Abstracts for Conference on Interactions of model theory with number theory and algebraic geometry

Pila and Wilkie's influential theorem, which bounds the density of rational and algebraic points lying on the transcendental parts of sets definable in o-minimal expansions of the real field, has already had far-reaching consequences for diophantine geometry. While their result gives the best bound possible for o-minimal structures in general, Wilkie has conjectured an improvement for sets definable in the real exponential field, namely that the bound could be improved to one involving some power of the logarithm of the height.

Let $G$ be an algebraic group over a field of characteristic $0$, $A$ an analytic (or formal) subgroup of $G$ and $V$ an algebraic subvariety of $G$. Ax proved that if the intersection of $A$ and $V$ is Zariski dense in $V$, then $A$ and $V$ tend to be in general position. I will discuss a theorem involving formal maps which implies Ax's theorem and also covers some cases in the positive characteristic situation.

We define a Grothendieck ring for basic real semialgebraic formulas, that is for systems of real algebraic equations and inequalities. In this ring the class of a formula takes into consideration the algebraic nature of the set of points satisfying this formula and contains as a ring the usual Grothendieck ring of real algebraic formulas.

Given an algebraic group (or more generally, a group scheme) $G$ over a field $K$, one may consider the category $C$ of $K$-linear representations of $G$. The classical Tannakian formalism describes the structure of such a category $C$ as an abstract (tensor) category. In particular, it shows that $G$ can be recovered from its category of representations.