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Series: Research Horizons Seminar

The derivation of the properties of macroscopic systems (e.g. the air in a
room) from the motions and interactions of their microscopic constituents is
the principal goal of Statistical Mechanics. I will introduce a simplified
model of a gas (the Kac model). After discussing its relation with more
realistic models, I'll present some known results and possible extension.

Series: Research Horizons Seminar

After some brief comments about the nature of mathematical modeling in
biology and medicine, we will formulate and analyze the SIR infectious
disease transmission model. The model is a system of three non-linear
differential equations that does not admit a closed form solution.
However, we can apply methods of dynamical systems to understand a great
deal about the nature of solutions. Along the way we will use this
model to develop a theoretical foundation for public health
interventions, and we will observe how the model yields several
fundamental insights (e.g., threshold for infection, herd immunity,
etc.) that could not be obtained any other way. At the end of the talk
we will compare the model predictions with data from actual outbreaks.

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.