Summary: UIC Model Theory Seminar, January 18, 2005
From pseudofinite fields to PAC substructures of stable
The background for my thesis research is a 40-year interaction between field
theory and model theory which began with work of James Ax on pseudofinite
fields. In 1991 Hrushovski introduced the notion of a bounded PAC (bPAC)
substructure of a model of a strongly minimal theory which generalized the
notion of pseudofinite fields to strongly minimal models.
Influenced by Hrushovski's work on definability in pseudofinite fields and
related structures, Kim and Pillay in the mid-1990's completed the foun-
dational work on simple theories which Shelah had begun much earlier.
The "bounded PAC substructures" of strongly minimal sets, studied by
Hrushovski, could now be seen as supersimple structures of SU-rank 1. The
basic aim of my research is to extend Hrushovski's theory from strongly
minimal structures to arbitrary stable structures where possible and, as a
consequence, to give a general method of building new simple theories.
This fits into a very basic theme of research over the past 10 years which
creates new stable/simple structures from old ones, for example adding a