Conference: Modern Perspectives in Representation Theory

This grant will support the participation of early-career US-based researchers at the research program "Modern Perspectives in Representation Theory," which is a six-week program May 5-June 13, 2025 based at the Sydney Mathematical Research Institute (SMRI). The program is focused on several overlapping aspects of interest in representation theory: combinatorial, number-theoretic, and new developments using machine learning. The program will bring together experts in these subjects, visiting Sydney in overlapping intervals of one to three weeks, with the aim of bringing together and encouraging new collaborations amongst a diverse international cohort of mathematicians working in different aspects of representation theory. Two workshops will be held during the six-week program. Senior scientists in residence will explore relations between the aforementioned aspects of representation theory through their expertise in the following topics: combinatorics of Coxeter groups, total positivity, and emerging geometric methods on the representation theory of p-adic groups. In more detail: (1) Combinatorics of Coxeter groups is one of the central interfaces between Lie theory and combinatorics. Of particular interest is the recent suggestion that techniques in machine learning and artificial intelligence can be applied to problems in Coxeter combinatorics. (2) Current interest in total positivity comes from multiple directions--the recent resolution of conjectures on the topology of Lusztig’s totally nonnegative flag varieties, the development of a theory of algebraic groups over a semifield, and the discovery of deep relations with scattering amplitudes in particle physics. (3) Progress in the theory of representation theory of p-adic groups has been fueled by geometric perspectives in representation theory. Of particular interest is the relationship between geometric and categorical approaches within the context of existing algebraic constructions of representations of p-adic groups. More information can be found at the program website https://sites.google.com/view/mod-perspect-representation/home. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.