Katharina Dupont: Finding Definable Valuations with the help of Definable Groups

Shalom,
Next week we start the regular meetings of the logic seminar.

The time is: November 25, Monday 10:30 – 12:00.
Place: Buiding 72 classroom 123.

Please find the title and abstract for the first two talks.
Speaker: Katharina Dupont (University of Konstanz).

Finding Definable Valuations with the help of Definable Groups

Abstract
For my PhD project I am working on the question “Under which conditions does an NIP field admit a non-trivial definable valuation?”.

In his preprint “Definable Valuations” Koenigsmann defines for a subgroup G of a field K the valuation ring O_G.

Assume q\neq char(K) is a prime such that G:=(K^x)^q\neq K^x.
Then O_G is (topologically equivalent to) a non-trivial definable valuation ring if and only if the set B_G:={\bigcap_{i=1}^n a_i (G+1) | n\in N, a_i\in K^x} fulfills the six axioms (V 1)-(V 6) of V-topologies, i.e. B_G is a basis of zero neighbourhoods of a topology induced by an absolute value or a valuation.

In the first part of my talk I will give a general introduction on definable valuations. I will explain Koenigsmann’s definition and the connection between V-topologies and (definable) valuations.

In the second part I will explain how under the assumptions that K does not have the independence property (is NIP) and 1<[K^x:(K^x)^q]<\infty we can use Keisler measures to show the axiom (V 1). If there is time I will talk about the other axioms as well.

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