Model theory of difference fields

Model Theory of Difference Fields

The purpose of this course is to learn the fundamental results about the model theory of difference fields, as developed in Chatzidakis and Hrushovski [10]. The goal is to get to a level that will allow reading that paper. In this section we give some basic definitions, and mention some of the applications of this theory. A more detailed description of the content of the course can be found in ...

Introduction to difference algebra and model theory of difference fields

Model Theory of Finite Difference Fields and Simple Groups

Asymptotic classes are classes of finite structures which have uniformly definable estimates for the cardinalities of their first-order definable sets akin to those in finite fields given by the Lang-Weil estimates. The goal of the thesis is to prove that the finite simple groups of a fixed Lie type and Lie rank form asymptotic classes. This requires the following: 1. The introduction describes...

A Survey on the Model Theory of Difference Fields

We survey the model theory of difference fields, that is, fields with a distinguished automorphism σ. After introducing the theory ACFA and stating elementary results, we discuss independence and the various concepts of rank, the dichotomy theorems, and, as an application, the Manin–Mumford conjecture over a number field. We conclude with some other applications. Difference field are fields wit...

Classical Model Theory of Fields

We begin with some thoughts on how model theory relates to other parts of mathematics, and on the indirect role of Gödel’s incompleteness theorem in this connection. With this in mind we consider in Section 2 the fields of real and p-adic numbers and show how these algebraic objects are understood model-theoretically: theorems of Tarski, Kochen, and Macintyre. This leads naturally to a discussi...