting directions, yet many fundamental questions remain open. One of the most basic ones can be phrased as follows: Can a surface of any given topology be realised as an embedded free boundary minimal surface in the 3-dimensional Euclidean unit ball? We will answer this question affirmatively for surfaces with connected boundary and arbitrary genus.

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Characterizing slopes for Legendrian knots

Dr. Marc Kegel, Humboldt-Universität zu Berlin

Abstract

From a given Legendrian knot $K$ in the standard contact $3$-sphere, we can construct a symplectic $4$-manifold $W_K$ by attaching a Weinstein $2$-handle along $K$ to the $4$-ball. In this talk, we will construct non-equivalent Legendrian knots $K$ and $K'$ such that $W_K$ and $W_K'$ are equivalent. On the other hand, we will discuss an example of a Legendrian knot $K$ that is characterized by its symplectic $4$-manifold $W_K$. This is based on joint work with Roger Casals and John Etnyre. No previous knowledge of contact geometry is assumed. We will discuss all relevant notions.

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Negatively curved Einstein metrics on quotients of 4-dimensional hyperbolic manifolds.

Dr. Bruno Premoselli, Université Libre de Bruxelles

Abstract

Few examples of closed Einstein manifolds with negative scalar curvature are known in dimensions larger than 4, and until recently it was believed that the only negatively curved ones were the trivial ones, ie closed quotients of (complex)-hyperbolic space. We will construct in this talk new examples of non-trivial closed negatively curved Einstein 4-manifolds. More precisely, we will show that Einstein metrics with negative sectional (and scalar) curvature can be found on quotients of ``large’’ closed hyperbolic 4-manifolds with symmetries. The proof is via a glueing procedure, starting from an approximate Einstein metric that is obtained as the interpolation between a ``black-hole – type’’ model metric near the symmetry locus and the hyperbolic metric at large distances. This is a joint work with J. Fine (ULB, Brussels).

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Wedge theorems for ancient mean curvature flows

Prof. Niels Martin Møller, University of Copenhagen

Abstract

We show that so-called "wedge theorems" hold for all properly immersed, not necessarily compact, ancient solutions to the mean curvature flow in $\mathbb{R}^{n+1}$. Such nonlinear parabolic Liouville-type results add to a long story, generalizing recent results for self-translating solitons, which in turn imply the minimal surface case (Hoffman-Meeks, '90) that contains the classical cases of cones (Omori '67) and graphs (Nitsche, '65). As an application we classify the convex hulls of the spacetime tracks of all proper ancient flows, without any of the usual curvature assumptions. The proofs make use of a linear parabolic Omori-Yau maximum principle for (non-compact) ancient flows. This is joint work with F. Chini.

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A two-valued Bernstein theorem in dimension four

Dr. Fritz Hiesmayr, University College London

Abstract

The Bernstein theorem is a classical result in geometric analysis, which states that entire minimal graphs are linear in all dimensions up to eight. Here we present a generalisation to so-called two-valued minimal graphs. These are graphs of two-valued functions, which are singular geometric objects that model the behaviour of minimal hypersurfaces near branch points. The two-valued Bernstein theorem we present proves that entire two-valued minimal graphs are linear in dimension four.

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Foliation of Asymptotically Schwarzschild Manifolds by Generalized Willmore Surfaces

Alexander Friedrich, University of Copenhagen

Abstract

\noindent From the perspective of general relativity asymptotically Schwarzschild, or more generally asymptotically flat, manifolds represent isolated systems. Here the idea is that in the absents of classical energy the spacetime should resemble the Minkowski space. The Hawing energy is a quasi local energy of general relativity that strong relation to the Willmore functional. It was introduced by S.W. Hawking in order to measure the gravitational energy of spacetimes that are classically empty. Starting from the Hawking energy we develop a notion of generalized Willmore functionals. Further, we construct a foliation of the asymptotically flat end of an asymptotically Schwarzschild manifold by large, area constrained spheres which are critical with respect to a generalized Willmore functional. This achieved via a perturbation argument starting from round centered spheres in Schwarzschild space. These foliations can be interpreted as a center of mass as measured by the generalized Willmore functional. Time permitting we will discuss the existence and regularity of critical points of generalized Willmore functionals in a more general setting. The results presented are based on the a corresponding analysis for the Willmore functional by T. Lamm, J. Metzger and F. Schulze, and are part of the authors Ph.D. thesis.

