April 24, 2014

Symposium Celebrating New Fellows of the Royal Society of Canada
March 25, 2008


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.