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
Toric integrable systems, also known as symplectic toric
manifolds, arise as examples in different contexts within geometry
and related areas. Semitoric integrable systems are a
generalization of toric integrable systems in dimension four. In
this...
For the past 25 years, Legendrian contact homology has played a
key role in contact topology. I'll discuss a package of new
invariants for Legendrian knots and links that builds on Legendrian
contact homology and is derived from rational symplectic...
The goal of this lecture series is to give you a glimpse into
the Langlands program, a central topic at the intersection of
algebraic number theory, algebraic geometry and representation
theory. In the first lecture, we will look at a celebrated...
Geometry and representation theory are intertwined in deep and
foundational ways. One of the most important instances of this
relationship was uncovered in the 1970s by Deligne and Lusztig: the
representation theory of matrix groups over finite...
