Across various applications, such as diffusion modeling, image processing, and mechanics, continuum models that incorporate nonlocal effects have seen greatly increased use in recent years. These models are characterized by partial integro-differential equations; that is, equations of integral operators that act on difference quotients of multi-variable functions. In this talk, we will discuss recent contributions to the mathematically rigorous underpinning of such nonlocal models across several different contexts. Such contributions include the properties of solutions to nonlocal equations, the robust nature of their discretizations, rigorous characterizations of long-range and other phenomena captured by the equations, and the consistency of nonlocal models with existing classical models in suitable asymptotic regimes. The contexts include continuum mechanics, semi-supervised learning, fractional PDEs, and coupled local-nonlocal equations.

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.

I will present a series of new developments in the subject of strong convergence, development in joint works: with David Gao, Aareyan Manzoor, Gregory Patchell on a new source of purely finite matricial fields; with David Gao on Toeplitz exactness; and with David Gao and Mahan Mj on strongly converging unitary representations of extensions by exact groups. These works feature a new systematic method for proving strong convergence of unitary representations, and answers various open problems in the field. As an example, we show that for any closed hyperbolic 3-manifold, there exists a sequence of unitary representations with finite images strongly converging to the left regular representation. In light of Antoine Song's approach towards Yau's conjecture on the existence of minimal surfaces of negative curvature in 3-spheres, our work can be accessed towards constructing minimal 3 dimensional submanifolds whose shape is hyperbolic 3-space, inside larger dimensional spheres.

I will give a (biased) survey on the role and importance of totally geodesic submanifolds, particularly in negative curvature. The goal will be to start with some broad strokes motivation in terms of differential geometry and symmetry, to move through older work where totally geodesic manifolds play a key role as a tool and then reach recent work of my own with various coauthors where finiteness results were proven.  This includes joint work with Bader, Elliott Smith, Filip, Lafont, Lowe, Miller, and Stover.  Elliott Smith and Miller were both UCSD undergrads.

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.

Graphs are ubiquitous for modeling relational data, appearing across social networks, biology, and chemistry. Measuring the similarity between graphs is central to tasks like graph classification and clustering, yet it poses significant computational challenges on large datasets. We introduce a matrix factorization framework for graph comparison. Viewing adjacency matrices as kernel matrices, we first define a pseudo-metric called MMFD that admits a simple closed-form solution without iterative optimization. We then generalize it to MFD, which more effectively exploits the factor structure of adjacency matrices. To handle large-scale clustering, we further develop a variant with linear time and space complexity in the number of graphs. Experiments on real-world datasets show that our methods substantially improve clustering performance and efficiency over existing approaches.

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.

Recent work of Bhatt-Lurie and Drinfeld has constructed a category of p-adic motives (aka prismatic F-gauges) for schemes in positive and mixed characteristic. Roughly, these correspond to a notion of 'variations of Hodge structures' in integral p-adic Hodge theory. In this talk, I will review this notion and explain how to completely describe it in the case of Frobenius liftable schemes in positive characteristic . This description is in terms of (big) Fontaine-Laffaile modules, a somewhat classical coefficients system in p-adic Hodge theory and is closely related to recent results of Ogus and Terentiuk--Vologodsky--Xu.
While the result is of a classical flavour, our techniques use some recent conceptual advances in derived geometry, due to various authors, which we will explain if time permits.