Integer partitions are ubiquitous in mathematics, arising in subjects as disparate as algebraic combinatorics, algebraic geometry, number theory, representation theory, to mathematics physics. Many of the deepest results on partitions have their origin in the work of Ramanujan. In this lecture, we will describe a completely new and unexpected role for partitions that also arises […]
The classical oddtown and eventown problems involve a collection of subsets of a finite set with an odd (resp. even) number of elements such that all pairwise intersections contain an even number of elements. In this talk, we will discuss these results as well as the following variants: We consider set sizes and pairwise intersection […]
Quandles are algebraic structures encoding the motion of knots through space. Quandle cocycle quivers categorify the quandle cocycle invariant. In this talk we will define a quiver representation associated to quandle cocycle quivers and use it to obtain new polynomial invariants of knots.
Let K be a compact convex set in the Euclidean space R^n. How many lights are needed to illuminate its boundary? A classical conjecture of Boltyanskii (1960) asserts that 2^n lights are sufficient to illuminate any such set K. While this is still open, an earlier observation of Hadwiger (1945) guarantees that if K has […]
Following Kalashnikov, we recover Givental’s small J function for CP^1 by viewing it as a quiver flag variety.
Following Kalashnikov, we recover Givental’s small J function for CP^1 by viewing it as a quiver flag variety.
We’ll first define the two-point gravitational correlators which appeared last week as descendant Gromov-Witten invariants. By request, we’ll then introduce Gromov-Witten invariants as they appear in the expository work https://arxiv.org/abs/2501.03232 and give CP^1 to demonstrate some of the identities which GW invariants satisfy. If time allows, we’ll also give the small and big quantum cohomology for CP^1.
I will present an integral — requiring no character twists — converse theorem for recognizing when is a Dirichlet series with algebraic integer coefficients equal to the L-function of a modular form. This refines the unbounded denominators conjecture of Atkin and Swinnerton-Dyer. Analogies with basic function field arithmetic then suggest a quantitative refinement which precludes a pair of GL(2) automorphic L-functions […]
A big area in combinatorics over the last several decades has been the study of pattern-avoiding permutations, whose enumeration is exciting and mysterious. Alternating sign matrices (ASMs) are a generalization of permutations whose study in combinatorics has also been exciting and mysterious. In this talk, I will explain some new asymptotic results involving the number […]
Large Language Models like ChatGPT rely on surprisingly familiar mathematics. This talk will explore how ideas from (linear) algebra, number theory and combinatorics appear — both directly and indirectly — in the structure and behavior of these models. Along the way, we’ll touch on themes like structure, symmetry, and scale, and consider how abstract mathematical […]
A Jacobian variety is a principally polarized abelian variety (PPAV) associated with a smooth complex algebraic curve. For dimensions less than or equal to 3, every PPAV is either a Jacobian or a product of Jacobians. The Schottky problem concerns dimensions 4 and greater: which PPAVs are Jacobians? The Schottky problem can also be posed […]
The recognition that theoretical models of natural language syntax have robust algebraic foundations is longstanding. Both the syntactic structures proposed (trees, semirings, etc.) and metrics developed to understand them (the Chomsky hierarchy, partial orders, and so forth) closely resemble structures and systems familiar to theoretical mathematicians (groups, rings, fields, ...). Despite the underlying mathematical tools, […]
