Géométrie et Théorie
des Modèles
Année 2020  2021
Organisateurs :
Zoé
Chatzidakis, Raf Cluckers.
Pour recevoir le programme par email, écrivez à :
zchatzid_at_dma.ens.fr. Pour les personnes ne
connaissant pas du tout de théorie des modèles, des notes introduisant les
notions de base (formules, ensembles définissables,
théorème de compacité, etc.) sont disponibles
ici. Elles peuvent aussi
consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed.,
Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.
Les notes de quelquesuns des exposés sont disponibles.
Vendredi 16 octobre, Zoom.
9h  10h20 : Dmitry Novikov (Weizmann
Institute), Complex Cellular
Structures
Real semialgebraic sets admit socalled cellular decomposition, i.e.
representation as a union of convenient semialgebraic images of standard cubes.
The GromovYomdin Lemma (later generalized by Pila and Wilkie) proves that the maps could be chosen of C^rsmooth norm
at most one, and the number of such maps is uniformly bounded for finitedimensional families.
This number was not effectively bounded by Yomdin or Gromov, but it
necessarily grows as r → ∞.
It turns out there is a natural obstruction to a naive holomorphic complexification of this result related to the natural hyperbolic metric of complex holomorphic sets.
We prove a lemma about holomorphic functions in annulii, a quantitative version of the great Picard theorem.
This lemma allowed us to construct an effective holomorphic version of the cellular decomposition results in all dimensions, with explicit polynomial bounds
on complexity for families of complex (sub)analytic and semialgebraic sets.
As the first corollary we get an effective version of YomdinGromov Lemma with polynomial bounds on the complexity, thus proving a longstanding Yomdin conjecture about tail entropy of analytic maps.
Further connection to diophantine applications will be explained in
Gal's talk.
Notes de l'exposé
et vidéo.
10h30  11h50 : Gal Binayamini (Weizmann Institute), Tame geometry and diophantine approximation
Tame geometry is the study of structures where the definable sets admit finite complexity. Around 15 years ago Pila and Wilkie discovered a deep connection between tame geometry and diophantine approximation, in the form of asymptotic estimates on the number of rational points in a tame set (as a function of height). This later led to deep applications in diophantine geometry, functional transcendence and Hodge theory.
I will describe some conjectures and a longterm project around a more
effective form of tame geometry, suited for improving the quality of the
diophantine approximation results and their applications. I will try to
outline some of the pieces that are already available, and how they
should conjecturally fit together. Finally I will survey some
applications of the existing results around the ManinMumford
conjecture, the AndreOort conjecture, Galoisorbit lower bounds in
Shimura varieties, unlikely intersections in group schemes, and some
other directions (time permitting).
Notes de l'exposé
et vidéo
(les deux premières minutes manquent).
Vendredi 13 novembre :
9h  10h20 : Will Johnson (Fudan U), The étaleopen
topology
Fix an abstract field K. For each Kvariety V, we will define an “étaleopen” topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudofinite fields, the étaleopen topology seems to be new, and has some interesting properties.
The étaleopen topology is mostly of interest when K
is large (also known as ample). On nonlarge fields, the
étaleopen topology is discrete. In fact, this property
characterizes largeness. Using this, one can recover some wellknown
facts about large fields, and classify the modeltheoretically stable
large fields. It may be possible to push these arguments towards a
classification of NIP large fields. Joint work with ChieuMinh Tran,
Erik Walsberg, and Jinhe Ye.
Notes de l'exposé
et vidéo
(les cinq premières minutes manquent).
Vendredi 27 novembre, 9h : suite et fin de
l'exposé. Notes et vidéo de
l'exposé.
