# BCAM-Severo Ochoa Courses: Model Theory in Bilbao

Date: Mon, Sep 11 - Fri, Sep 22 2023

Hour: 9:00-13:30

Location: Maryam Mirzakhani Seminar Room and UPV/EHU

Speakers: Pablo Cubides Kovasics (Universidad de los Andes), Christian D'Elbée (Bonn University) and Immi Halupczok (Düsseldorf University)

School in model theory with applications to algebra and geometry

The event is devoted to Ph.D. students, postdocs and researchers with interest in applications of model theory to algebra and geometry. In the first week solid foundations will be laid. In the second some selected applications will be discussed. Lectures will be complemented with exercise sessions.

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**Courses**

First week:

__Introduction to model theory (Pablo Cubides Kovasics), 14 hours__

Abstract: The aim of this course is to introduce the students to the basics of model theory. After a brief recall of first-order logic, we will revise the main definitions and results of a first course on this subject. A particular emphasis will be given to algebraic structures, providing most of the tools needed for the courses of the second week of the school.

Second week:

__Model theory of Henselian valued fields and Ax-Kochen-Ershov principle (Christian D'Elbée), 8 hours__

Abstract: The goal of this course is to prove the Ax-Kochen-Ershov (AKE) theorem. This classical result was proven by Ax and Kochen and independently by Ershov in the 1960s. AKE theorem, is considered as the starting point of the model theory of valued fields, and witnessed numerous refinements and extensions. To a certain measure, motivic integration can be considered as such. The AKE theorem is not only an important result in model theory, it yields a striking application to p-adic arithmetics. Artin conjectured that all p-adic fields are C_2 (every homogeneous polynomial with degree d and n>d^2 unknowns over the field of p-adic has a non trivial zero in the field). A consequence of AKE is that the p-adics are asymptotically C_2, meaning that for all but finitely many primes p, the field of p-adic is C2. We will see how this result follows from the AKE theorem.

__A non-archimedean approach to stratifications (Immi Halupczok), 8 hours__

Abstract: A stratification of a set X (e.g. an algebraic subset X ⊂ ℂ^n) is a partition of X into finitely many smooth subsets called strata which as smooth and which satisfy certain regularity conditions, which, in one way or another, express that X is "roughly translation invariant" along the strata. If one replaces ℂ by a bigger field K containing infinitesimal elements (with smaller norm than any non-zero complex number), one has new (natural) possibilities to make precise what "roughly translation invariant" means. I will explain how this approach yields new notions of stratifications. As an example application, I will show how this yields new information about certain Poincaré series, which, given a system of polynomial equations with integer coefficients, count the number of solutions of the system in ℤ/p^rℤ, for primes p. This main part of this is a collaboration with David Bradley-Williams. I might also present things which is in collaboration with Pablo Cubides Kovacsics.

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