June 25, 2018

Fields Institute Graduate School Information Day

November 22, 2003
12:30 - 4:00 p.m.


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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