Model Theory, Combinatorics and Valued fields
Courses, seminars, etc.
Warning: Always check where a particular event takes place
as the classroom but also the building may change depending on the weeks. See the
agenda for precise
information. The possible venues will be: the IHP (amphitheater Darboux
or Hermite); the ENSCP (same campus as the IHP); the ENS (at 45 rue
d'Ulm). Here are directions for the classrooms outside
the IHP.
Main courses:
Geometric stability theory and NIP, 18
hours. During the first and second periods of the trimester (Jan 15  26 and
Feb 5  March
2), Mondays and Wednesdays, 9  10:30 am (except during the first week). Lecturers:
Anand
Pillay (Notre Dame): I will give an introduction to notions and techniques of contemporary model theory, including imaginaries, typespaces, and saturated models,
with an eye to the later applications. I will in general focus quite a bit on type spaces (and the topology) and functions on type spaces. I will discuss local stability and local NIP. I also plan to mention Keisler measures, and the pseudofinite counting measure and its properties in certain contexts. I may also make some connections to combinatorics and
continuous logic if there is time. Of course these things will be
treated in detail by other lecturers later. Slides.
Pierre
Simon (Berkeley): A structure is NIP if all families of
uniformly definable sets have finite VCdimension. NIP structures
are generally of geometric origin (either algebraic, real, or
nonarchimedean). I will give an introduction to NIP structures,
focussing on the tools needed in applications to combinatorics and
valued
fields. Amador
MartinPizarro (Freiburg): A complete theory has finite Morley rank if there is a tame integervalued dimension function on definable sets, called Morley rank. We will introduce Morley rank and degree, and some basic properties of groups definable in theories of Morley rank. The goal is to present a full proof of the group configuration theorem, which is a fundamental tool in order to recover definable groups from a certain abstract configuration of points.
Model theoretic aspects of regularity lemmas, 12
hours. During the first period of the trimester (Jan 15  26 and
Feb 5  9), Mondays
and Wednesdays, 2  4pm (except during the first
week). Lecturer: Artem
Chernikov (UCLA).
This is a minicourse on applications of model theory in hypergraph combinatorics, concentrated on the regularity phenomena (Szemerédi's regularity lemma and its
relatives). We will give proofs of the regularity and removal lemmas for graphs and hypergraphs using probability measures on ultraproducts (the so called “nonstandard” method).
Then we will investigate the effect of various modeltheoretic tameness
assumptions on a structure (such as stability, NIP (= finite
VCdimension), distality, generalizations of Tao's algebraic
regularity lemma, group versions, etc.) on the regularity properties
of hypergraphs definable in it.
Notes around the course.
Model theory of motivic integration, 12
hours. Second period of the trimester (Feb 12  March 2), Mondays
and Wednesdays, 2  4pm (except on Wednesday February 14: 2:30pm  4:30pm). Lecturer:
Immi Halupczok
(Düsseldorf).
The field Q_p of padic numbers carries a natural Lebesgue measure, so that it makes sense to integrate functions (Q_p)^n > C. On other valued fields of interest, like k((t)), where k is a field of characteristic 0, no useful Lebesgue measure exists, the main problem being that certain sets need to have the cardinality of the residue field k as measure.
However, one can define a motivic measure, which takes values in a suitable Grothendieck ring R, which in particular contains an element [k] which formally plays the role of the cardinality of k. This then leads to motivic integration of motivic functions; the precise definition of a motivic function is somewhat technical, but one should think of it as a function (k((t)))^n → R.
In model theoretic approaches to motivic integration, one associates a
motivic measure to every definable set, and also the motivic functions
are defined in terms of definable objects. In this course, I will give
an introduction this kind of motivic integration, probably mainly
following the approach by HrushovskiKazhdan: This consists in
abstractly defining the most general measure possible using
abstract nonsense and then working out what the ring actually looks like
where that measure takes values.
In addition, we will have two weekly seminars (of 90 minutes each), a
junior seminar, and several
minicourses and/or working
groups. The precise programme of the minicourses and working groups will be
established as we go along. We also list some courses which may be
of interest and take place in Paris.
Minicourses
 Richness in Globally valued fields. Lecturer: Itaï Ben
Yaacov (Lyon 1). Amphitheater Darboux. Tuesday February 13, 2 
4pm; Friday February 16 and
Tuesday February 20.
In a joint project with E. Hrushovski we study globally valued fields (GVFs), namely fields in which (an abstract version of) the product formula holds.
Recall that the product formula for number fields and function fields of curves asserts that for any nonzero x in the field, the product of all the absolute values of x , normalised appropriately, is one, or equivalently, the sum of all valuations is zero.
Intuitively, a GVF consists of a field equipped with a measured family of valuations, such that for every nonzero x , the integral of the valuations of x vanishes. A somewhat better presentation of (almost) the same object is a triplet (K,G,v) , where K is a field, G is the latticeordered additive group of some (real) L^1 space, and v is a valuation with value group G , such that for all nonzero x , the integral of v(x) vanishes. Presented in this fashion, GVFs form an elementary class (in continuous logic). Natural modeltheoretic questions regarding this class are translated into interesting, and difficult, geometric questions.
