### Abstracts

__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.

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