The event takes place on Friday, April 26th 2019.

The talks are 80 min long, followed by 10 min for questions/discussion.

- 10.00 – 10.30:
*Welcome Coffee* - 10.30 – 12.00: Dimitri Wyss –
Non-archimedean and motivic integrals on the Hitchin fibration

- 12.00 – 13.30:
*Lunch* - 13.30 – 15.00: Nathanaël Mariaule –
On the decidability of the expansion of p-adic fields by multiplicative subgroups.

- 15.00 – 15.30:
*Coffee break* - 15.30 – 17.00: Pablo Cubides Kovacsics – Pairs and pro-definable spaces of types

## Below you can find the abstracts of the talks:

**Non-archimedean and motivic integrals on the Hitchin fibration ***(Dimitri Wyss)*

Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between ‘stringy’ Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using non-archimedean integrals on these moduli spaces, building on work of Denef-Loeser and Batyrev. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô.

In my talk I will outline the main arguments of the proofs and discuss the adjustments needed, in order to replace non-archimedean integrals by motivic ones. The latter is joint work with François Loeser.

**On the decidability of the expansion of p-adic fields by multiplicative subgroups.** *(**Nathanaël Mariaule)*

Let K be a finite algebraic extension of Qp or its algebraic closure and G be a multiplicative subgroup of Qp^* of finite rank.

In this talk I will discuss the question of the decidability of the theory of the pair (K, G). I will present an axiomatisation of the theory of this structure. We shall see that the question of decidablity is equivalent to the effectivity of the so-called Mann property. I will also discuss the expansion by two multiplicative subgroups. In the real case, P. Hieronymi proved that the expansion of R by two multiplicatively independent rank 1 subgroups defines Z. In the p-adic in some cases we also get that the ring of integer is definable but in some other cases the theory is decidable modulo the same issue of effective Mann property.

**Pairs and pro-de finable spaces of types ***(Pablo Cubides Kovacsics)*

Hrushovski and Loeser developed a theory providing a model-theoretic counterpart of the Berkovich analytification of algebraic varieties. Part of the novelty and the power of their construction resides both on the flexibility granted by working with definable types and on the fact that their spaces carry the structure of a strict pro-definable set. In this talk I will show this latter property is shared by other spaces of definable types even over structures such as o-minimal expansions of groups and real closed valued fields. Our approach is based on the model theory of pairs. This is a joint work with Jinhe Ye (Notre Dame).