We prove the Ax-Lindemann-Weierstrass theorem for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory of Schwarzian equations and machinery from the model theory of differentially closed fields. This result generalizes previous work of Pila-Tsimerman on the j function. (Joint work with James Freitag and Joel Nagloo)

I will present a bound for the number of F_q-points of bounded degree in a variety defined over Z, uniform in q. This generalizes work by Sedunova for fixed q. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.

Recall that a field K is large if it is existentially closed in K((t)). Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with M. Tressl we introduced and explored a differential analogue of largeness, that we conveniently call ``differentially large''. I will present some properties of such fields, and use a twisted version of the Taylor morphism to characterise them using formal Laurent series and […]

With every cover C -> P^1 of the projective line one can associate its so-called scrollar invariants (also called Maroni invariants) which describe how the push-forward of the structure sheaf of C splits over P^1. They can be viewed as geometric counterparts of the successive minima of the lattice associated with the ring of integers of a number field. In this talk we consider the following problem: how do the scrollar invariants of the Galois closure C' -> P^1 and of its various subcovers (the so-called resolvents of C -> […]

Joint work with Kobi Peterzil.Let G be a simple compact Lie group, for example G=SO_3(R). We consider the structure of definable sets in the subgroup G^{00} of infinitesimal elements. In an aleph_0-saturated elementary extension of the real field, G^{00} is the inverse image of the identity under the standard part map, so is definable in the corresponding valued field. We show that the pure group structure on G^{00} recovers the valued field, making this a bi-interpretation. Hence the definable sets in the group are as rich as possible.

Using the group configuration theorem, Hrushovski and Pillay showed that the law of a group definable in the reals or the p-adics is locally an algebraic group law, up to definable isomorphism. There are some natural expansions of these two theories of fields, by adding a predicate for a dense substructure, for example the algebraic reals or the algebraic p-adics. We will present an overview on some of the features of these expansions, and particularly on the characterisation of open definable sets as well as of groups definable in the […]