Abstracts

Bounds on the density of zeroes of the Riemann zeta function in strips

Abstract: 

We survey bounds about the density of zeroes of the Riemann zeta function and how such bounds relate to the distribution of primes. We discuss recent new work on the density of zeroes by Maynard and me.

Additive combinatorics from probabilistic and combinatorial perspectives 

Abstract:

Sumset is a fundamental object in additive combinatorics. Since Freiman’s celebrated theorem which characterizes the structure of integer sets with small sumsets, the structure theory of sets with small sumsets has gathered significant development, breakthroughs and applications. 

In this talk, I will discuss new combinatorial and probabilistic perspectives on sumsets. Motivated from the study of random Cayley graphs, these perspectives provide a new description of sets with small sumsets that is robust, flexible and quantitative. Emerging from this picture is the rich interaction between additive combinatorics and probabilistic combinatorics, which leads to new understanding of Cayley graphs and random Cayley graphs in general groups, as well as the resolution of longstanding questions in additive combinatorics.

Minimal submanifolds in Riemannian geometry

Abstract:

It has emerged over the past decades that the theory of stable minimal hypersurfaces is intimately connected to the study of Riemannian manifolds with scalar curvature lower bounds. This connection has led both to a better understanding of scalar curvature in Riemannian geometry and general relativity as well as to an understanding of the regularity theory of stable and min-max hypersurfaces. A corresponding theory for higher codimension submanifolds is largely unknown. In this talk we will give a brief survey of the codimension one theory and discuss recent progress and hopes for a higher codimension theory for surfaces. 

Diophantine Results for Shimura Varieties

Abstract:

Shimura Varieties are higher dimensional analogues of modular curves, and they play a foundational role in modern number theory. The most familiar Shimura varieties are the moduli spaces of Abelian varieties, and in this context we have a wealth of diophantine results, both in the number field and function field setting: Finiteness of S-rational points, the Tate conjecture, the Shafarevich conjecture, semisimplicity of Galois representations, and others. We focus on the exceptional setting for Shimura varieties, where the lack of a moduli interpretation makes matters more difficult. We explain some analogues of the aforementioned results. Crucial to this is the construction of canonical integral models, which we do at almost all primes. This is joint work with Ben Bakker and Ananth Shankar.