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.
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.
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.