Contribution list

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

Plenary speakers

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The Beat of Math

Prof. Alfio Quarteroni, Politecnico di Milano and EPFL

Abstract

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

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

Prof. Xavier Cabré, ICREA and Universitat Politecnica de Catalunya

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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Torsion in algebraic groups and problems which arise

Prof. Umberto Zannier, Scuola Normale Superiore

Abstract

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

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Minimal surfaces from a complex analytic viewpoint

Prof. Franc Forstnerič, University of Ljubljana

Abstract

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

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Statistical Learning: Causal-oriented and Robust

Prof. Peter Bühlmann, ETH Zurich

Abstract

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

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Bernoulli Random Matrices

Ms. Alice Guionnet, ENS Lyon

Abstract

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

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Escaping the curse of dimensionality in combinatorics

Prof. Janos Pach, Alfred Renyi Matematical Institute

Abstract

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

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Geometric Valuation Theory

Prof. Monika Ludwig, TU Wien

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, Technische Uninversität Berlin

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.

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Metric measure spaces and synthetic Ricci bounds

Prof. Karl-Theodor Sturm, Universität Bonn

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

Invited speakers

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

Abstract

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Packings of Partial Difference Sets

Dr. Shuxing Li, Simon Fraser University

Abstract

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

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

Dr. Peter Dukes, University of Victoria

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

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

Prof. Gloria Rinaldi, University of Modena and Reggio Emilia

Abstract

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

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Balancedly splittable orthogonal designs

Prof. Hadi Kharaghani, University of Lethbridge

Abstract

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

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Factorizations of infinite graphs

Dr. Simone Costa, University of Brescia

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

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

Dr. Tommaso Traetta, Università degli Studi di Brescia

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

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On the mini-symposium problem

Prof. Peter Danziger, Ryerson University

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

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An Evans-style result for block designs

Dr. Daniel Horsley, Monash University

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

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

Prof. Alessandro Montinaro, University of Salento

Abstract

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

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

Dr. Raul M. Falcon, Universidad de Sevilla

Abstract

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

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On the existence of large set of partitioned incomplete Latin squares

Ms. Cong Shen, Nanjing Normal University

Abstract

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

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Equitably $2$-colourable even cycle systems

Dr. Francesca Merola, Università Roma Tre

Abstract

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

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

Prof. Tao Feng, Beijing Jiaotong University

Abstract

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

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Kirkman triple systems with many symmetries

Prof. Marco Buratti, Università di Perugia

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

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On the classification of unitals on 28 points of low rank

Prof. Alfred Wassermann, University of Bayreuth

Abstract

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

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20 years of Legendre Pairs

Prof. ilias kotsireas, WLU

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

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Block designs constructed from orbit matrices using a modified genetic algorithm

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

Abstract

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

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On some periodic Golay pairs and pairwise balanced designs

Dr. Andrea Svob, University of Rijeka

Abstract

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

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

Dr. Kristijan Tabak, Rochester Institute of Technology - Croatia

Abstract

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

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Balanced Equi-$n$-squares

Dr. Trent Marbach, Ryerson University

Abstract

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

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Resolving sets and identifying codes in finite geometries

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

Abstract

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

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Weakly self-orthogonal designs and related linear codes

Prof. Vedrana Mikulić Crnković, University of Rijeka

Abstract

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

Complex Analysis and Geometry (MS - ID 17)

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A criterion of Nakano positivity

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

Abstract

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

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Metric geometry of domains in complex Euclidean spaces

Mr. Hervé Gaussier, Université Grenoble Alpes

Abstract

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

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Seiberg-Witten equations and pseudoholomorphic curves

Prof. Armen Sergeev, Steklov Mathematical Institute

Abstract

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

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

Dr. Sibel Sahin, Mimar Sinan Fine Arts University

Abstract

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

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The Density Property for Calogero--Moser Spaces

