# Model theory of difference fields, part I

## Alice Medvedev

### City College -- CUNY

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.

Prof. Medvedev is a model theorist, teaching at City College in the CUNY system. She received her doctorate from the University of California-Berkeley, under the supervision of Tom Scanlon, and subsequently held postdoctoral positions at Berkeley and at the University of Illinois-Chicago.