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 […]
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 […]
