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 $\mathbb Z^d$--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.