Title: Types, independence, classification, pseudo-finite fields -- a glimpse into model theory
Speaker: Tingxiang Zou (Institut Camille Jordan, Université Lyon 1, personal homepage https://sites.google.com/site/zoutingxiang2)
Date: Dec. 27th 16:00-18:00
There are two parts of this talk. The first part aims to give a brief (and very incomplete) introduction to model theory. We will start with Shelah's classification, which picks out several tame categories from all first-order theories. Among them are stable, simple and NIP theories. There are several ways to define, as well as to understand, these categories: from the number of types, from the existence of an abstract independence relation or from the inability to define some combinatoric configurations. We will finish with an algebraic example: the theory of pseudofinite fields. Pseudofinite fields are infinite fields elementary equivalent to ultra-products of finite fields. We will talk about their theories, the counting measure and dimension as well as their classification.
The second part of this talk will be devoted to the model theory of a generic expansion of pseudofinite fields: H-structures. This expansion preserves nice model theoretic properties. However, it is not clear that pseudofiniteness is also presevred. We will present the result which states that pseudofinite H-structures of pseudofinite fields exist.