This talk is a continuation of last week’s Logic Workshop. The necessary background will be summarized briefly at the beginning of this talk.

ACFA, the theory of difference closed fields, is a rich source of explicit examples of forking independence and nonorthogonality, the distinction between stable and simple theories, and the distinction between one-based and locally-modular groups. To present these examples, I will introduce “difference varieties”, which are the basic building blocks of definable sets in ACFA, and “sigma-varieties”, a more tractable special case of these.

I’ll begin by setting up the first-order language and axioms of difference, and give some interesting examples, including frobenius automorphisms of fields in positive characteristic and difference equations from analysis that give the subject its name. Difference-closed fields, a natural analog of algebraically closed fields, have a nice model theory, starting with almost-quantifier elimination. Further model-theoretic notions – algebraic closure, elementary equivalence, forking independence – all have elementary purely algebraic characterizations that I will explain. The model theory of difference fields has been used in arithmetic geometry in several exciting ways (Hrushovski’s results on the Manin-Mumford Conjecture; his twisted Lang-Weil estimates; several people’s work on algebraic dynamics) that I will probably not explain in detail.

This talk will be continued in the Model Theory Seminar the following week.