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This is the third lecture from the Mini-Course Model theory of algebraic vector fields by Rahim Moosa. The first lecture is given as a Colloquium talk, and the details for the second one are here.

Abstract

In this final lecture I will sketch the proof that the degree of nonminimality of a finite rank type in DCF is at most two, and deduce as a consequence one of the theorems stated in Lecture 1.

This is the second lecture from the Mini-Course Model theory of algebraic vector fields by Rahim Moosa. The first lecture is given as a Colloquium talk, and the third lecture is described here.

Abstract

In this talk I will discuss how one translates between notions coming from model theory and from differential-algebraic geometry. This should serve as an explanation for how model theory is involved in the results about algebraic vector fields that were discussed in Lecture 1 (colloquium).

Prof. Rahim Moosa will give a mini-course of three lectures, via the program for researchers from abroad. The first, introductory talk will be given as a Colloquium talk, followed by the second and third talks during the following week.

The School will focus on Polish Groups. General theory, examples, representations of discrete groups with Polish targets, and applications to discrete groups, geometry and dynamics. The three main courses will be:

Mini-Courses

1. General theory of Polish groups Jullien Melleray, Christian Rosendal and Todor Tsankov.
2. Algebraic groups over Polish fields. Jean Lécureux and Bertrand Rémy.
3. Infinite dimensional Lie groups and their symmetric spaces Bruno Duchesne.

In honor of Prof. Mikhail Klin, on the occasion of his retirement

Speaker: Antoine Ducros (Paris 6)

Title: Stability of Gauss valuations

Abstract:

A valued field $(k,|.|)$ is said to be stable (this terminology has no link with model-theoretic stability theory) if every finite extension $L$ of $k$ is defectless, i.e., satisfies the equality $\sum e_vf_v=[L:k]$, where $v$ goes through the set of extensions of $|.|$ to $L$, and where $e_v$ and $f_v$ are the ramification and inertia indexes of $v$. The purpose of my talk is to present a new proof (which is part of current joint reflexions with E. Hrushovski and F. Loeser) of the following classical fact (Grauert, Kuhlmann, Temkin,…) : let $(k,|.|)$ be a stable valued field, and let $(r_1,\dots,r_n)$ be elements of an ordered abelian group $G$ containing $|k^*|$. Let $|.|’$ be the $G$-valued valuation on $k(T_1,\dots,T_n)$ that sends $\sum a_I T^I$ to $\max_I |a_I|\cdot r^I$. Then $(k(T_1,\dots,T_n),|.|’)$ is stable too.

Our general strategy is purely geometric, but the proof is based upon model-theoretic tools coming from model theory (which I will first present; no knowledge of model theory will be assumed). In particular, it uses in a crucial way a geometric object defined in model-theoretic terms that Hrushovski and Loeser attach to a given $k$-variety $X$, which is called its stable completion; the only case we will have to consider is that of a curve, in which the stable completion has a very nice model-theoretic property, namely the definability, which makes it very easy to work with.

Joint Operator Algebras and Operator Theory Seminar, bringing together people from Ben-Gurion University, the Technion, Tel Aviv University and the University of Haifa