# Andres del Junco

University of Toronto

## Talk 1:

Title: Some applications of joinings in ergodic theory

This will be an introduction to the basic theory of joinings
in ergodic theory, for non-specialists. The notion of a joining
has come to play a central role in ergodic theory since it was introduced by
Furstenberg in 1970. Roughly speaking, a joining of two measure-preserving
transformations (henceforth ``systems'')
is an embedding of the two as factors in a common larger
system. The space of joinings can be viewed as a space of measures which
has a natural compact topology. Using only the most elementary properties
of joinings we will give several non-trivial applications, including a proof
that weak mixing implies weak mixing of all orders and a proof of the Halmos-
von Neumann discrete spectrum theorem. We will also introduce Veech's notion of
simplicity.

The remaining two talks will focus on more specialized applications and some
acquaintance with ergodic theory will be assumed.

## Talk 2:

Title: Gaussian systems

We will briefly introduce the notion of a
Gaussian system and then focus on the special case of Kronecker Gaussian
systems, where some beautiful ideas of Thouvenot bring joinings into
play to allow a detailed
analysis of the structure of such a system. Surprisingly these ideas have
have applications to general Gaussian systems. As an example we will sketch
a proof that simple systems are disjoint from Gaussians.

## Talk 3:

Title: Higher order independence, higher order mixing and the
semigroup of joinings

The basic question here is:
if three systems are embedded as factors in a larger system in such a way that
any two are independent when can we say that they are jointly independent? This
has a close connection with Rohlin's famous question on higher order mixing.
We will survey some recent progress in this area, especially work of Host and
Ryzhikov. We will also introduce a semigroup structure
on the space of joinings.
This is a promising new direction which has yet to be exploited systematically
but we will mention some surprising applications.