Algebra / Number Theory / Combinatorics Seminar

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 preserved under elementary equivalence.  As a result, we give the first example of a group that is inner amenable but not uniformly inner amenable.

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 of graph vertices chosen to capture the global behavior of functions from the vertex set to the real numbers. We first motivate and define graphical […]

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 field in question and can be attractive from both, theoretical and applied perspectives. We consider this construction when M is spanned by the set of […]

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 point transform f_S algebraically or analytically.  Here we focus on its analytic properties, showing that it is a complete invariant.   In fact, we prove that […]

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 derivative operator is replaced by the p-th power Frobenius operator. In this talk we will explain a relation between the mod p Riemann–Hilbert correspondence and […]

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 already very deep for algebraic curves defined over Q.  An intrinsic natural number associated with the curve, called its genus, plays an important role in […]

Let $C$ be a nice (smooth, projective, geometrically integral) curve over a number field $k$. The single most important geometric invariant of a curve is the genus, which can control various arithmetic properties of a curve. A celebrated result of Faltings implies that all points on $C$ come in families of bounded degree, with finitely […]

This talk explores elementary probability and statistics through the language of category theory. We introduce a category of Bundles and use it to reinterpret several results typically covered in an introductory course on probability and statistics. This approach naturally reveals the underlying geometric structures common to these results. The talk is accessible to anyone familiar […]

I will talk about some results concerning the non-vanishing of $L$-functions associated to fixed order characters $\ell$ at the central point over functions fields. Quadratic characters have been studied a lot over the years, and very good non-vanishing results are available in this case, due to work of Soundararajan. When focusing on cubic and higher […]

Hunter's theorem ensures that the complete homogeneous symmetric (CHS) polynomials of even degree are positive definite functions.  We provide new proofs of Hunter's theorem, applications to operator theory, and a noncommutative (NC) generalization that sheds light even on the commutative case.  Surprisingly, this work emerged from a problem in analytic combinatorics.