Members’ Colloquium

Erd\H{o}s unit distance problem asks the following: Let P

be a set of n distinct points in the plane, and let U(P)

denote the number of pairs of points in P that are at distance 1. How large can U(P) be? In 1946, Erd\H{o}s showed that for P=[n‾√]×[n‾√...

In their 1971 study of telephone switching circuitry, Graham and Pollak designed a novel addressing scheme that was better suited for the faster communication required by computers. They introduced the distance matrix of a graph, and used its...

The Prékopa-Leindler inequality (PL) and its strengthening, the Borell-Brascamp-Lieb inequality, are functional extensions of the Brunn-Minkowski inequality from convex geometry, which itself refines the classical isoperimetric inequality. These...

In the late 1800s, in the course of his study of classical problems of number theory, the young Hermann Minkowski discovered the importance of a new kind of geometric object that we now call a convex set. He soon developed a rich theory for...

The Poisson-Furstenberg boundary  is a measure space that describes asymptotics of infinite trajectories of random walks. The boundary is non-trivial if and only if the defining  measure admits non-constant bounded harmonic functions.

The origin of...

We travel the years in order to understand the relationship between Nilpotency and Riemannian geometry: including Gromov's almost flat theorem for manifolds with bounded curvature and Fukaya-Yamaguchi's almost nilpotency of spaces with lower...

This colloquium will explore some fundamental issues in the mathematics of plasmas, focusing on the stability and instability of solutions to Vlasov-type equations, which are crucial for describing the behavior of charged particles in a plasma. A...

Expansion in graphs is a well studied topic, with a wealth of applications in many areas of mathematics and the theory of computation.

High dimensional expansion is a generalization of expansion from graphs to higher dimensional objects, such as...

A promenade in Serre's paper, plus some related results of mine.

Joint work with Amir Mohammadi, Zhiren Wang, and Lei Yang

Let Q be an indefinite ternary quadratic form. In the 1980s Margulis proved the longstanding Oppenheim Conjecture, stating that unless Q is proportional to an integral form, the set of values...