Here are some of my areas of research:
Eigenvalues and Resonances. One of the problems I have studied pertains to the
existence of eigenvalues and resonances for the Laplacian on perturbed
cylinders with Neumann boundary conditions.
Such perturbed cylinders are known as waveguides,
and the Laplace operator on waveguides is used to model the
vibrations of certain physical systems (such as a car's
exhaust pipe). The existence of $L^2$ eigenvalues and resonances
are an obstruction to the efficient dissipation of the energy to
infinity for such vibrating systems.
For the unperturbed cylinder, it is well known that
there are no $L^2$ eigenvalues or resonances. However, for some local
perturbations it has been shown that eigenvalues can exist,
and for asymptotic perturbations the eigenvalues can
even have finite accumulation points. Nevertheless, for generic
perturbations, eigenvalues do not exist.
I have recently proven that an infinite set of resonances near the real
axis will exist
if the cylinder has a "stable, periodic billiard ball
trajectory". For a picture of
a stable, periodic billiard ball trajectory of length d, see this picture.
Never let
your car's exhaust pipe get bent into this shape!
I proved a couple of years ago that an infinite set of resonances near
the real axis will exist
if the cylinder has a "stable, periodic billiard ball
trajectory". For a picture of
a stable, periodic billiard ball trajectory of length d, see this picture.
Never let
your car's exhaust pipe get bent into this shape!
Control Theory I have also begun a collaboration with my
departmental colleague
Louis Tebou in the area of Control Theory. The basic problem here is the
following.
Given a vibrating system, can one devise a feedback mechanism that will
bring the system
to rest in a given period of time? This field has important applications
in Engineering,
where one is trying to control the vibrations of jet engines,
bridges, buildings, etc.
Voting Theory. Finally, I have been working with
Keith Dougherty from University of Georgia's
Political Science department on some mathematical aspects of voting
theory. The mathematical
problems arise here because minor changes in an assembly's voting rules
can change the
likelihood that a measure passes. To give a simple example, if a
proposal requires unanimous consent, then
it is much less likely to be approved then if it required a majority of
voters to vote in favour
of the proposal. Nonetheless, both Unanimity Rule and Majority Rule are
common voting rules
throughout the world. Which of these rules is better? Experts would argue
that it depends on
the context. For instance, Unanimity Rule would be unworkable in the UN
general assembly, but
it might be appropriate in the US Supreme court. The reason that a given
voting rule is appropriate
in one setting but not in another is due to certain properties that
can be quantified, using
either Game Theory or Probability Theory. Keith Dougherty and I have used
probabilistic methods
to compare various common voting rules. Hopefully this research might help
create assemblies that
better represent the preferences of their constituents. We still haven't
figured out a way to get Democrats
elected, though!
Some of my papers are available as pdf, ps or wpd files in
this link: |
