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CIRM – Centre International de Rencontres Mathématiques

Combinatorial statements, such as theorems of Caratheodory, Radon, Helly, Sperner, Tucker, Ky Fan, etc., are fundamental results of combinatorial (algebraic) topology, accessible to non-specialists, which are immediately applicable to mathematical economics, data science, game theory, graph theory, mathematical optimization, computational geometry, and other fields. The aim of our project is to study the “Ky Fan correspondence » between the topology of triangulated spherical bundles and combinatorics of generalized labelings (coloring) of the associated simplicial complexes. Guided by the review of De Loera et. al. (Bull. Amer. Math. Soc., 2019) we explore the consequences of this correspondence for the envy-free and fair division problem, Tverberg-type theorems, and other related problems of topological combinatorics.

CIRM – Centre International de Rencontres Mathématiques

Friezes, introduced 50 years ago by Coxeter, are patterns of integers defined by a simple, local condition. They consist of a finite (or infinite) number of rows written in a lattice, starting with a row of 0s and a row of 1s and satisfying a local SL2-rule: for any four neighbours a, b, c, d forming a diamond we require ad − bc = 1. Since the discovery of links to cluster algebras of type A, a plethora of generalisations have been studied in the last decades. With this workshop, we will provide a platform to establish new links between friezes and research areas in geometry, algebra and combinatorics.

Geometrie und Topologie,    Algebra

Simons Laufer Mathematical Sciences Institute (SLMath)

Random graphs are ubiquitous in modern probability theory. Besides their intrinsic mathematical beauty, they are also used to model complex networks. In the early 2000’s, I. Benjamini and O. Schramm introduced a mathematical framework in which they endowed the set of locally finite rooted connected graphs with the structure of a Polish space, called the local topology. The goal of this summer school is to introduce the framework of local limits of random graphs, the concepts of Benjamini-Schramm (or unbiased) limits and unimodularity, as well as the most important applications. The lectures will be delivered by Nicolas Curien (Prof. Paris-Saclay University) and Justin Salez (Prof. Université Paris-Dauphine) and will be complemented by many problem sessions, where students will work in small groups under the guidance of teaching assistants, who are researchers in the field.

Kurse und Veranstaltungen für Studenten der Mathematik,    Statistik, Spieltheorie