### Supervisor: Dr Sylvy Anscombe

Model Theory is a discipline within Mathematical Logic that views Mathematical structures through the prism of (mostly) first-order logic. One application of Model Theory is to the study of valued fields: fields equipped with a Krull valuation. Examples include the p-adic valuations on the field of rational numbers, and also on the fields of p-adic numbers, as well as valuations associated to rational places of function fields. The list goes on.

My own work has focused on Diophantine sets in valued fields: those sets of n-tuples that are defined by an existential formula, from the languages of rings or valued fields, or similar. Equivalently, Diophantine sets are projections of algebraic sets. Typically one aims to describe such sets to provide a stepping-stone to quantifier-elimination or model-completeness results. Other further goals include finding axiomatisations of theories of valued fields, and proving decidability or existential decidability (i.e. Hilbert’s Tenth Problem). Another feature of my work so far is that I usually work in positive characteristic. In this setting the issue of inseparability rears its head, and makes life difficult!

Broadly this project will study Diophantine sets in valued fields, both in the `local’ setting (e.g. local fields, large fields, henselian fields) and in the `global’ setting (e.g. global fields, function fields). Students with a good background in some of Commutative Algebra, Arithmetic Geometry, and Algebraic Number Theory are particularly encouraged to apply.