TASK IV - HENSELIAN FIELDS
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
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).