Finite-dimensional complex simple Lie algebras and affine Lie algebras
have played a prominent role in mathematics and physics. Kac-Moody Lie
algebras, Lie algebras graded by finite root systems, and extended
affine Lie algebras all are natural generalizations of them. Our
understanding of these Lie algebras has been greatly enriched by the
work of Canadian mathematicians. This talk will survey results on
these general classes of Lie algebras.
A difference field is a field with a distinguished automorphism. A
generic difference field is a difference field such that every system
of difference equations which has a solution in a difference field
extension, has a solution in the field.
In the first part of the talk I will state the main model-theoretic
results obtained on these fields and explain their significance and
importance. In the second part of the talk, I will mention some
applications obtained by Hrushovski to the solution of diophantine
problems (e.g., the Manin-Mumford conjecture and the Jacobi
conjecture for difference fields). I will also mention some intriguing
questions, which lie at the boundary of model theory and diophantine
GEOFFREY GRIMMETT, University of Cambridge, Statistical Laboratory, CMS,
Wilberforce Road, Cambridge CB3 0WB, UK
Conformality and universality
What is the Hausdorff dimension of the perimeter of Brownian motion?
What is the exponent in the formula for the number of self-avoiding
walks? Prove the physicists' conjectures for exact values of critical
exponents for percolation. Derive a rigorous theory of universality
for lattice systems. (All in two dimensions.)
Recent insights threaten to answer such challenges, and more. (The
answer to the first is definitely 4/3.) Schramm has shown how
Loewner's work on conformal maps provides a family of canonical random
processes which `should' be the limits of such spatial systems in the
limit of small mesh size. Smirnov has justified the `should' for site
percolation on the triangular lattice.
We shall outline these famous problems, and the (proposed) programme of
solution. This a beautiful story of the interaction of mathematical
disciplines, and involves probability, mathematical physics, conformal
functions, and combinatorics.
BARRY SIMON, Department of Mathematics, Caltech, Pasadena, California 91125,
Singular continuous spectra for Schrodinger operators
I'll begin by reviewing the decomposition of measures on the real line
into spectral types and then overview the recent literature on the
occurrence of ``exotic'' spectral types for one dimensional operators
of the form Hu = -u¢¢+qu.