250053 VO Model Theory of Valued Fields (2021S)

The concept of a valuation, which arose early in the 20th century to better understand the then-newfangled concept of p-adic number, now plays an important role not only in number theory but also in commutative algebra and various flavors of geometry: algebraic, semialgebraic, rigid analytic, tropical, to name a few. The general theory of (Krull) valuations is not usually part of the university curriculum in algebra, but is crucial for applications of model theory to these fields.

The aim of this course is to provide a thorough introduction to valued fields, with a particular emphasis on their model theory, leading up to the classical results by Ax-Kochen-Ershov and Macintyre on the model theoretic properties of the p-adics. This should prepare students to be able to study more recent developments, such as motivic integration or the model-theoretic investigation of fields with additional structure.

I will try to make the course accessible to students with varying backgrounds, and hence only assume a basic knowledge of algebra (groups, rings, fields) and logic (on the bachelor level). If in doubt about your preparation, ask me.