


Plenary Speakers / Conférenciers principaux
 PETER CAMERON, Queen Mary, University of London
Homogeneity, randomness, and homomorphisms

A graph (or more general relational structure) is homogeneous
if any isomorphism between finite induced subgraphs extends to an
isomorphism of the graph. Surprisingly, in view of the rigidity of
almost all finite graphs, a countably infinite random graph is
homogeneous with probability 1: indeed, it is isomorphic to a
particular graph, the Rado graph, with probability 1. All countable
homogeneous graphs are known; the others are not "random" in any
reasonable sense, though they are all residual in the sense of Baire
category (another interpretation of the phrase "almost
all". Recently, Jaroslav Nesetril and I have investigated the
analogue of homogeneity for graphs and posets, where "isomorphism"
is replaced by "homomorphism" (or "monomorphism" in the
definition. The talk will also touch on other kinds of structure.
 CRAIG FRASER, University of Toronto, IHPST, Victoria College, Toronto, M5S 1K7
Calculus of Variations and Mathematical Existence in the
Nineteenth Century

We tend to approach mathematics of the past from our perspective
today: understanding of earlier developments is shaped by what
happened later. Nevertheless, a better historical understanding of a
given subject will result from a comparison with what came before,
from a study of the origins and background of the development under
consideration. The subject of mathematical existence came to the
forefront of various branches of analysis in the nineteenth
century. The recurring interest in existence questions after 1830
represented a new theoretical tendency in mathematics, one that was
not present in the writings of eighteenthcentury masters of analysis.
In the calculus of variations there was the wellknown use of
variational arguments such as Dirichlet's principle to establish the
existence of solutions of boundaryvalue problems defined by partial
differential equations. Existence questions also came up in other
parts of this subject: in the theory of the second variation, in
Weierstrasss field theory and in the study of constrained
optimization. The lecture will examine the history of some technical
results that involved existence assumptions or raised questions
concerning the existence of mathematical objects. Included in this
survey will be the work of Adolph Mayer, Edmund Husserl, Weierstrass,
Adolf Kneser, Hilbert, Hadamard, and Oskar Bolza.
 MARK LEWIS, University of Alberta, Edmonton
Mathematical models for carnivore territories

In this talk I will propose a set of mechanistic rules that can be
used to understand the process of territorial pattern formation
through interactions with scent marks. The models are described as
systems of partial differential equations, coupled to ordinary
differential equations. Under realistic assumptions the resulting
territorial patterns include spontaneous formation of `buffer zones'
between territories which act as refuges for prey such as deer. This
result is supported by detailed radiotracking studies. In some cases,
energy methods can be applied to the system, and the lowest energy
solution corresponds to a spatial territory.
The model will also be analysed using game theory, where the objective
of each pack is to maximize its fitness by increasing intake of prey
(deer) and by decreasing interactions with hostile neighboring
packs. Predictions will compared with radio tracking data for coyotes
and wolves, including some new data from Yellowstone, where topography
and local prey density can be shown to affect movement behavior.
 ALAN C. NEWELL, University of Arizona
How Kolmogorov spectra are formed

In turbulence, the most interesting stationary and attracting solutions
are finite flux Kolmogorov spectra which describe an energy density
distribution that allows for a constant flux of energy from large
scale sources to small scale sinks. The manner in which such spectra
are realized can be surprising. I will discuss what happens for wave
turbulence situations and then speculate on what may very well be
generic behavior. What is most interesting is the introduction of the
notion of entropy, more accurately entropy production, which I will
show plays a meaningful role even in nonisolated systems. I will try
to make the ideas accessible to an audience with a broad background.
 PETER OLVER, University of Minnesota
Moving frames

The classical method of moving frames was developed by Elie Cartan
into a powerful tool for studying the geometry of curves and surfaces
under certain geometrical transformation groups. In this talk, I will
discuss a new foundation for moving frame theory based on equivariant
maps. The method is completely algorithmic, and can be readily
applied to completely general Lie group and even infinitedimensional
pseudogroup actions. The resulting theory and applications are
remarkably wideranging, including geometry, classical invariant
theory, differential equations, the calculus of variations, symmetry
and object recognition in computer vision, and the design of
symmetrypreserving numerical algorithms.
 FRANK T. SMITH, University College London
Multibranching and networks in biomedical applications

Recent and continuing studies on branching tube flows will be
described, the motivation coming from applications to the
cardiovascular system, lung airways and cerebral arteriovenous
malformations. The work is based partly on modelling for increased
flow rates, partly on direct numerical simulations and partly on the
various comparisons possible. Small differentials in pressure acting
across a multiple branching may be considered first, followed by
substantial pressure differentials in a side branching, in a multiple
branching or in a basic threedimensional branching. All of these
cases include a comparison of results between the modelling and the
direct simulations. Wall shear, pressure variation, influence lengths,
and separation or its suppression will be examined, showing in
particular sudden spatial adjustment of the pressure between mother
and daughter tubes, nonunique flow patterns and an almost linear
increase of flow rate with increasing number of daughters, depending
on the specific conditions. The extension to large networks of vessels
will also be addressed. The agreement between modelling and direct
simulations is found to be generally close at moderate flow rates,
suggesting their combined use in the biomedical applications.
 MIKHAIL ZAICEV, Moscow State University, Moscow, Russia
Group gradings on algebras

Groups and semigroups gradings on rings and algebras are studied very
intensively last years. One of the basic problems of the structure
theory of graded algebras is to describe all finite dimensional simple
objects. In particular, it is very important to describe all possible
gradings on finite dimensional simple algebras. For example, in case
of associative algebras over an algebraically closed field any simple
finite dimensional algebra is a matrix algebra. So, one of the first
step is the classification of all group gradings on matrix
algebras. The results concerning the classification of all group
gradinds on matrix algebras, on finite dimensional simple Lie and
Jordan algebras will be presented in the talk.

