# Talk presentations

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

### Plenary speakers

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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

Abstract

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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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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Geometric Valuation Theory

###### Prof. Monika Ludwig, TU Wien

Abstract

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, Ludwig Maximilian University of Munich

Abstract

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.

### Invited speakers

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Subset products and derangements

###### Prof. Aner Shalev, Hebrew University of Jerusalem

Abstract

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

Abstract

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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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

Abstract

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

Abstract

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

Abstract

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

Abstract

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

Abstract

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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Nonstandard finite elements for wave problems

###### Prof. Ilaria Perugia, University of Vienna

Abstract

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

Abstract

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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The dawn of formalized mathematics

###### Prof. Andrej Bauer, University of Ljubljana

Abstract

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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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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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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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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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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###### 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.

### EMS Prize Winners

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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.

### 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.

### Public speakers

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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.

### 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)

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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.

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

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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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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.

### Algorithmic Graph Theory (MS - ID 54)

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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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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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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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Clique-Width: Harnessing the Power of Atoms

###### Dr. Konrad Dabrowski, University of Leeds

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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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~$3$.

### Analysis of PDEs on Networks (MS - ID 26)

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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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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.

### Analysis on Graphs (MS - ID 48)

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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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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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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)

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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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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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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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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: \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. 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)

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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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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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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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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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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)

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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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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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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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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.

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

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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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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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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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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.

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

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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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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 $$\label{1} \left\{\begin{array}{ll}U_t=\mathcal{A} U +M P, \hbox{ in } H,\\ P_t=BP+ N U, \hbox{ in } X,\\ U(0)=U_0,P(0)=P_0, \end{array}\right.$$ 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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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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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.

### Combinatorial Designs (MS - ID 16)

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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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###### 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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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

Abstract

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A map on an orientable surface is called orientably-regular' if its automorphism group has a single orbit on arcs (incident vertex-edge pairs), and is then called reflexible' or `chiral' depending on w