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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

American Institute of Mathematics, Pasadena, California

This workshop, sponsored by AIM and the NSF, will be devoted to new developments in integral p-adic cohomology theories, focusing in particular on their applications to the study of integral models of Shimura varieties. The past decade has seen several innovations in p-adic cohomology, with the introduction, first of perfectoid geometry, and more recently, of prismatic cohomology and its refinements. The power of these general theories has been brought to bear on the problem of understanding the arithmetic behavior of Shimura varieties, both local and global, with applications to key number-theoretic frameworks like the Langlands and Kudla programs. The workshop’s goal is to push these applications further, and to explore new ones, with the eventual goal of obtaining a systematic and naturally functorial theory of integral models of Shimura varieties.

Simons Foundation

A conference marking the 50th anniversary of the publication of Barry Mazur's "Modular curves and the Eisenstein ideal" and Ken Ribet's "A modular construction of unramified p-extensions of Q(μ_p)". The talks will feature recent advances in areas of research that have been influenced by these two papers.

Institute for Computational and Experimental Research in Mathematics

A conjecture of Malle predicts an asymptotic formula for the quantity of number fields with a given Galois group and whose discriminant is bounded by a growing parameter. This conjecture is foundational to the area of Arithmetic Statistics and has received considerable interest over the past 20 years, notably including Bhargava receiving the Fields Medal in part for his work on quartic and quintic field extensions.

Tél.: [4018635030];     Email.: info@icerm.brown.edu