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Series: Research Horizons Seminar

I will review a little bit of the theory of algebric curves, which essentialy
amounts to studying the zero set of a two-variable polynomial. There are
several amazing facts about the number of points on a curve when the ground
field is finite. (This particular case has many applications to cryptography
and coding theory.) An open problem in this area is whether there exist
"supersingular" curves of every genus. (I'll explain the terminology, which has
something to do with having many points or few points.) A new project I have
just started should go some way toward resolving this question.

Series: Research Horizons Seminar

In this talk, I will use the shortest path problem as an example to illustrate how one can use optimization, stochastic differential equations and partial differential equations together to solve some challenging real world problems. On the other end, I will show what new and challenging mathematical problems can be raised from those
applications. The talk is based on a joint work with Shui-Nee Chow and Jun Lu. And it is intended for graduate students.

Series: Research Horizons Seminar

TBA

Series: Research Horizons Seminar

The question of the asymptotic order of magnitude of the
fluctuation of the Optimal Alignment Score of two random sequences
of length n has been open for decades. We prove a relation between
that order and the limit of the rescaled optimal alignment score
considered as a function of the substitution matrix.
This allows us to determine the asymptotic order of the fluctuation
for many realistic situations up to a high confidence level.

Series: Research Horizons Seminar

I will describe a class of mathematical models of composites and
polycrystals. The problems I will describe two research projects that are
well suited for graduate student interested in learning more about this area
of research.

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.