Pillay's conjecture and its solution—a survey

Abstract

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups. All the results discussed here have already appeared in print (mainly). The goal of the notes is to put the results together and to provide a direct path through the proof of the conjecture, avoiding side-tracks and generalizations which are not needed for the proof. This is especially true for the last paper in the list which was often written with an eye towards generalizations far beyond o-minimality. The last section of this paper has gone through substantial changes in the final stages of the writing. Originally, it contained several open questions and conjectures which arose during the work on Pillay's Conjecture. However, most of these questions were recently answered in a paper of Hrushovski and Pillay, in which the so-called Compact Domination Conjecture has been solved. In another paper, the assumptions for Pillay's Conjecture were weakened from o-minimal expansions of real closed fields to o-minimal expansions of groups.