# Video Lectures

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).

Since work of Montgomery and Katz-Sarnak, the eigenvalues of random matrices have been used to model the zeroes of the Riemann zeta function and other L-functions. Keating and Snaith extended this to also model the distribution of values of the L...

In this lecture, we will review recent works regarding spectral statistics of the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices.

Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \la_2\geq\la_3\cdots\geq \la_N$...

I’ll speak about new joint work with Rachel Greenfeld and Marina Iliopoulou in which we address some classical questions concerning the size and structure of integer distance sets. A subset of the Euclidean plane is said to be an integer distance...

I'll discuss spectral gaps in the following contexts:

- d-regular graphs

- locally symmetric spaces e.g. hyperbolic manifolds

- finite dimensional unitary representations of discrete groups
e.g. free groups, surface groups

Of particular interest are...