Video Lectures

To study the asymptotic behavior of orbits of a dynamical system, one can look at orbit closures or invariant measures. When the underlying system has a homogeneous structure, usually coming from a Lie group, with appropriate assumptions a wide...

Starting with the "Leibniz" formula for π

π/4=1−1/3+1/5−1/7+…

the special values of Dirichlet L-functions have long been a source of fascination and frustration. From Euler's solution in 1734 of the Basel problem to Apery's proof in 1978 that zeta(3...

Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups of number...

The Higher order Fourier uniformity conjecture asserts that on most short intervals, the Mobius function is asymptotically uniform in the sense of Gowers; in particular, its normalized Fourier coefficients decay to zero.  This conjecture is known to...

What is the densest lattice sphere packing in the d-dimensional Euclidean space? In this talk we will investigate this question as dimension d goes to infinity and we will focus on the lower bounds for the best packing density, or in other words on...

Discrete subgroups of PSL(2,C) are called Kleinian groups and they are fundamental groups of complete oriented hyperbolic 3-manifolds/orbifolds. Except for countably many conjugacy classes, all Kleinian groups have infinite co-volume in PSL(2, C).

The Mobius function is one of the most important arithmetic functions. There is a vague yet well known principle regarding its randomness properties called the “Mobius randomness law". It basically states that the Mobius function should be...

Bounds for Dirichlet polynomials play an important role in several questions connected to the distribution of primes. For example, they can be used to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to...

Bounds for Dirichlet polynomials play an important role in several questions connected to the distribution of primes. For example, they can be used to bound the number of zeroes of the Riemann zeta function in vertical strips, which is relevant to...

We'll discuss problems where bounds for L-functions have arisen as inputs and where techniques for estimating them through their integral representations have been useful (all of which have been shaped and influenced by Peter Sarnak’s work).