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Large area-constrained Willmore spheres in initial data sets

Thomas Koerber, University of Vienna

Abstract

Area-constrained Willmore spheres are surfaces that are particularly well-adapted to the Hawking mass, a local measure of the gravitational field of initial data sets for isolated gravitational systems. In this talk, I will present recent results (joint with M. Eichmair) on the existence and uniqueness of large area-constrained Willmore surfaces in such initial data sets. In particular, I will describe necessary and sufficient conditions on the scalar curvature of the initial data set that guarantee the existence of a unique asymptotic foliation by large area-constrained Willmore spheres. I will also discuss recent results on the geometric center of mass of this foliation.

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Applications of virtual Morse--Bott theory to the moduli space of $\operatorname{SO}(3)$ Monopoles

Prof. Thomas Leness, Florida International University

Abstract

We describe some recent joint work with P. Feehan on the Morse theory of a function on the moduli space of $\operatorname{SO}(3)$ monopoles on a smooth four-manifold and sketch some applications to the topology of smooth four-manifolds.

Geometric-functional inequalities and related topics (MS - ID 23)

Partial Differential Equations and Applications

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Fractional Orlicz-Sobolev spaces

Prof. Andrea Cianchi, Università di Firenze

Abstract

Optimal embeddings for fractional-order Orlicz–Sobolev spaces are presented. Related Hardy type inequalities are proposed as well. Versions for fractional Orlicz-Sobolev seminorms of the Bourgain-Brezis-Mironescu theorem on the limit as the order of smoothness tends to 1 and of the Maz’ya-Shaposhnikova theorem on the limit as the order of smoothness tends to 0 are established.

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Blaschke--Santal\'o inequalities for Minkowski endomorphisms

Prof. Franz Schuster, Vienna University of Technology

Abstract

In this talk we explain how each monotone Minkowski endomorphism of convex bodies gives rise to an isoperimetric inequality which directly implies the classical Urysohn inequality. Among this large family of new inequalities, the only affine invariant one -- the Blaschke--Santal\'o inequality -- turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible. Moreover, for functional analogues of monotone Minkowski endomorphisms, a family of analytic inequalities for log-concave functions is established which generalizes the functional Blaschke--Santal\'o inequality.

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A liquid-solid phase transition in a simple model for swarming

Prof. Rupert Frank, Caltech / LMU Munich

Abstract

We consider a non-local shape optimization problem, which is motivated by a simple model for swarming and other self-assembly/aggregation models, and prove the existence of different phases. In particular, we show that in the large mass regime the ground state density profile is the characteristic function of a round ball. An essential ingredient in our proof is a strict rearrangement inequality with a quantitative error estimate. The talk is based on joint work with E. Lieb.

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Classical multiplier theorems and their sharp variants

Dr. Lenka Slavikova, Charles University

Abstract

The question of finding good sufficient conditions on a bounded function $m$ guaranteeing the $L^p$-boundedness of the associated Fourier multiplier operator is a long-standing open problem in harmonic analysis. In this talk we recall the classical multiplier theorems of H\"ormander and Marcinkiewicz and present their sharp variants in which the multiplier belongs to a certain fractional Lorentz-Sobolev space. The talk is based on a joint work with L. Grafakos and M. Masty\l o.

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Reverse superposition estimates, lifting over a compact covering and extensions of traces for fractional Sobolev mappings

Prof. Jean Van Schaftingen, UCLouvain

Abstract

When \(u \in W^{1, p} (\mathbb{R}^m)\), then \(\lvert u\rvert \in W^{1, p} (\mathbb{R}^m)\) and \[ \int_{\mathbb{R}^m} \vert D u\vert^p = \int_{\mathbb{R}^m} \vert D \vert u\vert \vert^p; \] this provides an a priori control on \(u\) by \(\lvert u \rvert\) in first-order Sobolev spaces. For fractional Sobolev spaces when \(sp > 1\), we prove a reverse oscillation inequality that yields a control on \(u\) by \(\lvert u \rvert\) in \(W^{s, p} (\mathbb{R}^m)\). As another consequence of the reverse oscillation estimate, given a covering map \(\pi : \widetilde{\mathcal{N}} \to \mathcal{N}\), with \(\widetilde{\mathcal{N}}\) compact, we prove any \(u \in W^{s, p} (\mathbb{R}^m, \mathcal{N})\) has a lifting, that is, can be written as \(u = \pi \circ \widetilde{u}\), with \(\widetilde{u} \in W^{s, p} (\mathbb{R}^m, \widetilde{\mathcal{N}})\). This completes the picture for lifting of fractional Sobolev maps and implies the surjectivity of the trace operator on Sobolev spaces of mappings into a manifold when the fundamental group is finite.

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An optimization problem in thermal insulation

Prof. Cristina Trombetti, Università degli Studi di Napoli Federico II

Abstract