**ABSTRACTS**

**Victor LeBlanc**, Mathematics, University
of Ottawa

*Euclidean Symmetry and the Dynamics of Spiral Waves*

As the name suggests, spiral waves are waves which propagate through
some excitable medium, with the wave front having the shape of a spiral.
Spiral waves occur in many different physical contexts: certain types
of chemical reactions, slime-mold aggregates, and the electrical potential
of cardiac tissue. In the last case, spiral waves are a "bad thing",
since they are believed to be the precursor to potentially fatal conditions
such as ventricular fibrillation and tachycardia. Thus, a thorough understanding
of the way these waves propagate (and especially, how they can be controlled
or eliminated) has important potential applications.

The mathematical models for the many different phenomena in which spiral
waves are observed are usually reaction-diffusion partial differential
equations. In an appropriate mathematical setting, these can be viewed
as an (infinite-dimensional) dynamical system with nice symmetry properties:
the flow commutes with an action of the Euclidean group of all planar
rotations and translations. I will show in this talk how it is possible,
using just these symmetry properties and a few reduction theorems, to
explain many of the experimentally-observed dynamics and bifurcations
of spiral waves, and even make some predictions about how they should
behave under certain circumstances.

My colleague Dr. Yves Bourgault and I have founded the University of
Ottawa Numerical Heart Laboratory, which includes researchers in the
Faculty of Science, the Faculty of Engineering, as well as clinicians,
medical imaging experts and biomedical engineers at the University of
Ottawa Heart Institute. This group is working towards the development
of a completely integrated and coupled bio-mechanical and electrophysiological
numerical model of the cardiovascular system. The group currently has
a Beowulf cluster of Pentium-based computers for parallel code development,
and has access to the High-Performance Virtual Computing Laboratory.
My research within this group contributes to the theoretical study and
the development and implementation of anisotropic bidomain models of
cardiac electrophysiological waves.

**Sam Roweis**, Computer Science, University
of Toronto

*The Mathematics of Computer Science*

Research in modern Computer Science uses quite a lot of sophisticated
mathematics. For example, work on network design, internet communication
protocols, cryptography, machine learning, computer graphics & vision
and many other interesting and complex problems can involve theoretical
analyses of complexity, proofs of correctness/security and optimizations
over various measures of performance, utility or fairness. These analyses
often employ techniques very similar to those used in applied math or
statistics: inductive proofs, reductions to standard forms, expectations
over probabilistic outcomes, counting objects of a certain type, maximizing
over functions using multivariate calculus or over functionals using
variational techniques.

The exciting twist in Computer Science is that the results of these
analysis often allow us to write computer programs which can do amazing
things. Many of you know that work on factoring in number theory led
to secure connections in your web browser. Similarly, a host of other
mathematical results have been responsible for things like scheduling
sports tournaments, setting prices in financial markets, landing airplanes,
operating the telephone network, compressing music into smaller files,
and so forth. Computer science often lies at the interface between powerful
but abstract mathematics and practical but tedious engineering implementations,
and as such, it can be a very fun place to work as an aspiring math
or stats researcher.

In this talk, I will review some of the general areas of research in
our department that involve substantial mathematical work and relate
them to the resulting applications in the real world.

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