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
