In a classic paper from 1990, Beilison, Lusztig and MacPherson gave a geometric realization of the quantized enveloping algebra of gl_n by defining a convolution product on the space of invariant functions over the variety of pairs of n-step partial flags over a finite field. This construction was generalized by Rosso to the mirabolic setting by modifying the points on the variety to include the additional data of a vector. A presentation for this "mirabolic quantum group" in terms of generators and relations was recently given by Fan, Zhang and Ma. I will describe this construction of the mirabolic quantum group and discuss its representation theory. Time permitting, I will also discuss a mirabolic quantum Schur-Weyl duality that this algebra satisfies with a mirabolic version of the Hecke algebra of Type A.

This Ph.D. thesis concerns optimal transport and effective resistance on finite weighted graphs. We investigate a number of directions, including applications of these topics to geometric graph theory and combinatorial optimization, as well as extensions of them to graphs with matrix-valued edge weights. We conclude with a number of results elucidating their connections.

It was observed by Kollár that the moduli functor of stable varieties in characteristic p>0 is no longer proper when one considers varieties of dimension ≥ 3. The key point is the existence of families of plt good minimal models of general type for which taking the relative canonical model does not commute with base change. I am going to illustrate an example showing that this kind of pathological behavior is not limited to the relative canonical model, but can indeed occur for any step of the relative MMP.

While Weierstrass’ Approximation Theorem guarantees that continuous functions can be uniformly approximated by polynomials, it provides no information about the rate of this convergence. Bernstein’s Lethargy Theorem (BLT) classically addresses this gap by proving that the error of best polynomial approximation can decay at an arbitrarily slow, prescribed rate. This talk explores the evolution of BLT from its roots in classical approximation theory to its broad applications in functional analysis. We will discuss extensions of BLT to abstract Banach spaces and Frechet spaces. Building on this framework, we will investigate the deep connections between lethargy phenomena and operator ideals, the influence of Banach space reflexivity on the existence of lethargic convergence, and the interplay between BLT and interpolation theory via the Peetre K-functional.

I will show how to use multiple Dirichlet series techniques to prove new asymptotics for the number of G-extensions with bounded discriminant, inspired by their use in the study of moments of $L$-functions. In particular, assuming the generalized Lindelof Hypothesis we prove the existence of an asymptotic whenever $G$ has nilpotency class $2$. This work is joint with Alina Bucur.

[pre-talk at 3:00PM by Justine Dell]

 A central mystery in deep learning is why gradient-based optimizers reliably find good solutions despite training a nonconvex loss function. Most theoretical work either proves favorable properties under strong assumptions or gives worst-case bounds that are too loose to be useful in practice. In this talk, I take a different approach: rather than analyzing large networks asymptotically, I study the loss landscape of a small, concretely specified network where every critical point can be computed exactly using tools from algebraic geometry. The findings are sharp: across all data realizations and all three optimizers, the dynamically accessible critical points are in exact bijection with the local minima of the loss, as independently confirmed by Hessian eigenvalue analysis. All saddle points are completely inaccessible, with empirical basin measure zero. I also show that removing the network's scaling symmetry via an affine chart systematically degrades all three optimizers, a phenomenon explained by the fiber connectivity structure of the parameterization map. Finally, I will discuss how these findings position algebraically-certified small networks as a rigorous testbed for optimizer theory, and outline extensions to wider architectures and polynomial activation functions.

One can use the Riemann zeta function evaluated at a parameter s>1 to create a probability distribution on the positive integers. If X(s) is a random integer with this distribution, one might ask whether one can produce a natural stochastic process in the parameter s. Using an idea of Lloyd this is possible and reveals a predominant Poisson behavior in X(s). In addition, we can use mod-Poisson convergence of Jacod, Kowalski and Nikeghbali to prove precise large deviation estimates for the number of prime divisors of X(s) as s goes down to 1. These ideas apply more generally to integers selected via Dirichlet series, polynomials with coefficients in a finite field or ideals selected in a Dedekind domain. This talk is based on joint work with Jingyuan Chen and Mariia Khodiakova.

One of the central problems in arithmetic statistics is counting number field extensions of a fixed degree with a given Galois group, ordered by discriminant. In this talk, we focus on extensions with Galois group of the form C2 ≀ H over an arbitrary base field. We begin by discussing the historical development of results in counting such extensions, including the work of Jürgen Klüners, who established the main term in this setting. We then turn to the problem of obtaining explicit power-saving error terms. Using Tauberian methods, we describe how such savings can be achieved, and present an alternative approach that leads to improved power-saving error terms in greater generality. We conclude with a brief discussion of possible directions for future work.

In this talk, a selection of results from my thesis will be presented. The results will concern the application of ultraproduct methods in operator algebras and surrounding fields. They will be organized around three themes: applications to the structure theory of operator algebras, continuous model theory of tracial von Neumann algebras, and approximation theory of groups and group actions.

Syntomic cohomology, extracted from the Frobenius fixed points of prismatic cohomology, is a basic motivic invariant of schemes in mixed and positive characteristic. Recent work of Bhatt--Lurie and Drinfeld geometrizes this theory and defines coefficients for syntomic cohomology as quasi-coherent sheaves on certain stacks. In this talk, I will explain how to completely describe these stacks, and therefore their categories of sheaves, in terms of Fontaine--Laffaille--Faltings modules in the special case of Frobenius liftable schemes. This result is closely related to recent results of Ogus, Terentiuk--Vologodsky--Xu and an announced result of Madapusi--Mondal, although a precise relationship remains elusive. While our result is of a classical flavour, the techniques involved use some recent conceptual advances in derived geometry, due to several authors, which I will also explain if time permits.

In Part I, we provide an explicit version of the Eichler--Selberg trace formula for Shimura curves with level structure over the rationals. As an application, we provide an algorithm to compute the isogeny decomposition of the Jacobian of Shimura curves into modular abelian varieties using the method that Rouse--Sutherland--Zureick-Brown developed for classical modular curves. We also give a trace formula for definite quaternionic modular forms over the rationals.

In Part II, we compile a complete list of isomorphism class representatives of curves of genus 6 over $\mathbb{F}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_6$, due to Mukai in the generic case. Our computed value of $\#\mathcal{M}_6(\mathbb{F}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom--Canning--Petersen--Schmitt.

We also report progress on compiling a corresponding list in genus 7 over $\mathbb{F}_2$ (for which explicit descriptions of canonical curves in each stratum of the Brill--Noether stratification of the moduli space $\mathcal{M}_7$ are also available) and genus 5 over $\mathbb{F}_3$, where the censuses are complete in all except for the generic strata in both cases.