For example, one can ask whether the theory GVF (which is inductive) admits a modelcompanion, or equivalently, whether the class of existentially closed GVFs is elementary. In other words, one would want to characterise GVFs K which are “rich enough” inside any GVF extension L/K (technically: L can always be embedded in an ultrapower of K ), and then show that this characterisation can be stated as a logical formula.
There are two main approaches for this:
I. Describe GVFs, and GVF extensions, using “global” information (intersection numbers), in order to obtain such a characterisation.
II. Start with a single additional axiom, called Fullness, which allows us to obtain global existence properties from local data, and show that any full GVF is existentially closed.
In this minicourse I shall discuss the second approach. It will be organised as follows:
1. Basic definitions. I shall define GVFs, and motivate the fullness axiom via the question of amalgamation of GVFs.
2. Valued vector spaces. These allow us to restate fullness in a more useful way, and setup a strategy for finding points in a full GVF, thereby showing (hopefully) that it is rich.
3. Baby case: In order to show the mechanics of this strategy, we will consider in a first time the easy case of finding a special point on the projective line.
4. Preliminaries for the general case: Chow forms and rudimentary aspects of stability in normed fields.
5. The general case: how to find special rational points in a projective variety over a full GVF.
6. The last part will be dedicated to consequences of the previous
results. In particular, we shall go back to the question of
amalgamation, showing that any two GVF extensions of a full GVF can be
amalgamated (a “weak richness” property, short of existential
closedness). I shall give a few additional arguments which justify my
belief that a proof that full GVFs are existentially closed is within
reach.
Preprint on Arxiv, giving the proof of the (local)
asymptotic estimate of volumes.
 Surreal models of the reals with exponentiation. Lecturer: Alessandro
Berarducci (Pisa).
Tuesday 6 February, 10am  12pm (Darboux); Thursday February 8, 2  4pm
(Darboux).
I plan to speak about techniques to construct models of T_exp (the theory of the reals as an exponential field)
based on generalized power series, and then consider the question of when there is a good derivation on them (good includes the fact that it respects exp and commutes with infinite sums).
There is a uniform approach which includes all kinds of transseries and also the surreal numbers.
The approach is inspired by the surreal numbers, but one does not need to know (or introduce) the surreals: all the the relevant facts can be presented in a purely algebraic way, well suited for model theorists.
I will show that if a certain condition is satisfied, then there is a derivation (and the surreals satisfy the condition).
I will also present an example without a derivation.
Along the way I will make various observation concerning the rich interplay between valuations and exponentiation.
Joint work with Kuhlmann, Mantova, Matusinski.
Slides.
 Pseudofinite structures. Lecturer: Dario Garcia
(Leeds). Thursday 8 and Friday 9 February, 10:30 am  12:30pm.
The purpose of this series of lectures is to present some of the main definitions on model theory of pseudofinite structures. I will review the concept of “pseudofinite dimension” (cf. [3]) and its relationship with forking and modeltheoretic dividing lines on ultraproducts of finite structures as presented in [1]. I will also present results appearing in [5] where some counting arguments on pseudofinite structures and the existence of Zilber polynomials are shown to imply that every strongly minimal pseudofinite structure is locally modular.
If time permits, I will discuss some topics on asymptotic classes of finite structures (cf. [3],[4])
Contents
Session 1: Basic definitions and examples of pseudofinite structures.A language for counting definable sets and cardinality comparison. Fine/coarse pseudofinite dimensions. Fine pseudofinite dimension and forking. Pseudofinite structures, simplicity and stability.
Session 2: Celldecomposition in strongly minimal theories. Strongly minimal pseudofinite structures. Unimodularity, and related problems. Asymptotic classes of finite structures and measurable structures.
References
[1] D. García, D. Macpherson, C. Steinhorn. Pseudofinite structures and simplicity. J. Math. Log. 15, (2015)
[2] R. Elwes. Asymptotic classes of finite structures. J. Symbolic Logic 72 (2007)
[3] E. Hrushovski. On pseudofinite dimensions. Notre Dame J. Formal Logic. Volume 54, Number 34 (2013)
[4] D. Macpherson, C. Steinhorn. Onedimensional asymptotic classes of finite structures. Trans. Amer. Math. Soc. 360 (2008)
[5] A. Pillay. Strongly minimal pseudofinite structures. Available at
http://arxiv.org/pdf/1411.5008v2.pdf (2014)
Notes
 Structural limits. Lecturers: Jaroslav Nesetril (Charles
University Prague) and Patrice Ossona de Mendez (EHESS Paris).
Thursday 18 January, 10am12pm (Hermite); Tuesday 23 January, 10:45 am 
12:45pm (Darboux); Friday 26 January, 10am  12pm (Darboux).
In the series of lectures we plan to cover the following topics:
 Classical limits of structures (elementary, left limits and local limits)
 Sparsity (in many of its characterisations)
 Structural convergence (the representation theorem, various fragments of interests)
 Near the limit analysis (1point lift theorem, expanding and clustering)
 Modeling limits (non standard approach and existence via Friedman logic)
Slides of Course 1 and
of Course 3.
