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:

Starting from the model theory of tame fields developed in

    Kuhlmann, F.-V.: The algebra and model theory of tame valued fields, J. reine angew. Math. 719 (2016), 1-43,

one main question is how the condition of tameness can be relaxed while preserving some structure theory that still could lead to good model theoretic results. A first step was made in

    Kuhlmann, F.-V. - Pal, K.: The model theory of separably tame fields, J. Alg. 447 (2016), 74-108,

but these fields are still very close to tame fields. My approach to the question is to use a classification of the defects of valued field extensions. One of the two types types of defect has been identified as the more harmful, so the question arises: which classes of valued fields avoid the harmful defects? It turns out that a slight generalization of the deeply ramified fields does the job. We will introduce this class of valued fields as well as the deeply ramified fields and the so-called semitame fields.

To cover all deeply ramified fields, it was crucial to generalize the classification of defects that was originally given only in the equal characteristic case to the mixed characteristic case. In a new development, this can now be done by a unified definition that does not need a case distinction, by identifying an interesting invariant of defect extensions of prime characteristic.

In my talk I will also describe a challenging open problem: prove by purely valuation theoretical means the known fact that algebraic extensions of deeply ramified fields are again deeply ramified.

All this contains no model theoretic results so far (in contrast to the theory of tame and separably tame fields). The aim is to build a structure theory that then, hopefully, can be used by model theorists to push beyond tame fields. I will mention in passing a relatively new result that indirectly is connected with model theory: valued fields with only finitely Artin-Schreier extensions (as are the NTP2 valued fields) are deeply ramified (and even semitame).

Notes 1.47 MB application/pdf

Deeply Ramified Fields And Their Relatives