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American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to further developing the method of flag algebras and its applications. Flag algebras, developed by Razborov in 2007, allows one to solve problems in combinatorics via streamlined calculations that combine elements from computer engineering and optimization. It led to many recent breakthroughs on long-standing open problems of Erdős, Sós, Turán, Gromov and Zarankiewicz, to name a few. The technique is versatile and can be applied in other settings than graphs and hypergraphs including permutations, oriented graphs, point sets, embedded graphs, and phylogenetic trees.

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to studying arithmetic dynamics of multiple maps. In classical arithmetic dynamics, we consider the iteration of a single endomorphism of a variety defined over a field of arithmetic interest — typically a number field or the function field of a curve. An exciting new direction in arithmetic dynamics that holds particular promise for striking new results is what we call "dynamics of multiple maps": dynamical behavior arising from the interaction of two or more endomorphisms on the same space. This includes iteratively applying rational functions on P1 chosen at random from a family according to some probability distribution; a correspondence from a variety X to itself; and forming words from a non-commuting family of involutions on a K3-surface.

Institute for Computational and Experimental Research in Mathematics, Brown University, Providence, RI (ICERM)

Webs are diagrammatic tools for representing complex calculations graphically. These diagrams first arose from the representation theory of classical groups, and they have since become important in disparate areas of mathematics. In representation theory, they encode morphisms of quantum groups. In topology, webs give rise to powerful link invariants. In algebra and geometry, Kuperberg's \(mathrm{sl}(3)\) web bases have important relationships with the theory of cluster algebras and affine buildings. In combinatorics, they explain certain dynamics on Young tableaux. Recent work by Gaetz--Pechenik--Pfannerer--Striker--Swanson introduced an \(\mathrm{sl}(4)\) web basis that has exploited and extended exciting connections between webs, plabic graphs, and crystals. There are further connections to total positivity, duality conjectures for cluster algebras and mirror symmetry.

Théorie des graphes et combinatoire,    Théorie des groupes