Working groups
 Perfectoid
spaces. Very informal working group, following the notes
of Conrad's
seminar. Wednesdays, 9:15  11am, Amphitheater Darboux or
Hermite depending on the weeks. Next session, Wednesday March 14.
 Reading seminar On pseudofinite dimensions. Thursdays, 10:30am 
12pm, amphitheater Darboux or Hermite depending on the weeks. Will
concentrate on Ehud Hrushovski's paper On pseudofinite
dimensions, Notre Dame Journal of Formal Logic,
Volume 54, Numbers 34, 2013, 463  495. Next session, Thursday March 15.
Let
us also mention the following outside courses and seminars, which may be of interest:
Timothy Gowers (Cambridge, FSMP, PSL)  Introduction à la
combinatoire additive. Tuesdays and Thursdays, 2pm  4pm, in amphitheater Rataud, ENS. 8
January  23 March.
Additive Combinatorics is a branch of mathematics with roots in combinatorics, additive number theory, harmonic analysis, and ergodic theory. This course will cover some of the most important results in the area, such as Roth's theorem on arithmetic progressions, Szemeredi's regularity lemma, and Ruzsa's proof of Freiman's theorem. However, the emphasis will be more on the techniques of proof than on the results themselves.
Announcement
Adrien Deloro (IMJPRG)  Théorie des
modèles des groupes affines I* (in French). 8 January 
16 February 2018. Tuesdays and Thursdays, 2  4pm; Fridays, 4  6pm.
Le but de ce cours (en deux parties) est de montrer en quoi la théorie des modèles, c'estàdire le versant mathématique de la logique, est un langage pertinent pour l'étude des groupes algébriques.
La première partie présentera divers phénomènes géométriques (élimination, transfert) sous l'angle modèlethéorique, avant d'embrayer sur une invitation aux groupes de rang de Morley fini, vaste généralisation logique des groupes algébriques affines.
More details
on the course.
Classrooms. Course
notes.
Adrien Deloro (IMJPRG)  Théorie des
modèles des groupes affines II* (in French). 26 February
 6 April 2018.
Cette seconde partie exposera des applications aux groupes algébriques avant de s'orienter vers des questions (ouvertes) spécialisées.
More details on the course.
Matthew Morrow (IMKPRG)  Espaces adiques et espaces
perfectoïdes (in French). 26 February  6 April.
Le but du cours est d'introduire l'approche de Huber à la géométrie analytique rigide en termes d'espaces adiques, qui étend la théorie des schémas à une grande classe d'anneaux topologiques. Un cas particulier est les espaces perfectoïdes introduits par Scholze en 2011, qui jouent un rôle fondamental dans beaucoup de progrès récents en géométrie arithmétique padique.
26 February  6 April.
Web
page of the course.
Todor Tsankov, Logique continue (in French). Starting 11
January. Thursdays, 8:30  10:30am, room 253E, Halle aux
Farines (10 rue Françoise Dolto, 75013
Paris). Web
page
of the course.
La logique continue est une version de la logique du premier ordre où les structures sont des espaces
métriques et les formules prennent des valeurs dans les nombres réels (à la place de {0, 1}). Toutes les notions
principales de la théorie des modèles se généralisent à ce cadre mais souvent de nouveaux phénomènes
apparaissent qui n'ont pas d'analogue dans le cadre classique. Cette logique est particulièrement adaptée à
l'étude des objets qui proviennent de l'analyse comme les espaces de Banach, les algèbres d'opérateurs et les
probabilités.
C'est une branche nouvelle de la théorie des modèles qui est actuellement en plein développement.
Le cours va développer les bases de la logique continue (formules, espaces de types, structures omégacatégoriques,
etc.) et une attention particulière sera portée aux exemples.
Il n'y a pas de prérequis particuliers sauf des connaissances de
base en analyse.
Séminaire général de logique, Mondays 3:10 
4:10 pm, room 2015, Sophie Germain.
Séminaire Structures algébriques
ordonnées, Tuesdays 2  3:45 pm, room 1016, Sophie Germain.
Directions
Ecole Normale Supérieure de Chimie de Paris (ENSCP). 11 rue
Pierre et Marie Curie. Building to the left when looking at the
IHP.
Amphitheater Chaudron. After the main hall, take the corridor on
the left to the end. There are several doors, one of them has the name
of the amphitheater.
Amphitheater Moissan. Cross the main hall, and take the stairs
down one floor. The name of the amphitheater is indicated.
Ecole Normale Supérieure (ENS), 45 rue d'Ulm. 7 minutes walk from the
IHP. Map.
Amphithéâtre Rataud or Galois. It is located in
the new building, at level 1: go through or around the main building
of ENS, you will see the new building; take the entrance to the right of the
main entrance (one needs to press on the button to open the door), go
down the stairs, the amphitheatre is in front of you on the left.
Salle Actes. Enter the main building, turn right to reach
Staircase (Escalier) A. Go up to the first floor, the room is on the
right.
Amphitheater Dussane. Enter the main building, turn left and
continue until the end of the corridor. Nr 9
on the map.
