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 […]
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 […]
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 […]
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 […]
Virtual links can be represented as equivalence classes of Gauss diagrams under Reidemeister moves. The Forbidden Moves are moves which look plausible but change the virtual isotopy class of the […]
We will examine the multiplicative structure of two skein algebras--- the usual Kauffman bracket skein algebra of a surface (generated by loops) and a generalization of it due to Roger-Yang […]
This is a talk in two parts covering two projects that the speaker mentored over the summer of 2025. The first project deals with the study of polytopes that arise […]
The classical Siegel's lemma (1929) asserts the existence of a nontrivial integer solution to an underdetermined integer homogeneous linear system, whose "size" is small as compared to the size of […]
Hecke algebras play a central role in both number theory and representation theory. While some Hecke algebras have explicit descriptions in terms of generators and relations, others are understood through […]
A key problem in computer proofs based on solutions from copositive optimization, is checking whether or not a given quadratic form is completely positive or not. In this talk we describe the first known algorithm for arbitrary rational input. It is based on a suitable adaption of Voronoi's Algorithm and the underlying theory from positive definite […]
A low autocorrelation binary sequence of length $\ell$ is an $\ell$-tuple of $+1$s and $-1$s that does not strongly resemble any translate of itself. Such sequences are used in communications and remote sensing for synchronization and ranging, where translation represents time delay. A single number that indicates how good a sequence is for such purposes, […]
In this talk, I'll attempt to give a friendly introduction to tropical linear series and explore their relationship to matroid theory. Along the way, we'll stop to admire the beautiful view from enumerative geometry and combinatorics. This is joint work with Chih-Wei Chang, Matthew Dupraz, Hernan Iriarte, David Jensen, Sam Payne, and Jidong Wang, and also with […]
