Research talks are meant to show you what older graduate students do for a living. They will each last approximately one half hour, and will take place on Wednesday, September 23 in Eckhart 203. The morning session begins at 10:00am and the afternoon session at 1:00pm. There will be a half hour break in the middle of the afternoon session.

**Morning, 10:00am**

__Timur Akhunov__

*Well-posedness of non-linear dispersive equations*

Well-posedness of partial differential equations (PDE) is a study
of a question if a given equation have a decent unique solution, with your
favorite definition of decent. It is one of the most basic questions one can
ask, once an equation is formulated, but it is quite non-trivial with at
least one of the Hilbert's problems being of this type. Dispersive equations
is a relatively young and active branch of PDEs and non-linear equations
have fewer tools available to study them and hence are richer in open
problems.

__Edward Wallace__

*Noise and nonlinearity: a messy combination*

Some biological systems can be modelled as a collection of probabilistic elements which interact with each other in nonlinear ways. I'll introduce the particular one I'm studying, a model for neural network dynamics, and tell you about the interesting things it does and what we think is causing it. Then I'll give a couple of well-understood examples of how noise and nonlinearity can produce surprising and interesting dynamics: bistable 1-d systems, and coherence resonance.

__Tam Nguyen Phan__

*Rigidity*

Let M be a compact manifold. A self homeomorphism/diffeomorphism/isometry
of M induces an automorphism of the fundamental group of M. The reverse
statement is not true in general. However, it is true for certain classes
of manifolds, that is, any automorphism of the fundamental group of M is
induced by a homeomorphism/diffeomorphism/isometry of M. Such manifolds
are called rigid. I’ll state some of the main results in the field and
talk a bit about the part of my research that involves rigidity.

**Afternoon, 1:00pm**

__Thomas Zamojski__

*Problems in Dynamics on Homogeneous Spaces*

Dynamics on homogeneous spaces attracted much attention in the
last 30 years after fields medalist G.Margulis proved the long standing
Oppenheim conjecture in number theory using dynamical methods. Since then,
the subject has enlarged considerably.
I will give an overview of current important problems, avoiding technical
statements of theorems. I will then move on to some applications to number
theory, focusing on my current research: counting rational points on
varieties and Manin's conjecture.

**Half Hour Break, approximately 2:30-3**

__Tom Church__

*2-dimensional topology: Surface bundles, hyperbolic structures, and configurations of curves*