November 1, 2002
Stu Whittington, Department of Chemistry, University of Toronto
Randomly coloured self-avoiding walks: A model of random copolymers
Copolymers are polymer molecules made up of at least two types of monomers. If the sequence of monomers is determined stochastically they are called random copolymers, and can be thought of as primitive models of important biopolymers like DNA and proteins. One possible model of a random copolymer is a radomly coloured self-avoiding walk on a lattice. A sequence of colours is determined by some random process and then the vertices of the self-avoiding walk inherit these colours. The set of walks models the possible conformations of the polymer and the relative weights of different walks (and hence different polymer
conformations) depend on the sequence of colours. The talk will describe these models and their application to several physical situations. Although some rigorous results are known there are many open questions and these will be introduced and discussed during the talk.

February 7, 2003
John Sipe, Department of Physics, University of Toronto
Effective field theories for nonlinear optics in artificially structured materials
The study of nonlinear optical pulse propagation using effective field equations, such as the nonlinear Schroedinger equation and the nonlinear coupled mode equations, has been an active area of research in 1D photonic crystals, or "gratings," for the past fifteen years. These techniques are now being extended to higher dimensional photonic crystals and more general artificially structured materials. Unfortunately, while reasonably rigourous derivations of such effective field equations for structures with large variations in their linear optical properties have been performed, they are complicated and do not allow a simple understanding of the conservation laws that the final equations exhibit.

After a review of experimental and theoretical work on 1D structures, we consider a new approach to the derivation of a general class of effective field equations based directly on a canonical formulation of the underlying Maxwell fields. This makes some progress towards easier and clearer derivations, results in effective theories that can then be immediately quantized, and allows for an identification of the physical significance of conserved quantities.

This research field is full of challenges, both with respect to identifying the correct effective field equations and in characterizing their solutions. Some of these will be highlighted.

March 21, 2003
Ray Kapral, Chemistry, University of Toronto
Twisting Filaments in Oscillatory Media

Scroll waves are one of the most commonly observed patterns in three-dimensional oscillatory and excitable systems. They play a role in physical systems like heart where they are believed to be responsible for flutter and fibrillation. The locus of the core of a scroll wave is a vortex filament and it organizes the structure of the pattern. If twist is applied to such a vortex filament it may undergo a series of bifurcations as the twist density is increased. Some of the bifurcations are akin to those seen in elastic rods. The filament bifurcates via supercritical Hopf bifurcations to a helix, and subsequently to a super-coiled helix. Further increases in the twist density lead to more complex structures. These features are analyzed using results from the topology of ribbon curves.

March 28, 2003
Eugene Fiume, Computer Science, University of Toronto
Signal theoretic characterisation of three dimensional polygonal geometry
Computer graphics abounds in the shameless theft of techniques from the mathematical sciences. In some cases, various aspects of the field can strongly benefit from more systematic mathematical treatment. In the past ten years, collections of polygons have become the "normal form" of geometric representations for computer graphics. Operations such as smoothing, enhancement, compression and decimation that are performed on such meshes very strongly suggest the desirability of representing polygonal meshes as three-dimensional signals so that, for example, frequency domain representations might be realisable. In this talk, I will speak of my collaboration with Richard Zhang on the signal theoretic representation of polygonal meshes, why it is important to find one, and our progress in the development of the equivalent of a Discrete Fourier Transform for such 3D geometry.