This talk is an overview of my thesis work, which consists of 3 projects exploring the effect of multiscale structure on a class of interacting particle systems called weakly interacting diffusions. In the absence of multiscale structure, we have a collection of N particles, with the dynamics of each being described by the solution to a stochastic differential equation (SDE) whose coefficients depend on that particle's state and the empirical measure of the full particle configuration. It is well known in this setting that as N approaches infinity, the particle system undergoes the ``propagation of chaos,'' and its corresponding sequence of empirical measures converges to the law of the solution to an associated McKean-Vlasov SDE. Meanwhile, in our multiscale setting, the coefficients of the SDEs may also depend on a process evolving on a timescale of order 1/\epsilon faster than the particles. As \epsilon approaches 0, the effect of the fast process on the particles' dynamics becomes deterministic via stochastic homogenization. We study the interplay between homogenization and the propagation of chaos via establishing large deviations and moderate deviations results for the multiscale particles' empirical measure in the combined limit as N approaches infinity and \epsilon approaches 0. Along the way, we derive rates of homogenization for slow-fast McKean-Vlasov SDEs.

Every knot in S^3 bounds a PL disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? In the joint work with Hom and Stoffregen, we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. The proof uses Heegaard Floer homology. More specifically, the obstruction comes from knot cobordism maps by Zemke and the construction uses recent filtered mapping cone formula for cables of the knot meridian.

We discuss the DGA map induced by an exact Lagrangian cobordism, and an analytic strategy to lift the map to integer coefficients, introduced by Fukaya, Oh, Ohta and Ono and further adapted by Ekholm, Etnyre, and Sullivan and Karlsson respectively. We then explain how this strategy can be applied to find a concrete combinatorial formula for a mini-dipped pinch move, thereby completely determining the integral DGA maps for all decomposable, orientable Lagrangian cobordisms. If time permits, we will show how to obtain this formula in a model case. We will also go into future potential work, including applications to Heegaard Floer Homology and nonorientable cobordisms.

Circadian rhythms are biological oscillations that align our physiology and behavior with the 24-hour environmental cycles conferred by the Earth’s rotation. In this talk, I will discuss two projects that focus on circadian clock cells in the brain and the entrainment of circadian rhythms to the light-dark cycle. Most of what we know about the electrical activity of circadian clock neurons comes from studies of nocturnal (night-active) rodents, hindering the translation of this knowledge to diurnal (day-active) humans. In the first part of the talk, we use data assimilation and patch-clamp recordings from the diurnal rodent Rhabdomys pumilio to build the first mathematical models of the electrophysiology of circadian neurons in a day-active species. We find that the electrical activity of circadian neurons is similar overall between nocturnal and diurnal rodents but that there are some interesting differences in their responses to inhibition. In the second part of the talk, we use tools from dynamical systems theory to study the reentrainment of a model of the human circadian pacemaker following perturbations that simulate jet lag. We show that the reentrainment dynamics are organized by invariant manifolds of fixed points of a 24-hour stroboscopic map and use these manifolds to explain a rapid reentrainment phenomenon that occurs under certain jet lag scenarios.

Markoff triples were introduced in 1879 and have a rich history spanning many branches of mathematics. In 2016, Bourgain, Gamburd, and Sarnak answered a long standing question by showing there exist infinitely many composite Markoff numbers. Their proof relied on showing the connectivity for an infinite family of graphs associated to Markoff triples modulo p for infinitely many primes p. In this talk we discuss what happens for the projective analogue of Markoff triples, that is surfaces W in P^1 x P^1 x P^1 cut out by the vanishing of a (2,2,2)-form that admit three non-commuting involutions and are fixed under coordinate permutations and double sign changes. Inspired by the work of B-G-S we investigate such surfaces over finite fields, specifically their orbit structure under their automorphism group. For a specific one-parameter subfamily W_k of such surfaces, we construct finite orbits in W_k(C) by studying small orbits that appear in W_k(F_p) for many values of p and k. This talk is based on joint work with E. Fuchs, J. Silverman, and A. Tran.

The classification of discrete series is one main subject in Langlands program with numerous applications. We first explain the result on the classification of discrete series of odd GSpin groups, generalizing the Mœglin-Tadi ́c classification for classical groups. Note that our approach will give alternate proof for classical groups. This is a joint work with Ivan Mati ́c. We also explain its application on the generic local Langlands correspondence via Langlands-Shahidi method. If time permits, we will explain possible generalization of those to other groups, which is work in progress

In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.