Model Theory of Valued Fields

My research project is concerned with application of model theory to valuation theory. Valuation theory is an area of algebra which stems from the generalization of the concept of absolute value in a field to the concept of a valuation. An absolute value consists of a map, satisfying certain properties, from a field to the additive group of real numbers. If we consider maps satisfying the same properties, but we allow to replace the real numbers with an arbitrary abelian group, we come to the more general notion of valuation. The theory of valued fields has been developed during the past century, and it has several applications to other branches of mathematics, such as number theory and algebraic geometry. Model theory is an area of mathematical logic. It can be considered as the study of mathematical structures with respect to the formulas which are satisfied in them, or as the study of formulas and sets of formulas with regard to the structures in which those are satisfied. The formulas in consideration are usually expressed in first-order logic, or in logics related to the latter. Mathematical structures are involved in plenty of branches of mathematics, and this is the reason why model theory is the part of mathematical logic which has most connections with the rest of mathematics. In particular, as algebraic structures are particular cases of structures satisfying first-order theories, model theory has naturally several applications to algebra. In the past decays model theory has played an important role in the study of algebraic theories. For example, the theory of algebraically closed fields has been understood in much more depth thanks to the fact that it has been studied from a model theoretic point of view. Since the properties which define valuations are also expressible in first-order logic, their study is also suitable in a model theoretic setting. Over the last years, there have been some advances in modern model theory for applications to the study of valued fields. In light of these developments, the aim of my research is to enhance the bridge between model theory and valuation theory. This would increase our understanding of valued field, and the results which will be proved in my work could help obtaining a larger amount of applications to other areas of mathematics. A first step into this direction consists of addressing classical model theoretic questions, such as definability or decidability of valued fields. Further, my research may involve the development of further model theoretic tools, which would help in the pursue of investigating valued fields, but which might also turn out to be useful in other contexts.

This project falls within the EPSRC Mathematical Logic research area