We will discuss recent results about a class of Hamilton-Jacobi-Bellman (HJB) equations in spaces of probability measures that arise in the study of stochastic optimal control problems for systems of particles with common noise, interacting through their empirical measures. We will present a procedure to show that the value functions $u_n$ of particle problems, when converted to functions of the empirical measures, converge as $n\to\infty$ uniformly on bounded sets in the Wasserstein space of probability measures to a function , which is the unique viscosity solution of the limiting HJB equation in the Wasserstein space. The limiting HJB equation is interpreted in its "lifted" form in a Hilbert space, a technique introduced by P.L. Lions. The proofs of the convergence of $u_n$ to use PDE viscosity solution techniques. An advantage of this approach is that the lifted function of is the value function of a stochastic optimal control problem in the Hilbert space. We will discuss how, using Hilbert space and classical stochastic optimal control techniques, one can show that is regular and there exists an optimal feedback control. We then characterize as the value function of a stochastic optimal control problem in the Wasserstein space. The talk will also contain an overview of existing works and various approaches to partial differential equations in abstract spaces, including spaces of probability measures and Hilbert spaces.

On November 4th, 2019, the following talk was given: "Junior Topology: The Volume Conjecture by Casandra Monroe in RLM 12.166", with an abstract stating: "In 1995, Kashaev proposed a striking relationship between evaluations of the n-colored Jones polynomial of a knot and the hyperbolic volume of its knot complement, called the Volume Conjecture. In this (notably not a Senior Topology) talk, we will discuss both halves of this alleged equality and the knots we know that satisfy it." Since then, a lot of things have changed, both mathematically and otherwise. I'd like to revisit this subject, correct some old mistakes, and probably make some new ones. I cannot fully predict what will be in this talk because it is partially in your (the audience's) hands, but I am fairly certain it will contain: allusions to Borges and the Braid Group, the wisdom of the directing duo Daniels and also of Daniel S. Freed, and of course, knots and geometry.

The fundamental group of the link of algebraic singularities is an important invariant. In the 60's, Mumford proved that the triviality of this fundamental group characterizes smoothness for normal surface singularities. In 2014, Kollar and Kapovich proved that every finitely presented group appears as the fundamental group of a 3-dimensional normal singularity. In the case of the singularities of the minimal model program, we have some positive results. In 2021, Braun proved that the fundamental group of a klt singularity is finite. However, much less is known in the log canonical case. In this talk, we will discuss some recent results regarding the fundamental groups of log canonical singularities of dimension at most 4. This is joint work with Fernando Figueroa.

Studying the spectra of graph matrices gives us a way to measure and understand various graph properties. One interesting measurement is the gap between the largest and second largest eigenvalues of a finite graph, which can give us bounds on how well a graph expands (a measure of connectivity). It was conjectured by Noga Alon in ’86 and proved by Joel Friedman in ’03 that uniformly random d-regular graphs have asymptotically almost surely maximal spectral gap, or almost optimal expansion properties, as the number of vertices tends to infinity. There are still many open questions regarding sequences of graphs and their spectral properties. In this talk, I will discuss some results about the spectra of regular graphs and some of the techniques used in Friedman’s (and later, Charles Bordenave’s) proof. I will also discuss graph convergence and some known results about the spectra of sequences of graphs that converge. Finally, I will discuss a work in progress with Lewis Bowen about the spectral gap of a regular graph sequence that is not uniformly chosen. This is a candidacy talk.

We consider general two-dimensional autonomous velocity fields and prove that their mixing and dissipation features are limited to algebraic rates. As an application, we consider a standard cellular flow on a periodic box, and explore potential consequences for the long-time dynamics in the two-dimensional Euler equations.