organized by Giulio Peruginelli and Franz-Viktor Kuhlmann

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Pierre Touchard
University of Campania 'Luigi Vanvitelli', Caserta, Italy
**

gave a talk

In the 40's, Krasner defined the notion of Corpoid, notably in order to describe the structure of certain sets emmerging from the study of valued fields $K$, namely $K^\star/1+\mathfrack{m}$ where $\mathfrack{m}$ is the maximal ideal of the valuation ring. This notion of corpoid had many parallel developments (gradued rings, skeletons and I suppose hyperfields) and in particular in model theory of valued fields (amc-structures, RV-sort). I will discuss one of them, called the RV-sort of the valued field $K$. The precise definition that I will use is that of Flenner; it consists of equipping $K^\star/1+\mathfrack{m}$ with a kind of addition $\oplus$ and its natural multiplication. Notably I will explain some recent use of this structure to produce transfer principles (or `AKE-like principles'). Those can have the following form: A certain (model theoretic) property P holds for the theory of a valued field $K$ if and only if

- P holds for the theory of the value group,

- P holds for the theory of the residue field,

- K satisfies a certain algebraic condition.

We will see some examples of such transfers for the following properties P: quantifier elimination (Flenner), distallity (Aschenbrenner-Chernikov-Gehret-Ziegler), NIPness and Burden (Chernikov -Simon, T.) while emphasing the role of the RV-sort in these study.

Monday, October 18, 2021, 15:00:

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Franz-Viktor Kuhlmann
Institute of Mathematics, University of Szczecin, Poland**

gave a talk on

Immediate approximation types were introduced in the article

Kuhlmann, F.-V. - Vlahu, Izabela:

and used as a replacement for pseudo Cauchy sequences. An element in an immediate extension of valued fields is the limit of many pseudo Cauchy sequences, but has a unique immediate approximation type.

Pseudo Cauchy sequences cannot efficiently describe extensions of a valuation v from a field K to the rational function field K(x) if the extensions are not immediate. Therefore, a tool is necessary that provides more information. For this purpose, we introduce and classify general approximation types. We prove:

1) Every approximation type over (K,v) is the approximation type of $x$ for a suitable extension of v to K(x).

2) Under certain natural conditions, (K(x),v) can be chosen inside an elementary extension of (K,v).

3) There is a bijection between all approximation types over (K,v) and all extensions of v to K(x) if K is algebraically closed or lies dense in its algebraic closure. In addition, the sort of extension can be read off from the sort of the approximation type, and vice versa.

4) The bijection is preserved in the case of general K when the class of all approximation
types and the class of all extensions are suitably restricted. This is interesting in
particular for the restriction to all pure extensions of v, a notion defined and used in
the article

Kuhlmann, F.-V.: *Value groups, residue fields and bad places of rational function
fields*, Trans. Amer. Math. Soc. 356 (2004), 4559-4600.

Friday, July 2, 2021, 15:00:

gave a talk on

For many questions in arithmetic algebraic geometry, it would be useful to know that every valuation ring is a filtered direct limit of regular rings, as follows from the local uniformization conjecture. In this talk I will review this and related questions.

Friday, June 25, 2021, 15:00:

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Arpan Dutta
Department of Mathematics, IISER Mohali, India
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gave a talk on

In this talk, we establish a connection between minimal pairs of definition and implicit constant fields for valuation transcendental extensions.

Friday, June 18, 2021, 15:00:

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Dario Spirito
Dipartimento di Matematica, Universita' di Padova, Italy**

gave a talk on

We introduce the class of pseudo-monotone sequences, a generalization of pseudo-convergent sequences, in order to generalize Ostrowski's Fundamentalsatz to valuation domains of arbitrary rank.

This is a joint work with Giulio Peruginelli.

Friday, June 11, 2021, 15:00:

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Giulio
Peruginelli
Dipartimento di Matematica, Universita' di Padova, Italy**

gave a talk on

Given a rank one valuation V on a field K, Ostrowski's approach to the study of rank one extensions of V to the field of rational functions K(X) was based on a special kind of sequences of elements of K that he introduced and called pseudo-convergent. We will review his ideas and how his method has been recently generalized to characterize extensions to K(X) of a valuation of K of any rank.

We will show some results about two relevant subspaces of the Zariski-Riemann space of valuation domains of K(X) arising from pseudo-convergent sequences.

This is joint work with Dario Spirito.

Friday, June 4, 2021, 15:00:

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Anna Rzepka
Institute of Mathematics, University of Silesia at Katowice, Poland**

gave a talk

The investigation of valued fields and related areas has shown the importance of a better understanding of the structure of defect extensions of valued fields. An important task is to give necessary and sufficient conditions for a valued field to admit no defect extensions. Ramification theoretical methods show that a central role in the issue of defect extensions is played by towers of Galois defect extensions of prime degree. We classify separable defect extensions of prime degree into dependent and independent ones. We also use the classification of defect extensions to give conditions for valued fields toadmit no defect extensions.

Further, we introduce and study several classes of valued fields, like semitame fields, deeply ramified fields and generalized deeply ramified (gdr) fields. All the classes under consideration can be seen as generalizations of the class of tame valued fields. In particular, we investigate which types of defect extensions such fields admit and give conditions for generalized deeply ramified fields to admit no defect extensions.

This is joint work with Franz-Viktor Kuhlmann.

Friday, May 21, 2021, 15:00:

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Andrei Bengus-Lasnier
Institut de Mathématiques de Jussieu - Paris Rive Gauche, France**

gave a talk on

*Last update: November 6, 2021
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