Event Detail

Fri Mar 1, 2024
Evans 60
4–6 PM
Logic Colloquium
Mariana Vicaría (UCLA)
Model theory of valued fields

Model theory is a branch of mathematical logic that studies structures (that is sets equipped with relations, functions and constants) and their definable sets, that is the subsets of various cartesian powers that can be defined in terms of these distinguished constants, relations and functions via the logical connectives and quantifiers. There is a more general class of subsets that one could study, called the interpretable sets, obtained by taking the quotient of a definable set by a definable equivalence relation. A natural question is: given a structure can one classify the interpretable sets in that structure?

A valued field is a field K equipped with a distinguished subset O, a valuation ring. Examples of valued fields are the p-adic field Q_p or the Laurent series over the complex numbers C((t)). Given O a valuation ring of a field and M its maximal ideal, we commonly refer to the additive quotient O/M as the residue field, while the multiplicative quotient KxOx is an ordered abelian group and it is called the value group.

One of the most striking results in the model theory of valued fields is the Ax-Kochen/Ershov theorem which roughly states that the first order theory of an un-ramified henselian valued field is completely determined by the first order theory of its residue field and its value group. A principle follows from this theorem: the model theory of a valued field is controlled by its residue field and its value group.

In this talk I will make a brief description of valued fields and their model theory. I’ll present how the problem of classifying interpretable sets in henselian valued fields can be approached in an Ax-Kochen style: What obstructions come from the residue field? and from the value group? I will conclude by presenting the classification of the interpretable sets in valued fields obtained in joint work with Rideau-Kikuchi, building on [1] and [2].