Logic Seminar

I will discuss an application to probability theory (more precisely, to the theory of exchangeable random variables) of recent developments in affine logic, the fragment of continuous logic in which only affine connectives are allowed.

An affine theory is simplicial if its type spaces, which are always compact convex sets, form Choquet simplices. A remarkable dichotomy holds: for a complete simplicial theory, either all type spaces are Bauer simplices (i.e., the set of extreme points is closed) or they are all Poulsen simplices (i.e., the extreme points are dense).

On the other hand, simplices of invariant measures arising in ergodic theory and probability theory can be represented as *quantifier-free* type spaces of appropriate universal affine theories. I will discuss a quantifier-free adaptation of the dichotomy result mentioned above and its application to the geometry of certain simplices of invariant measures arising in exchangeability theory. Relative property (T) for Polish groups will play an important role.