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
Meg Shulman
Timur Akhunov
Edward Wallace
Tam Nguyen Phan
Keerthi Madapusi Sampath
Ian Shipman
Thomas Zamojski Half Hour Break, approximately 2:30-3
Tom Church
Calculating with RO(G)-graded cohomology theories
Algebraic topologists like to study topological spaces by associating
algebraic invariants to them, such as homotopy and homology groups, and
cohomology rings. If you start out with a space X having an action by a
finite group G, then Bredon cohomology, which is naturally graded on
RO(G) instead of the integers, is a good candidate for an invariant. It
is also extremely difficult to calculate, meaning that there are many
open questions in the field. I'll talk a little bit about the
computations that have been done, and some possible approaches to doing
more.
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.
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.
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
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.
2-dimensional topology: Surface bundles, hyperbolic structures, and configurations of curves