Past Events

Abstract:  For many Diophantine equations or systems, the number of solutions within a box of side length N can grow like a power of N. Obtaining a nontrivial upper bound for the exponent is crucial for various problems. Recently, an analytic method called ``decoupling'' has been successful…

We will discuss the density of closed geodesics on hyperbolic manifolds and the trace of the resolvents as applications of Selberg's trace formula. Time permitting, we may also define Selberg's zeta function and/or prove the prime geodesic theorem.

Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic…

A striking phenomenon in probability theory is universality, where different probabilistic models produce the same large-scale or long-time limit. One example is the Kardar-Parisi-Zhang (KPZ) universality class, which contains a wide range of natural models, including growth processes modeling…

We consider the advection-diffusion equation on T^2 with a Lipschitz and time-periodic velocity field that alternates between two piecewise linear shear flows. We prove enhanced dissipation on the timescale |logν|, where ν is the diffusivity parameter. This is the optimal decay rate as ν→0 for…

Suppose that some subset of the entries of an m x n matrix are filled in with generic complex numbers. If we are allowed to fill in the remaining entries in any way that we want, what is the smallest rank that we can make it? I will explain some techniques to address this problem and what little…

Dehn surgery is an important construction in low-dimensional topology that describes all three-manifolds. The Cosmetic Surgery Conjecture predicts two different Dehn surgeries on the same knot always gives different three-manifolds. We discuss how a certain object from gauge theory, the Chern-…

Call an integer y-smooth if all its prime factors are less than or equal to y. We will take counting linear equations in y-smooth numbers as the starting point, and the primary objective is to construct a majorant of a significant subset of y-smooth numbers which can be orthogonal to exponential…

Markov chains give natural approaches to sample from various distributions on independent sets of graphs. Considering the uniform distribution on independent sets of a fixed size k in a graph G, the corresponding Markov chain is the "down-up" walk. The down-up walk starts at an arbitrary…

We will discuss joint work with Si-Ying Lee on generalizing the torsion-vanishing results of Caraiani-Scholze and Koshikawa for the cohomology of Shimura varieties. This is accomplished by combining a variety of geometric methods based on the Fargues-Fontaine curve. In the process, we also…