Nigel Pynn-Coates : Monotone T-convex T-differential fields

Posted by James Freitag , part of the Logic Seminar.

At

Sept. 27, 2022, 4 p.m.

In

636 SEO

Abstract

Let T be a complete, model complete, power bounded o-minimal theory extending the theory of real closed fields. A T-convex T-differential field is an expansion of a model of T by a valuation and a derivation, each of which is compatible with the o-minimal structure, the former in the T-convex sense of van den Dries--Lewenberg and the latter in the sense of Fornasiero--Kaplan. When T is the theory of the real field with restricted analytic functions, we can expand an ordered differential Hahn field to a T-convex T-differential field, in which case the derivation is monotone, i.e., weakly contractive with respect to the valuation (monotone differential Hahn fields were studied earlier by Scanlon and Hakobyan). I will describe joint ongoing work with Kaplan on monotone T-convex T-differential fields, achieving among other results an Ax--Kochen/Ershov type theorem for such structures. A key step is isolating an appropriate analogue of henselianity in this setting.