## Symposium Celebrating New Fellows of the Royal Society of Canada

March 25, 2008

## Abstracts

**David Brydges**, University of British
Columbia

*Statistical Mechanics and the Renormalisation Group*

A very long random walk, seen from so far away that individual
steps cannot be resolved, is the continuous random path called Brownian

motion. This is a rough statement of Donsker's theorem and it is
an example of how models in statistical mechanics fall into equivalence

classes classified by their scaling limits. The renormalisation
group is a program whose objective is to identify equivalence classes
that

arise in statistical mechanics. In some simple cases where the scaling
limit is close to Gaussian the renormalisation group can be formulated

precisely and used to prove theorems. The proofs are based on a
special way to decompose the Greens function for an elliptic operator
into a sum of positive definite functions with finite range.

**Walter Craig**, McMaster University

*Bounds on Kolmogorov spectra for the Navier - Stokes equations*

Let $u(x,t)$ be a (possibly weak) solution of the Navier - Stokes
equations on all of ${\mathbb R}^3$, or on the torus ${\mathbb R}^3/
{\mathbb Z}^3$. The {\it energy spectrum} of $u(\cdot,t)$ is the
spherical integral \[E(\kappa,t) = \int_{|k| = \kappa} |\hat{u}(k,t)|^2
dS(k)

,\qquad 0 \leq \kappa < \infty , \] or alternatively, a suitable
approximate sum. An argument involking scale invariance and dimensional
analysis given by Kolmogorov in 1941, and subsequently refined by
Obukov, predicts that large Reynolds number solutions of the Navier
- Stokes equations in three dimensions should obey \[E(\kappa, t)
\sim C\kappa^{-5/3} ,\] at least in an average sense. I will explain
a global estimate on weak solutions in the norm $|{\cal F}\partial_x
u(\cdot, t)|_\infty$ which gives bounds on a solution's ability
to satisfy the Kolmogorov law. The result gives rigorous upper and
lower bounds on the inertial range, and an upper bound on the time
of validity of this regime. This is joint work with Andrei Biryuk.

**Lisa Jeffrey, **Department of Computer
and Mathematical Sciences, University of Toronto at Scarborough

*Flat connections on Riemann surfaces *

Several noteworthy results in mathematics have recently been obtained
by studying gauge theory on a two-dimensional spacetime (or Riemann
surface); these results (formulas for correlation functions in two-dimensional
Yang-Mills theory) were discovered by E. Witten (1991-92). Yang-Mills
theory involves a gauge field (or connection), and the Yang-Mills
action is the norm squared of its curvature. This gauge field is
a generalization of the vector potential from electromagnetism,
and the curvature corresponds to the tensor formed from the electric
and magnetic fields. The two-dimensional case is a prototype for
the four-dimensional case, and can be solved exactly.

The mathematical interpretation of Witten's results is that they
can be used to obtain formulas for the multiplication in the cohomology
of certain moduli spaces (spaces parametrizing flat connections
on Riemann surfaces). For an abelian gauge group the moduli space
is a quotient of the first cohomology group of the Riemann surface.
We will talk about the nonabelian case which arises in Witten's
work.

Witten's formulas have been given a mathematically rigorous proof
(L. Jeffrey and F. Kirwan 1998) using methods from symplectic geometry,
which is the natural mathematical framework for the Hamiltonian
formulation of classical mechanics. We will outline these results
as well as some more recent developments.