# Scientific

## Algebraic Z^d-actions

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 --actions, i.e. of several commuting automorphisms of a probability space. After some introductory remarks on more general -actions the lectures focused on ‘algebraic’ -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.

## Entropy and Orbit Equivalence

## On Hilbert's Tenth Problem

## Projective Modules in Classical and Quantum Functional Analysis

## On the Chromatic Number of Graphs and Set Systems

**Fundamenta**in 2000 will only be stated in the second or the third part of these lectures. Meanwhile I will introduce the main concepts and state—--and sometimes prove—--simple results about them.

## Self-Interacting Walk and Functional Integration

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 -dimensional simple cubic lattice as the number of steps grows. When the end-to-end distance has been conjectured to grow as Const. where is the number of steps. We include a theorem, obtained in joint work with John Imbrie, that validates the conjecture in the simplified setting known as the ``Hierarchial Lattice.''

## Convex Optimization

## Actions of Z^k associated to higher rank graphs

We construct an action of on a compact zero-dimensional space obtained from a higher graph satisfying a mild assumption generalizing the construction of the Markov shift associated to a nonnegative integer matrix. The stable Ruelle algebra is shown to be strongly Morita equivalent to . Hence is simple, stable and purely infinite, if satisfies the aperiodicity condition.

*Ergodic Theory Dynam. Systems*

**23**(2003), no. 4, 1153-1172.

## Exponential Sums Over Multiplicative Groups in Fields of Prime Order and Related Combinatorial Problems

Let be a prime. The main subject of my talks is the estimation of exponential sums over an arbitrary subgroup of the multiplicative group :

These sums have numerous applications in additive problems modulo , pseudo-random generators, coding theory, theory of algebraic curves and other problems.