Zilber conjectured that the complex exponential field Cexp is quasiminimal. He later showed that if Schanuel's conjecture is true and Cexp is strongly exponentially-algebraically closed then his conjecture holds. I will report on work-in-progress showing that Schanuel's conjecture can be dropped as an assumption, and strong exponential-algebraic closedness can be weakened to exponential-algebraic closedness. This is joint work with Martin Bays. Hide abstract

Classical model theory of non-archimedean local fields has regained momentum in recent years. We will highlight some of the new developments on issues of definability and decidability originating from old open problems, but with a view towards new arithmetically compelling challenges. Hide abstract

The Pila-Wilkie counting theorem counts rational points in a set X definable in an o-minimal structure. In this talk, I will present an extension of this theorem (due to Pila) which counts intersections between X and affine spaces defined over the rational numbers. I will then discuss applications of Pila's theorem to unlikely intersections in tori, abelian varieties and Shimura varieties. Hide abstract

The aim of this project is to attach a geometric structure to the ring of integers and to understand Spec (Z) from the point of view of stability theory. In his paper "The notion of dimension in geometry and algebra" Y.I. Manin asked several questions about the dimension of Spec(Z). Recently A.Connes and C.Consani published an important paper which introduces a much more complex structure called the arithmetic site which includes Spec(Z). Our approach for the same purpose is based on the generalization of constructions applied by Boris Zilber in non-commutative (and commutative) algebraic geometry. We describe a category of certain representations of integral extensions of Z and establish its tight connection with the space of elementary theories of pseudo-finite fields. From model-theoretic point of view the category of representations is a multi-sorted structure which we prove to be super-stable with pre-geometry of trivial type. It comes as some surprise that a structure like this can code a rich mathematics of pseudo-finite fields. We formally have answered two of Manin's questions about Dimension of Spec(Z) being 1 and infinity. Hide abstract

We prove the stability under integration and under Fourier transform of a concrete class E of functions, containing all globally subanalytic functions and their complex exponentials. The class E is a system of algebras of which we describe explicitly the generators. The methods of proof pertain to o-minimality (in particular, subanalytic resolution of singularities) and to the theory of almost periodic functions. This provides an example of a fruitful interaction between singularity theory, o-minimality and analysis. Joint work with R. Cluckers, G. Comte, D. Miller and J.-P. Rolin. Hide abstract