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
The deadline for application for ASL travel funds is passed.
List of participants. Timetable.
(taken on Friday 12 January).
to the CIRM.
Organising and Scientific Committee
Zoé Chatzidakis (CNRS - ENS), Dugald MacPherson (Leeds), Sergei Starchenko (Notre Dame), Frank Wagner (Lyon 1).
- Introduction to model theory. Lecturers: Katrin Tent (Münster) and Charlie Steinhorn
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
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
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
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.
Parts I, II
- 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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