Georgia Institute of Technology College of Sciences

We will try to address a few universality questions for the behavior of large random matrices over finite fields, and then present a minimal progress on one of these questions.

A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air. When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps. The double-bubble minimizes total surface area among all sets enclosing two fixed volumes. This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s. The analogous case of three or more Euclidean sets is considered difficult if not impossible. However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems. We also use the calculus of variations. Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking. http://arxiv.org/abs/1901.03934

We study delocalization properties of null vectors and eigenvectors of matrices with i.i.d. subgaussian entries. Such properties describe quantitatively how "flat" is a vector and confirm one of the universality conjectures stating that distributions of eigenvectors of many classes of random matrices are close to the uniform distribution on the unit sphere. In particular, we get lower bounds on the smallest coordinates of eigenvectors, which are optimal as the case of Gaussian matrices shows. The talk is based on the joint work with Konstantin Tikhomirov.

The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations(i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under theGromov-Hausdorff convergence (literally, there is no such continuityin general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimensionand diameter and lower bound on sectional curvature. Points of thiscompactification are pairs: an Alexandrov space and a constructible(in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.