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Series: Research Horizons Seminar

I will discuss how one can solve certain concrete problems in
number theory, for example the Diophantine equation 2x^2 + 1 = 3^m, using
p-adic analysis. No previous knowledge of p-adic numbers will be assumed.
If time permits, I will discuss how similar p-adic analytic methods can be
used to prove the famous Skolem-Mahler-Lech theorem: If a_n is a sequence of
complex numbers satisfying some finite-order linear recurrence, then for any
complex number b there are only finitely many n for which a_n = b.

Series: Research Horizons Seminar

The hyperplane conjecture of Kannan, Lovasz and Simonovits asserts that the
isoperimetric constant of a logconcave measure (minimum surface to volume
ratio over all subsets of measure at most half) is approximated by a
halfspace to within an absolute constant factor. I will describe the
motivation, implications and some developments around the conjecture and an
approach to resolving it (which does not seem entirely ridiculous).

Series: Research Horizons Seminar

Contact geometry is a beautiful subject that has important
interactions with topology in dimension three. In this talk I will give a
brief introduction to contact geometry and discuss its interactions with
Riemannian geometry. In particular I will discuss a contact geometry analog
of the famous sphere theorem and more generally indicate how the curvature
of a Riemannian metric can influence properties of a contact structure
adapted to it.

Series: Research Horizons Seminar

In this talk we will connect functional analysis and analytic
function theory by studying the compact linear operators on Bergman
spaces. In particular, we will show how it is possible to obtain a
characterization of the compact operators in terms of more geometric
information associated to the function spaces. We will also point to
several interesting lines of inquiry that are connected to the problems in
this talk. This talk will be self-contained and accessible to any
mathematics graduate student.

Series: Research Horizons Seminar

One of the most outstanding problems in differential geometry is
concerned with flexibility of closed surface in Euclidean 3-space: Is it
possible to continuously deform a smooth closed surface without
changing its intrinsic metric structure? In this talk I will give a
quick survey of known results in this area, which is primarily concerned
with convex surfaces, and outline a program for studying the general
case.

Series: Research Horizons Seminar

Consider electrostatic plasmas described by Vlasov-Poisson with a fixed ion background. In 1946, Landau discovered the linear decay of electric field near a stable homogeneous state. This phenomena has been puzzling since the Vlasov-Poisson system is time reversible and non-dissipative. The nonlinear Landau damping was proved for analytic perturbations by Mouhot and Villani in 2009, but for general perturbations it is still largely open. I will discuss some recent results with C. Zeng on the failure of nonlinear daming in low regularity neighborhoods and a regularity threshold for the existence of nontrivial invariant structures near homogeneous states. A related problem to be discussed is nonlinear inviscid damping of Couette flow, for which the linear decay was first observed by Orr in 1907.

Series: Research Horizons Seminar

In the talk we will start from examples of open surfaces, such as the complex plane minus a Cantor set, review their classification, and then move to higher dimensions, where we discuss ends of manifolds in the topological setting, and finally in the geometric setting under the assumption of nonpositive curvature.

Series: Research Horizons Seminar

Orthogonal polynomials turn out to be a useful tool in analyzing
random matrices. We present some old and new aspects.

Series: Research Horizons Seminar

We survey research spanning more than 20 years on what starts out
to be a very simple problem: Representing a poset as the inclusion order
of circular disks in the plane. More generally, we can speak of spherical
orders, i.e., posets which are inclusion orders of balls in R^d for some d.
Surprising enough, there are finite posets which are not sphere orders.
Quite recently, some elegant results have been obtained for circle orders,
lending more interest to the many open problems that remain.

Series: Research Horizons Seminar

Recently, there has been a lot of interest in estimation of
sparse vectors in high-dimensional spaces and large low rank
matrices based on a finite number of measurements of randomly
picked linear functionals of these vectors/matrices. Such
problems are very basic in several areas (high-dimensional
statistics, compressed sensing, quantum state tomography, etc).
The existing methods are based on fitting the vectors (or the matrices)
to the data using least squares with carefully designed complexity
penalties based on the $\ell_1$-norm in the case of vectors and
on the nuclear norm in the case of matrices. Proving error bounds
for such methods that hold with a guaranteed probability is based on several
tools from high-dimensional probability that will be also discussed.