Seminar

Valuation theory plays a major part in the interaction between number theory and logic. In this seminar, a variety of topics from valuation theory, and in particular connections to model theory, will be discussed. There will be a strong emphasis on results involving the definability of valuations.

To participate in this seminar, please register here: https://www.msri.org/seminars/25206

Abstract:

Abstract: 1-dependent theories better known as NIP theories are the first class of the hierarchy of n-dependent structures. The random n-hypergraph is the canonical object which is n-dependent but not (n-1)-dependent. Thus the hierarchy is strict. But so far no strict n-dependent fields for n>1 have been found. We conjectured that in fact no such fields exists and started to generalized results on 1-dependent fields to this wider context. In the center of this investigation is the following well known conjecture by Shelah: Let (K,v) be an infinite n-dependent valued field. Then v is henselian. We will start by introducing the n-dependent hierarchy and present the recent results on n-dependent fields with a focus on (multi)valued ones.

N-Dependent (Valued) Fields