Prof. Rafael Andrist, American University of Beirut

Abstract

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

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

Dr. Ngoc Son Duong, Universität Wien

Abstract

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

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

Dr. Jujie Wu, Sun Yat-Sen University

Abstract

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

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Images of CR manifolds

Prof. Jiri Lebl, Oklahoma State University

Abstract

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

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

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

Abstract

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

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Parabolic Fatou components with holes

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

Abstract

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

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Fekete Configurations, Products of Vandermonde Determinants and Canonical Point Processes

Dr. Jakob Hultgren, University of Maryland

Abstract

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

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

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

Abstract

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

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

Joaquim Ortega Cerdà, Universitat de Barcelona

Abstract

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

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

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Non-commutative polynomial in quantum physics

Prof. Antonio Acin, ICFO

Abstract

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

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Tensor Decompositions on Simplicial Complexes: Existence and Applications

Mr. Tim Netzer, University of Innsbruck

Abstract

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

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Hilbert's 17th problem for noncommutative rational functions

Dr. Jurij Volčič, Texas A&M University

Abstract

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

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Quantum de Finetti theorems and Reznick’s Positivstellensatz

Ion Nechita, CNRS, Toulouse University

Abstract

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

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Optimizing conditional entropies for quantum correlations

Dr. Omar Fawzi, Inria

Abstract

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

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

Dr. David Papp, North Carolina State University

Abstract

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

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Optimization over trace polynomials

Dr. Victor Magron, LAAS CNRS

Abstract

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

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Fast Semidefinite Optimization with Latent Basis Learning

Abstract

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

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Efficient noncommutative polynomial optimization by exploiting sparsity

Dr. Jie Wang, LAAS-CNRS

Abstract

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

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

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

Abstract

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

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

Dr. Hamza Fawzi, University of Cambridge

Abstract

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

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Quantum Isomorphism of Graphs: an Overview

Dr. Laura Mančinska, University of Copenhagen

Abstract

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

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The tracial moment problem on quadratic varieties

Dr. Aljaž Zalar, University of Ljubljana

Abstract

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

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Positive maps and trace polynomials from the symmetric group

Dr. Felix Huber, Jagiellonian University

Abstract

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

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Semidefinite relaxations in non-convex spaces

Abstract

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

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SOS relaxations for detecting quamtum entanglement

Dr. Abhishek Bhardwaj, CNRS

Abstract

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

Configurations (MS - ID 81)

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

Dr. Leah Berman, University of Alaska Fairbanks

Abstract

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

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Barycentric configurations in real space

Prof. Hendrik Van Maldeghem, Ghent University

Abstract

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

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Combinatorial Configurations and Dessins d'Enfants

Abstract

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

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Transitions between configurations

Prof. Gábor Gévay, University of Szeged

Abstract

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

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Splittability of cubic bicirculants and their related configurations

Dr. Nino Bašić, University of Primorska

Abstract

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

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Taxonomy of Three-Qubit Doilies

Dr. Metod Saniga, Slovak Academy of Sciences

Abstract

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

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Hirzebruch-type inequalities and extreme point-line configurations

Prof. Piotr Pokora, Pedagogical University of Cracow

Abstract

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

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Strongly regular configurations

Prof. Vedran Krčadinac, University of Zagreb Faculty of Science

Abstract

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

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Configurations from strong deficient difference sets

Dr. Marién Abreu, Università degli Studi della Basilicata

Abstract

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

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Highly symmetric configurations

Dr. Jurij Kovič, UP FAMNIT Koper and IMFM Ljubljana

Abstract

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

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4-lateral matroids induced by $n_3$-configurations

Dr. Michael Raney, Georgetown University

Abstract

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

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Jordan schemes

Dr. Sven Reichard, Dresden International University

Abstract

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

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

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On the complex hypothesis of Banach

Prof. Luis Montejano, Instituto de Matematicas

Abstract

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

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The $L_p$ Minkowski problem and polytopal approximation

Prof. Karoly Boroczky, Alfred Renyi Institute of Mathematics

Abstract

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

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On bodies floating in equalibrium in every direction

Prof. Dmitry Ryabogin, Kent State University

Abstract

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

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Strengthened inequalities for the mean width

Mr. Ferenc Fodor, University of Szeged

Abstract

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

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A new look at the Blaschke-Leichtweiss theorem

Prof. Karoly Bezdek, University of Calgary

Abstract

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

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Pea Bodies of Constant Width

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

Abstract

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

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The Golden ratio and high dimensional mean inequalities

Dr. Bernardo González Merino, Universidad de Murcia

Abstract

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

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Colorful Helly-type Theorems for Ellipsoids

Mrs. Viktória Földvári, Alfréd Rényi Institute of Mathematics

Abstract

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

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A cap covering theorem

Mr. Alexandr Polyanskii, MIPT

Abstract

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

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

Dr. Zsolt Langi, Budapest University of Technology

Abstract

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

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Covering techniques in Integer \& Lattice Programming

Mr. Moritz Venzin, École polytechnique fédérale de Lausanne

Abstract

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

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Rigidity of compact Fuchsian manifolds with convex boundary

Mr. Roman Prosanov, TU Wien

Abstract

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

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Coverings by homothets of a convex body

Dr. Nora Frankl, Carnegie Mellon University

Abstract

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

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Functional John and L\"owner Ellipsoids

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

Abstract

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

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Extremal sections and local optimization

Dr. Gergely Ambrus, Alfréd Rényi Institute of Mathematics

Abstract

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

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Cells in the box and a hyperplane

Prof. Imre Bárány, Alfréd Rényi Institute of Mathematics

Abstract

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

Current topics in Complex Analysis (MS - ID 32)

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

Prof. Stanisława Kanas, University of Rzeszow

Abstract

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

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Almost periodic functions revisited

Dr. Juan Matías Sepulcre, University of Alicante

Abstract

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

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Harmonic quasiconformal mappings and hyperbolic type metrics

Prof. Vesna Todorcevic, FON and MI SASA

Abstract

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

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Extremal decomposition of the complex plane

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

Abstract

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

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

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

Abstract

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

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Asymptotic estimates for one class of homeomorphisms

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

Abstract

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

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Generalizations of Hardy type inequalities via new Green functions

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

Abstract

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

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Recent studies on the image domain of starlike functions

Ms. Sahsene Altinkaya, Beykent University

Abstract

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

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

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

Abstract

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

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

Dr. Edyta Trybucka, University of Rzeszów

Abstract

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

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

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

Prof. Jan Slovak, Masaryk University

Abstract

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

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Differential geometry of submanifolds in flag varieties via differential equations

Mr. Boris Doubrov, Belarusian State University

Abstract

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

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Contact CR submanifolds in odd-dimensional spheres: new examples

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

Abstract

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

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

Prof. Emilio Musso, Politecnico di Torino

Abstract

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

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Similarlity geometry revisited: Differential Geometry and CAGD

Prof. Jun-ichi Inoguchi, University of Tsukuba

Abstract

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

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

Prof. Emilija Nešović, University of Kragujevac Faculty of Science

Abstract

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

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

Dr. Ljiljana Primorac Gajcic, University of Osijek

Abstract

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

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Simple closed geodesics on regular tetrahedra in spaces of constant curvature

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

Abstract

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

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

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

Abstract

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

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

Prof. Zlatko Erjavec, University of Zagreb

Abstract

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

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

Dr. Miloš Petrović, University of Niš, Serbia

Abstract

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

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

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

Abstract

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

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Topologically Embedded Pseudospherical Surfaces

Prof. Lorenzo Nicolodi, Università di Parma

Abstract

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

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

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

Prof. Davide Guidetti, Università di Bologna

Abstract

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

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

Prof. Anar Assanova, Institute of mathematics and mathematical modeling

Abstract

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

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Generalized concepts of almost periodicity

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

Abstract

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

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Decay estimates for parabolic equations with inhomogeneous density in manifolds

Prof. Daniele Andreucci, Sapienza Università di Roma

Abstract

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

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

Abstract

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

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

Prof. Henryk Zoladek, University of Warsaw

Abstract

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

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Aspects of nonlinear differential equations

Dr. Galina Filipuk, University of Warsaw

Abstract

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

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

Dr. Michele Bufalo, University of Bari

Abstract

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

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

Prof. Federico Zullo, Università di Brescia

Abstract

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

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

Dr. Simone Creo, Sapienza Università di Roma

Abstract

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

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Integrability and asymptotic behaviour of a matrix lattice equation

Dr. Pilar R Gordoa, Universidad Rey Juan Carlos

Abstract

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

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

Dr. Andrew Pickering, Universidad Rey Juan Carlos

Abstract

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

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Evolution problems with dynamical boundary conditions of Wentzell-type

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

Abstract

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

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Evolution problems for fractional operators in irregular domains

Prof. Maria Rosaria Lancia, Sapienza

Abstract

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

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Thermoelastic Bresse system with dual-phase-lag model

Dr. Ivana Bochicchio, Università di Salerno

Abstract

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

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

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

Abstract

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

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Fractional Cauchy problem on random snowflakes

Prof. Raffaela Capitanelli, SAPIENZA-ROMA

Abstract

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

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

Dr. Ewa Stróżyna, Warsaw University of Technology

Abstract

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

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

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An overview of EU-MATHS-IN and its activities

Prof. Wil Schilders, TU Eindhoven

Abstract

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

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

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

Abstract

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

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

Ms. Maume-Deschamps Véronique, AMIES

Abstract

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

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

Mr. Johannes Leuschner, University of Bremen

Abstract

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

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MACSI and industrial mathematics in Ireland

Dr. Kevin Moroney, University of Limerick

Abstract

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

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

Prof. Zoltán Horváth, Széchenyi István University

Abstract

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

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

Prof. José Luis Santos, University of Coimbra

Abstract

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

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

Dr. Aleksander Grm, Univesity of Ljubljana

Abstract

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

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Simulation, optimal management and infrastructure planning of gas transmission networks

Dr. Ángel Manuel Rueda, University of A Coruña

Abstract

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

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Suboptimal scheduling of a fleet of AGVs to serve online requests

Mr. Markó Horváth, Institute for Computer Science and Control

Abstract

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

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

Prof. Christophe Prud'homme, Université de Strasbourg

Abstract

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

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

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Fracture in random heterogeneous particle systems

Prof. Anja Schlömerkemper, University of Würzburg

Abstract

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

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A derivation of Griffith functionals from discrete finite-difference models

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

Abstract

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

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Higher order far-field developments around lattice defects

Mr. Julian Braun, University of Warwick

Abstract

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

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Variational Modeling of Paperboard Delamination under Bending

Prof. Patrick Dondl, Albert-Ludwigs-Universität Freiburg

Abstract

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

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

Dr. Marco Morandotti, Politecnico di Torino

Abstract

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

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Geometric linearization of theories for incompressible elastic materials and applications

Prof. Bernd Schmidt, Universitaet Augsburg, Germany

Abstract

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

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Asymptotic analysis of rigidity constraints modeling fiber-reinforced composites

Dr. Antonella Ritorto, Katholische Universität Eichstätt-Ingolstadt

Abstract

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

Extremal and Probabilistic Combinatorics (MS - ID 20)

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Prof. Dhruv Mubayi, University of Illinois at Chicago

Abstract

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

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Dr. Luke Postle, University of Waterloo

Abstract

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

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Progress towards Nash-Williams' Conjecture on Triangle Decompositions

Dr. Michelle Delcourt, Ryerson University

Abstract

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

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Prof. Oleg Pikhurko, University of Warwick

Abstract

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

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On a problem of M. Talagrand

Dr. Jinyoung Park, Institute for Advanced Study

Abstract

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

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

Dr. Richard Montgomery, University of Birmingham

Abstract

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

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On graph norms

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

Abstract

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

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Extremal density for sparse minors and subdivisions

Dr. Jaehoon Kim, KAIST

Abstract

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

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Counting transversals in group multiplication tables

Dr. Rudi Mrazović, University of Zagreb

Abstract

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

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Packing $D$-degenerate graphs

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

Abstract

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

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Geometric constructions for Ramsey-Tur\'an theory

Dr. Hong Liu, University of Warwick

Abstract

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

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

Prof. Yury Person, Technische Universität Ilmenau

Abstract

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

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New results for MaxCut in $H$-free graphs

Dr. Stefan Glock, ETH Zurich

Abstract

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

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Progress on intersecting families

Mr. Andrey Kupavskii, Moscow Institute of Physics and Technology

Abstract

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

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Universal phenomena for random constrained permutations

Mr. Jacopo Borga, University of Zurich

Abstract

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

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How does the chromatic number of a random graph vary?

Dr. Annika Heckel, LMU Munich

Abstract

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

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

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Applications of Floer homology to clasp number vs genus.

Prof. Tomasz Mrowka, Massachusetts Institute of Technology

Abstract

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

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A Local Singularity Analysis for the Ricci Flow

Prof. Reto Buzano, Università degli studi di Torino

Abstract

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

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Spaces of constrained positive scalar curvature metrics

Prof. Alessandro Carlotto, ETH

Abstract

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

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Recent progress in Lagrangian mean curvature flow of surfaces

Prof. Jason Lotay, University of Oxford

Abstract

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

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

Dr. Nikolai Saveliev, University of Miami

Abstract

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

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

Dr. Graeme Wilkin, University of York

Abstract

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

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Alternating links, rational balls, and tilings

Brendan Owens, University of Glasgow

Abstract

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

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

Sašo Strle, University of Ljubljana

Abstract

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

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A Brakke type regularity for the Allen-Cahn flow

Mr. Shengwen Wang, Queen Mary University of London

Abstract

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

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A Positive Mass Theorem for Fourth-Order Gravity

Prof. PAUL LAURAIN, Université de Paris

Abstract

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

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Free boundary minimal surfaces in the unit ball

Dr. Mario Schulz, Queen Mary University of London

Abstract

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

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Characterizing slopes for Legendrian knots

Dr. Marc Kegel, Humboldt-Universität zu Berlin

Abstract

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

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

Dr. Bruno Premoselli, Université Libre de Bruxelles

Abstract

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

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Wedge theorems for ancient mean curvature flows

Prof. Niels Martin Møller, University of Copenhagen

Abstract

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

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A two-valued Bernstein theorem in dimension four

Dr. Fritz Hiesmayr, University College London

Abstract

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

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Foliation of Asymptotically Schwarzschild Manifolds by Generalized Willmore Surfaces

Alexander Friedrich, University of Copenhagen

Abstract

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

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Large area-constrained Willmore spheres in initial data sets

Thomas Koerber, University of Vienna

Abstract

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

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

Prof. Thomas Leness, Florida International University

Abstract

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

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

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Fractional Orlicz-Sobolev spaces

Prof. Andrea Cianchi, Università di Firenze

Abstract

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

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Blaschke--Santal\'o inequalities for Minkowski endomorphisms

Prof. Franz Schuster, Vienna University of Technology

Abstract

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

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

Prof. Rupert Frank, Caltech / LMU Munich

Abstract

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

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Classical multiplier theorems and their sharp variants

Dr. Lenka Slavikova, Charles University

Abstract

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

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

Prof. Jean Van Schaftingen, UCLouvain

Abstract

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

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An optimization problem in thermal insulation

Abstract