Logic Colloquium 2008
Anand Pillay Compact spaces and definability
This tutorial will hopefully have some appeal to both a general audience of logicians as well as to researchers in model theory. The aim is to describe various interactions between definability in model theory and compact spaces, with reference to current research.
In the introductory first lecture I will discuss the category of definable sets (for a given first order theory), as well as notions such as hyperdefinability. I will also introduce various notions of strong type and state a conjecture relating the Lascar group of a first order theory to the complexity of Borel equivalence relations.
In the second lecture I will discuss Keisler measures, or probability measures on type spaces, and give certain consequences, for theories without the independence property, of the Vapnis-Chervonenkis theorem (uniform law of large numbers).
In the third lecture I will introduce the notion of a definable set (or group) being dominated by a compact space (or group), and describe some recent results.
The second and third lectures are related to joint work with Ehud Hrushovski, as well as being closely related to Peterzil's tutorial in LC '07.