Research School Model Theory, Combinatorics and Valued Fields
8 - 12 January 2018, CIRM, Luminy, France
The Institut Henri Poincaré will host a trimester entitled Model theory, combinatorics and valued fields during the period 8 January - 6 April 2018. The main themes for this programme aim to develop both the internal model theory of tame structures and their recent applications. The proposed research school is intended to be the introductory pre-school of this programme.
Organising and Scientific CommitteeZoé Chatzidakis (CNRS - ENS), Dugald MacPherson (Leeds), Sergei Starchenko (Notre Dame), Frank Wagner (Lyon 1).
This course, concentrated at the beginning of the school, will present the very basic notions and results of model theory: structures, languages, satisfaction, definability, compactness theorem, saturation, ultraproducts, as well as an introduction to NIP and VC dimension.
This course will start by recalling the definitions and properties of valued fields (with value group not necessarily archimedean). It will then introduce some of the classical results on the model theory of valued fields: quantifier-elimination for theories of valued fields, Ax-Kochen-Ershov type theorems. Familiarity with these results is needed for the courses given at the IHP.
Erdös geometry studies finite sets within algebraic geometry. Landmark samples of this field include the sum-product phenomenon of Erdös and Szemerédi, or the Szemerédi-Trotter theorem about the number of incidences of a finite set of points and lines in the plane. The course will present the basic results.
This course will introduce classical Berkovich spaces, their motivation, and how they are used. It will give their translation in terms of space of types. If time permits, we will also define Huber spaces.
The regularity lemma is one of the cornerstones of modern graph theory. Roughly, it says that the vertex set of every sufficiently large graph can be partitioned into a constant number of parts such that the graph between almost every pair of parts is random-like. The purpose of this lecture series will be to discuss the regularity lemma, its variants and a selection of applications in graph theory, number theory and theoretical computer science.
Preregistration at the CIRMWeb page at the CIRM, and preregistration at the CIRM.