Consider an ordinary isogeny class of elliptic curves over a finite, prime field. Inspired by a random matrix heuristic (which is so strong it’s false), Gekeler defines a local factor for each rational prime. Using the analytic class number formula, he shows that the associated infinite product computes the size of the isogeny class.

I’ll explain a transparent proof of this formula; it turns out that this product actu- ally computes an adelic orbital integral which visibly counts the desired cardinality. Moreover, it the new perspective allows a natural generalization to higher-dimensional abelian varieties.

This is joint work with Julia Gordon and S. Ali Altug. 

We prove a result about the Galois module structure of the Fermat curve using commutative algebra, number theory, and algebraic topology. Specifically, we extend work of Anderson about the action of the absolute Galois group of a cyclotomic field on a relative homology group of the Fermat curve. By finding explicit formulae for this action, we determine the maps between several Galois cohomology groups which arise in connection with obstructions for rational points on the generalized Jacobian. Heisenberg extensions play a key role in the result. This is joint work with R. Davis, V. Stojanoska, and K. Wickelgren. 

Abstract: The combinatorial properties of large cardinals tend to clash with those satisfied by G\"odel's constructible universe, especially the square property (denoted $\square_\kappa$) isolated by Jensen in the seventies. Strong cardinal axioms refute the existence of square, but it is possible with some fine-tuning to produce models that exhibit some large cardinal properties together with weakenings of square. In this talk we will exhibit some results along these lines and will outline the techniques used to produce them.

A FBMS in the unit Euclidean ball is a critical point of the area functional among all surfaces with boundaries in the unit sphere, the boundary of the ball. The Morse index gives the number of distinct admissible deformations which decrease the area to second order. In this talk, we explain how to compute the index from data of two simpler problems. The first one is the corresponding problem with fixed boundary condition; and the second is associated with the Dirichlet-to-Neumann map for Jacobi fields. We also discuss applications to a conjecture about FBMS with index 4.

A number of diverse biological systems involve diffusion in a randomly switching environment. For example, such processes arise in brain biochemistry, insect respiration, intracellular trafficking, and biochemical reaction kinetics. These processes present interesting mathematical subtleties as they combine two levels of randomness: Brownian motion at the individual particle level and a randomly switching environment.

In this talk, we will demonstrate that these systems (a) arise naturally in several biological applications and (b) are mathematically rich. Special attention will be given to establishing mathematical connections between these classes of stochastic processes. In particular, we will use these connections to study certain random PDEs by analyzing the local time of a Brownian particle in a random environment.