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.

The school's activities will start at 9am on Monday 8 January and end at around 4pm on Friday 12 January.
Participants will be able to present a poster.

This meeting is sponsored by the ASL (Info). The deadline for application for ASL travel funds is passed.

List of participants. Timetable.   NEW: Some photos: 1, 2, 3 (taken on Friday 12 January).

Directions to the CIRM.

Organising and Scientific Committee

Zoé Chatzidakis (CNRS - ENS), Dugald MacPherson (Leeds), Sergei Starchenko (Notre Dame), Frank Wagner (Lyon 1).

Mini courses

  • Introduction to model theory. Lecturers: Katrin Tent (Münster) and Charlie Steinhorn (Vassar).
    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.

    Reference for the course: A Course in Model Theory, K. Tent, M. Ziegler. Lecture Notes in Logic Nr 40, CUP 2012.

    Pre-requisites: we strongly recommend to all participants to have read at least once the definitions of: languages, structures, formulas, satisfaction, elementary equivalence, quantifier elimination. Possible references are:
    - Chapter 1 of the book by Tent and Ziegler (see above).
    - Introduction to model theory, by Elisabeth Bouscaren. Chapter 1 of Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696. Available on the Springer web site.
    - The first few pages of Introduction to model theory, by Zoé Chatzidakis.
    - D. Marker, Introduction to Model Theory, pp. 15-35 in Model Theory, Algebra, and Geometry, D.Haskell, A.Pillay, and C.Steinhorn (Editors), Mathematical Sciences Research Institute Publications 39. Cambridge: Cambridge University Press, 2000. Can also be found here.
    Slides of Steinhorn: Parts I and II

  • Model theory of valued fields. Lecturers: Deirdre Haskell (McMaster) and Lou van den Dries (UI Urbana-Champaign).
    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. Then we will look at some of the results concerning imaginaries in valued fields. This ties into the lectures on Berkovich space. Familiarity with these results is needed for some of the courses given at the IHP.
    Slides: Parts I, II and III.
  • Introduction to Erdös geometry. Lecturer: Emmanuel Breuillard (Orsay/Münster)
    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. References for the course.
  • Introduction to Berkovich spaces. Lecturer: Antoine Ducros (UPMC).
    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.
  • Introduction to Szemerédi regularity and applications. Lecturer: David Conlon (Oxford)
    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.

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