Georgia Institute of Technology College of Sciences

Mean-field particle systems are well-understood by now. Typical results involve obtaining a McKean-Vlasov equation for the fluid limit that provides a good approximation for the particle system over compact time intervals. However, when the driving vector field lacks a gradient structure or in the absence of convexity or functional inequalities, the long-time behavior of such systems is far from clear. In this talk, I will discuss two such systems, one arising in the context of flocking and the other in the context of sampling (Stein Variational Gradient Descent), where there is no uniform-in-time control on the discrepancy between the limit and prelimit dynamics. We will explore methods involving Lyapunov functions and weak convergence which shed light on their long-time behavior in the absence of such uniform control.

Based on joint works with Amarjit Budhiraja, Dilshad Imon (UNC, Chapel Hill), Krishnakumar Balasubramanian (UC Davis) and Promit Ghosal (UChicago).

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.

 Let A be a subset of the integer lattice of positive upper density. Roth' theorem in this setting states that there are points x,x+y,x+2y in A with the length of the gap y arbitrary large. We show that the lengths of the gaps y contain an infinite arithmetic progression, as long as one measures the length in lp for p>2 even, while this not true for the Euclidean distance.

Such results have been previously obtained in the continuous settings for measurable subsets of Euclidean spaces using methods of time-frequency analysis, as opposed our approach is based on some ideas from additive combinatorics such as uniformity norms and arithmetic regularity lemmas. If time permits, we discuss some other results that can be obtained similarly.

There is a rich history of domino tilings in two dimensions.  Through a variety of techniques we can answer questions such as: how many tilings are there of a given region or what does a random tiling look like?  These questions and their answers become significantly more difficult in dimension three and above.  Despite this curse of dimensionality, I will discuss recent advances in the theory.  I will also highlight problems that still remain open.