Probability

Phase Transitions for Interacting Diffusions

Author: 
Frank den Hollander
Date: 
Wed, Jan 18, 2006
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
In the present talk we focus on the ergodic behavior of systems of interacting diffusions.

Algebraic Z^d-actions

Author: 
Klaus Schmidt
Date: 
Fri, Nov 1, 2002
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

This is a written account of five Pacific Institute for the Mathematical Sciences Distinguished Chair Lectures given at the Mathematics Department, University of Victoria, BC, in November 2002. The lectures were devoted to the ergodic theory of $ \mathbb Z^d $--actions, i.e. of several commuting automorphisms of a probability space. After some introductory remarks on more general $ \mathbb Z^d $-actions the lectures focused on ‘algebraic’ $ \mathbb Z^d $-actions, their sometimes surprising properties, and their deep connections with algebra and arithmetic. Special emphasis was given to some of the very recent developments in this area, such as higher order mixing behaviour and rigidity phenomena.

Notes: 

Entropy and Orbit Equivalence

Author: 
Daniel J. Rudolph
Date: 
Fri, Oct 1, 2004
Location: 
University of Victoria, Victoria, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 
In these notes we first offer an overview of two core areas in the dynamics of probability measure preserving systems, the Kolmogorov-Sinai theory of entropy and the theory of orbit equivalence. Entropy is a nontrivial invariant that, said simply, measures the exponential growth rate of the number of orbits in a dynamical system, a very rough measure of the complexity of the orbit structure. On the other hand, the core theorem of the orbit theory of these systems, due to Henry Dye, says that any two free and ergodic systems are orbit equivalent, that is to say can be regarded as sitting on the same set of orbits. The goal we set out to reach now is to explain and understand the seeming conflict between these two notions.
Notes: 

Self-Interacting Walk and Functional Integration

Author: 
David Brydges
Date: 
Thu, Sep 14, 2000
Location: 
University of British Columbia, Vancouver, Canada
Conference: 
PIMS Distinguished Chair Lectures
Abstract: 

These lectures are directed at analysts who are interested in learning some of the standard tools of theoretical physics, including functional integrals, the Feynman expansion, supersymmetry and the Renormalization Group. These lectures are centered on the problem of determining the asymptotics of the end-to-end distance of a self-avoiding walk on a $ D $-dimensional simple cubic lattice as the number of steps grows. When $ D=4 $ the end-to-end distance has been conjectured to grow as Const. $ n^{1/2}\log^{1/8}n, $ where $ n $ is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the $ D=4 $ conjecture in the simplified setting known as the ``Hierarchial Lattice.''

Notes: 

PDE from a probability point of view

Author: 
Richard Bass
Date: 
Thu, Jan 1, 2004
Location: 
UBC, Vancouver, Canada
Abstract: 
These notes are for a course I gave while on sabbatical at UBC. The topics covered are: stochastic differential equations, solving PDEs using probability, Harnack inequalities for nondivergence form elliptic operators, martingale problems, and divergence form elliptic operators. This course presupposes the reader is familiar with stochastic calculus; see the notes on my web page for Stochastic Calculus, for example. These notes for the most part are based on my book Diffusions and Elliptic Operators, Springer-Verlag, 1997.
Notes: 
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