The seminar will introduce to the model theory of pseudofinite structures. The first half of the seminar will be devoted to pseudofinite fields, the second half to pseudofinite groups, both very active research areas in the model theory of algebraic structures.
In 1968, James Ax determined the theory of all finite fields and showed that it is decidable. To understand this theory, it is crucial to study its infinite models, the so-called pseudofinite fields. Pseudofinite fields allow for a very elegant axiomatization. They are just those perfect fields which have a unique extension of degree n, for every natural number n, and which are pseudo-algebraically closed. Starting with the foundational work of Ax, we will study various aspects of the model theory of pseudofinite fields and in particular see the construction of the Chatzidakis-van den Dries-Macintyre measure for definable sets in pseudofinite fields, which is an important tool in many applications.
We will then study pseudofinite groups from various perspectives, starting with measurable groups which are defined through the existence of a measure on definable sets mimicking the Chatzidakis-van den Dries-Macintyre measure,
The material treated in the seminar may serve as a basis for a Bachelor or a Master thesis.