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 […]
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 […]
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 — […]
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 […]
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, […]
Biquandle arrow weights invariants are enhancements of the biquandle counting invariant for oriented virtual and classical knots defined from biquandle-colored Gauss diagrams using a tensor over an abelian group satisfying certain properties. In this talk, we categorify the biquandle arrow weight polynomial invariant using biquandle coloring quivers, obtaining new infinite families of polynomial invariants of […]
An inner amenable group is one in which there is a finitely additive conjugation-invariant probability measure on the non-identity elements. In this talk, we show that inner amenability is not […]
A graphical design is a quadrature rule for a graph inspired by classical numerical integration on the sphere. Broadly speaking, that means a graphical design is a relatively small subset […]
A Z-module M in a number field K gives rise to a lattice in the corresponding Euclidean space via Minkowski embedding. Such lattices often carry inherited structure from the number […]
Given any finite set of integer points S, there is an associated function f_S that encodes S, which we call its integer point transform. One can think of this integer […]
The Riemann–Hilbert correspondence relates algebra to differential equations on complex algebraic varieties. In characteristic p, there is an analogous correspondence due to Emerton–Kisin and later generalized by Bhatt–Lurie, where the […]
It is a fundamental question to find rational solutions to a given system of polynomials, and in modern language this translates into finding rational points in algebraic varieties. It is […]
