**TASK IV - HENSELIAN FIELDS**

**Description**

This is an area of research concerned primarily
with the model theory of the *p*-adics and fields of formal power
series. Elimination of
quantifiers play a major role, and has been at the source of various
applications (for example, in motivic integration and in the theory
of zeta functions of finitely generated groups).

**
List of specific problems**

**IV.1. p-adics (Model theory and applications).**

a) Develop geometric model theory
for finite extensions of the padics (with extra sorts).

b) Compute 2-variable zeta functions
for pro-p groups, and make connections between uniform p-adic cell decomposition
and 2-variable Dirichlet series encoding normalizer structure of a nilpotent
group.

c) Study of p-adic integration on definable
sets, including the subanalytic case; investigate uniformity issues; connections
with the motivic framework.

d) Obtain a p-adic version of the triangulation
of bounded definable closed subsets of real affine n-space (simplexes have
to be replaced in this padic version by another class of simple
definable bounded closed sets).

e) Study the (restricted) exponential
on the field of p-adic numbers (or on various complete extensions of it):
in particular, model-theoretic properties and Schanuel s conjecture.

**IV.2. Other valued fields**

a) Classify semisimple groups definable
in algebraically closed valued fields (ACVF); classify interpretable simple
groups; prove cell decomposition in ACVF (possible applications to arc spaces;
prove elimination of imaginaries for other important valued structures (e.g.
the padics or ACVF with subanalytic structure).** **