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 […]