MODNET
Research Training Network in Model Theory
Research > Task IV: Henselian fields

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 p-adics or ACVF with subanalytic structure).   
b) Prove decidability of the field of formal Laurent series over a finite field of prime order.


Progress

Year 1, Year 2, Year 3, Year 4.


Last updated: September 9 2009 15:49 Please send your corrections to: