School of Mathematics

Cohen, Lenstra, and Martinet gave conjectural distributions for the class group of a random number field. Since the class group is the Galois group of the maximum abelian unramified extension, a natural generalization would be to give a conjecture...

The restriction conjecture, one of the most central problems in harmonic analysis, studies the Fourier transform of functions defined on curved surfaces; specifically, it claims that the level sets of such Fourier transforms are relatively small...

The lifeblood of interactive proof systems is randomness, without which interaction becomes redundant. Thus, a natural long-standing question is which types of proof systems can indeed be derandomized and collapsed to a single-message NP-type...

We consider a general system of interacting random loops which includes several models of interest, such as the spin O(N) model, the double dimer model, random lattice permutations, and is related to the loop O(N) model and to the interacting Bose...

In this talk, we will discuss the behavior of the Yamabe flow on an asymptotically flat (AF) manifold. We will first show the long-time existence of the Yamabe flow starting from an AF manifold and discuss the uniform estimates on manifolds with...

Algorithms for understanding data generated from distributions over large discrete domains are of fundamental importance.  In this talk, we consider the sample complexity of *property testing algorithms* that seek to to distinguish whether or not an...

The interchange process \sigma_T is a random permutation valued process on a graph evolving in time by transpositions on its edges at rate 1. On Z^d, when T is small all the cycles of the permutation \sigma_T are finite almost surely. In dimension d...

The Cohen-Lenstra heuristics give predictions for the distribution of the class groups of a random quadratic number field. Cohen and Martinet generalized them to predict the distribution of the class groups of random extensions of a fixed base field...

Lower scalar curvature bounds on spin Riemannian manifolds exhibit remarkable rigidity properties determined by spectral properties of Dirac operators. For instance, a fundamental result of Llarull states that there is no smooth Riemannian metric on...