Our seminar this semester will focus on the model theory of difference fields. We meet in 891 (note the change in location!) Evans, Wednesdays 3-4pm. The Course Control Number is 15443.
The main sources are the following research papers.
- Zoé Chatzidakis and Ehud Hrushovski, Model Theory of Difference Fields, Trans. AMS, 351, no. 8, pp. 2997 – 3071, April 8, 1999
- Zoé Chatzidakis, Ehud Hrushovski, and Ya’acov Peterzil, Model theory of difference fields II: Periodic ideals and the trichotomy in all characteristics, Proc. LMS (3) 85 (2002) 257 – 311
- Zoé Chatzidakis and Ehud Hrushovski, Revisiting virtual difference ideals, Model Theory 3 no. 2 (2024) 285 – 304.
The model companion ACFA of the theory of a fields with a distinguished endomorphism was one the first non-trivial examples of a simple unstable theory to be studied in detail. We will recount some of the development of simplicity through the lens of difference fields.
Looking to applications outside of model theory, ACFA has been used in the study of special point problems and in algebraic dynamics. We will look at some of these applications. References include the following papers.
- Zoé Chatzidakis and Ehud Hrushovski, Difference fields and descent in algebraic dynamics I, JIMJ 7 (2008) no. 4, 653 – 686.
- Zoé Chatzidakis and Ehud Hrushovski, Difference fields and descent in algebraic dynamics II, JIMJ 7 (2008) no. 4, 687 – 704.
- Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, APAL 112 (2001), no. 1, 43 – 115.
- Alice Medvedev and Thomas Scanlon, Invariant varieties for polynomial dynamical systems, Annals of Mathematics (2) 179 (2014), no. 1, 81 – 177
- Thomas Scanlon, A positive characteristic Manin-Mumford theorem, Compos. Math. 141:6 (2005) 1351 – 1364
- Thomas Scanlon, Local André-Oort conjecture for the universal abelian variety, Inv. Math. 163 (2006) no. 1, 191 – 211