Model theory of pseudofinite structures

SoSe2021 seminar run by Martin Hils and Katrin Tent.

The seminar will introduce 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 a general treatment of asymptotic classes and measurable structures which are defined through the existence of a measure on definable sets mimicking the Chatzidakis-van den Dries-Macintyre measure. This will be followed by an investigation of measurable groups and continued by further works on pseudofinite groups, depending on the interest of the participants.

The material treated in the seminar may serve as a basis for a Bachelor or a Master thesis.

If you are interested in participating, please contact Martin Bays <baysm@wwu.de>, or Martin Hils, or Katrin Tent.

Bibliography

[Ax68] J. Ax, The Elementary Theory of Finite Fields, Annals of Math. 88(2) (1968), 239-271.

[Cha05] Z. Chatzidakis, Notes on the model theory of finite and pseudo-finite fields, Course Notes, 2005.

[ChDrMa92] Z. Chatzidakis, L. van den Dries and A. Macintyre, Definable sets over finite fields, J. Reine Angew. Math. 427 (1992), 107-135.

[ElJaMaRy11] R. Elwes, E. Jaligot, D. Macpherson and M. Ryten, Groups in supersimple and pseudofinite theories, Proc. Lond. Math. Soc. (3) 103 (2011), 1049-1082.

[ElMa08] R. Elwes and D. Macpherson, A survey of asymptotic classes and measurable structures. Model theory with applications to algebra and analysis. Vol. 2, 125-159, London Math. Soc. Lecture Note Ser., 350, Cambridge Univ. Press, Cambridge, 2008.

[ElRy08] R. Elwes and M. Ryten, Measurable groups of low dimension, Math. Log. Q. 54 (2008), 374-386.

[Mac18] D. Macpherson, Model theory of finite and pseudofinite groups, Arch. Math. Logic 57 (2018), 159-184.

[Mil15] J.S. Milne, Fields and Galois Theory, Course notes, 2015.

[OuPo13] A. Ould Houcine and F. Point, Alternatives for pseudofinite groups, J. Group Theory 16 (2013), 461-495.

[Lan72] S. Lang, Introduction to algebraic geometry, 3rd printing, with corrections. Addison-Wesley, 1972.
[TeZi12] K. Tent and M. Ziegler, A Course in Model Theory, Lecture Notes in Logic, CUP, 2012.