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Lothar Sebastian Krapp (Konstanz): Valuations Definable in the Language of Ordered Rings

Donnerstag, 05.12.2019 11:00 im Raum SR 1D
Mathematik und Informatik

Let L_r := {+,-,*,0,1} be the language of rings and let L_or := {+,-,*,0,1,<} be the language of ordered rings. The study of definable valuations (i.e. valuations whose corresponding valuation ring is a definable set) in certain fields is motivated by the general analysis of definable subsets of fields as well as by recent conjectures on the classification of NIP fields. There is a vast collection of results giving conditions on L_r-definability of henselian valuations in a given field, many of which are from recent years (cf. [Fehm-Jahnke19]). So far, not much seems to be known about L_or-definable valuations in ordered fields. Since L_or is a richer language than L_r, it is natural to expect further definability results in the language of ordered rings. In my talk, I will outline some progress in the study of L_or-definable valuations from my joint work with S. Kuhlmann and G. Lehéricy. In this regard, I will present sufficient topological conditions on the value group and the residue field of a henselian valuation v on an ordered field such that v is L_or-definable. Moreover, I will show how the study of L_or-definable valuations connects to ordered fields dense in their real closure as well as above mentioned conjectures regarding NIP fields. All valuation and model theoretic notions will be introduced.



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