Contributions to the model theory of valued fields

This research focuses on model theory of valued fields. The starting point for this subject can be seen as the work by A. Robinson around 1950, where he showed the model completeness of the theory of algebraically closed valued fields. Another important result is the so-called Ax-Kochen Theorem proved by J. Ax and S. Kochen in the 1960s, determining the first order theory of Henselian fields of characteristic zero, and obtaining an asymptotic version of a conjecture of Artin as a corollary. Valuations are also the key tool in interpretations in number fields, such as J. Koenigsmann's theorem defining the integers in the rational field by a universal formula; following work by J. Robinson, Poonen and others.

We describe below an initial technical goal of the present research.

Traditionally in mathematical logic, statements are either false or true. We also say that their truth value is either 0 or 1. There are however logical settings of continuous logic, developed by Chang-Keisler and many others, where statements can take arbitrary values in a compact space, or even in the real line. Many basic structures of analysis, such as Hilbert spaces, and L^1-lattices, become in this way accessible to model theoretic analysis.

When working with valued fields, one can consider a real valued setting, i.e. a valued field with the value group being a subgroup of the real numbers, or an arbitrary value group or higher rank. The former is used in the theory of Berkovich spaces, whereas the Hrushovski-Loeser model-theoretic version treats the value group as a definable object. The Ben-Yaacov - Hrushovski theory of globally valued fields is based on continuous logic, which is therefore restricted to rank one valuations. For this there is no first-order analogue at the moment.

We will attempt to determine in the local setting whether this restriction is necessary, or whether a theory of higher rank valuationsis compatible with continuous logic. A positive answer would open the way to similar questions for globally valued fields. As the latter question is wide open, and even the much more modest local question has not been studied before, it is quite possible that further investigation will lead to unforeseen changes of direction.

This project falls within the EPSRC Mathematical sciences research areas of Logic and Algebra, with various connections to number theory and algebraic geometry. No companies or collaborators are currently involved.