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Simons Laufer Mathematical Sciences Institute (SLMath)

Many exciting breakthroughs in combinatorics involve innovative applications of techniques from a wide range of areas such as harmonic analysis, polynomial and linear algebraic methods, spectral graph theory, and representation theory. This workshop will present recent developments in this area and facilitate discussions of research problems.

extremal combinatorics, extremal graph theory, probabilistic combinatorics, discrete geometry, additive combinatorics, combinatorial geometry, incidence geometry, arithmetic progressions, Discrete analysis

Graphentheorie und Kombinatorik,    Analysis

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to interactions between homological stability and asymptotic questions in number theory over function fields. In recent years, homological stability of suitable Hurwitz spaces has been used to make great progress on function field cases of Cohen-Lenstra heuristics (Ellenberg-Venkatesh-Westerland), Malle's conjecture (Ellenberg-Tran-Westerland), the Conrey-Farmer-Keating-Rubinstein-Snaith predictions (Bergström-Diaconu-Petersen-Westerland, Miller-Patzt-Petersen-Randal-Williams), and heuristics on Selmer ranks (Ellenberg-Landesman). The goal of the workshop is to bring together researchers both from analytic number theory and topology, and to further explore the connections between the two fields.

The workshop will focus on geometric and homological aspects of categorification and symplectic duality in the context of representation theory. The conference will concentrate on various representation theoretic aspects of the Langlands program.

In recent years, there has been an explosion of activity surrounding algebraic points on curves, from many different perspectives. These include the study of measures of irrationality, isolated and parametrized points, computational methods to determine algebraic points, and the arithmetic statistics of algebraic points. In this workshop, we aim to bring together researchers from these diverse perspectives, with the particular goal of developing bridges between them. The workshop will include overview talks on the various perspectives, research talks, an open problem session, and structured time for collaboration.

Zahlentheorie, Arithmetik,    Geometrie und Topologie

American Institute of Mathematics, Pasadena, California (AIM)

This workshop, sponsored by AIM and the NSF, will be devoted to studying a nascent bridge between commutative algebra and symplectic geometry, with an emphasis on developing Macaulay2 software for homological computations at the interface of these two fields. Recent breakthrough work of Hanlon-Hicks-Lazarev and Favero-Huang employs symplectic techniques to build line bundle resolutions over toric varieties, resolving several conjectures in toric geometry and multigraded commutative algebra. These results have illuminated a striking new connection between commutative algebra and symplectic geometry: this workshop will bring together experts in these fields with the goal of increasing our computational power to study the interplay between them.

For many (affine) algebraic varieties appearing in geometric representation theory and algebraic combinatorics, their coordinate algebra admits the structure of a cluster algebra. These algebraic varieties include (in order of increasing generality) the basic affine space, (double) Bruhat cells, positroid varieties, Richardson varieties, and braid varieties. Roughly speaking, this means that the corresponding algebraic variety comes with a collection of coordinate toric charts, with explicit coordinates and transformation rules between the charts that can be codified via quiver combinatorics. Conversely, the quiver and the coordinates can be read from combinatorial and geometric properties of the variety itself, though this is more of an art than a science. The goal of this workshop will be to understand the combinatorics and geometry underlying these cluster structures in the case of braid varieties, which include all of the cases described above and was established quite recently. The combinatorics can be encoded in two very different objects: weaves and 3D plabic graphs. Both appear in a variety of mathematical settings, including Soergel calculus and connections to high-energy physics. These rich combinatorial objects and their ties to other areas of mathematics inspire many questions about braid varieties, their cluster structures, and beyond.

The Institute for Mathematical and Statistical Innovation (IMSI)

This week-long conference is in celebration of the 10-year anniversary of AATRN, the Applied Algebraic Topology Research Network. It will be the first time that AATRN meets in person, bringing together researchers from different backgrounds—mathematics, statistics, computer science, physics, biology, etc. In other words, AATRN will be “geometrically realized” at the Institute for Mathematical and Statistical Innovation in Chicago, USA!

Geometrie und Topologie,    Verbandstreffen für Mathematiker

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

Linear Algebra over finite fields is a building block for several applications including data storage, error detection and correction, and public-key cryptography. These applications enable the security and possibility of our daily digital lives. This workshop aims to expose and engage junior researchers in the foundations and applications of linear algebra over finite fields.

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.

Graphentheorie und Kombinatorik,    Gruppentheorie