Problems in Model Theory and Number Theory

As the proof of the geometric Mordell-Lang conjecture for function fields by Hrushovski shows, deep connections between logic, number theory and algebraic geometry have already been established. In particular, the branches that have become known as stability theory and geometric model theory have found many applications in abstract algebra. On the Diophantine side, the techniques obtained by applying the theory of o-minimality to algebraic structures have led to many landmark results, including Pila's unconditional proof of the André-Oort conjecture for powers of the modular curve. The Zilber-Pink conjecture, which constitutes a generalisation of André-Oort, as well as other problems in transcendental number theory, are currently being attacked using tools of a similar nature. Research concerning Hilbert's tenth problem has been carried on by Koenigsmann using tools coming from logic. Koenigsmann has obtained many results on Grothendieck's program of anabelian geometry as well: for example, the first proof of the birational variant of the section conjecture for all smooth projective curves of genus >1 over the field of p-adic numbers is due to him, as well as the disproof of the conjecture and of its birational variant for the field of p-adic algebraic numbers.

As the examples above show, model theoretic tools, methods and insights can be used to study properties of algebraic structures. In particular, Model Theory can be used to highlight relationships between objects which seem to be very different at first glance. This is crucial to our understanding of how these objects behave, and logic and algebra can really complement one another. The context of this project is at the overlap of model theory and number theory. The aim is to use viewpoints which come from both areas, in order to study objects at the intersection of the two disciplines. This project falls within the EPSRC Mathematical sciences, Logic and Combinatorics research area.