10h30  11h50 : Jinhe (Vincent) Ye
(Sorbonne Université), Belles
paires of valued fields and analytification
In their work, Hrushovski and Loeser proposed the space V̂ of
generically stable types concentrating on V to study the homotopy
type of the Berkovich analytification of V. An important feature
of V̂ is that it is canonically identified as a projective
limit of definable sets in ACVF, which grants them tools from
model theory. In this talk, we will give a brief introduction to
this object and present an alternative approach to internalize
various spaces of definable types, motivated by Poizat's work on
belles paires of stable theories. Several results of interest to
model theorists will also be discussed. Particularly, we recover
the space V̂ is strict prodefinable and we propose a
modeltheoretic counterpart Ṽ of Huber's
analytification. Time permitting, we will discuss some comparison
and lifting results between V̂ and Ṽ. This is a joint
project with Pablo Cubides Kovacsics and Martin Hils.
Vidéo
et notes.
Vendredi 11 décembre :
9h : Annalisa Conversano (Massey
University), Groups definable in ominimal
structures and algebraic groups
Groups definable in ominimal structures have been studied by many
authors in the last 30 years and include algebraic groups over
algebraically closed fields of characteristic 0, semialgebraic
groups over real closed fields, important classes of real Lie
groups such as abelian groups, compact groups and linear
semisimple groups. In this talk I will present results on groups
definable in ominimal structures, demonstrating a strong
analogy with topological decompositions of linear algebraic
groups. Limitations of this analogy will be shown through
several examples.
Vidéo
(les deux premières minutes manquent)
et Notes
10h30 : Pablo
CubidesKovacsics (Düsseldorf), Cohomology of algebraic varieties over nonarchimedean fields
I will report on a joint work with Mário Edmundo and Jinhe Ye in
which we introduced a sheaf cohomology theory for algebraic
varieties over nonarchimedean fields based on HrushovskiLoeser
spaces. After informally framing our main results with respect to
classical statements, I will discuss some details of our
construction and the main difficulties arising in this new
context. If time allows, I will further explain how our results
allow us to recover results of V. Berkovich on the sheaf
cohomology of the analytification of an algebraic variety over a
rank 1 complete nonarchimedean field.
Vidéo
et Notes
Vendredi 15 janvier
2021 :
9h30 : Alessandro Berarducci
(Pisa), An application of surreal
numbers to the asymptotic analysis of certain exponential
functions
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing
the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set.
This set of germs is well ordered and Skolem conjectured that its order type is epsilonzero.
Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x).
They did so by studying the possible limits at infinity of the quotient f(x)/g(x) of two functions in the fragment:
if g is kept fixed and f varies, the possible limits form a discrete set of real numbers of order type omega.
Using the surreal numbers, we extend the latter result to the whole class of Skolem functions
and we discuss some additional progress towards the conjecture of Skolem.
This is joint work with Marcello Mamino
(http://arxiv.org/abs/1911.07576,
to appear in the JSL).
Vidéo et Notes.
11h10 : Daniel Palacin
(Freiburg/Complutense), Solving equations in finite groups and complete amalgamation
Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation x+z=2y has a solution in A.
Finitary versions of Roth's theorem study subsets A of {0, ... , N}, and ask whether the same holds for sufficiently large N, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, or even a system of equations. For instance, for the equation xy=z, Gowers (2008) showed that any subset of a finite simple nonabelian group will contain many solutions to this equation, provided it has sufficiently large density.
We will report on recent work with Amador MartinPizarro on how to find
solutions to the above equations in the context of pseudofinite groups,
using techniques from model theory which resonate with (a group version
of) the independence theorem in simple theories due to Pillay, Scanlon
and Wagner. In this talk, we will not discuss the technical aspects of
the proof, but present the main ideas to a general audience.
Vidéo.
Vendredi 12 février. Orateurs
: 9h : Yatir Halevi (Ben Gurion),
et 10h30 : Franziska Jahnke (Münster), On dpfinite fields Shelah's conjecture predicts that any infinite NIP field is
either separably closed, real closed or admits a nontrivial henselian
valuation. Recently, Johnson proved that Shelah's conjecture holds for
fields of finite dprank, also known as dpfinite fields. The aim of these two talks is to give an introduction to dprank in some algebraic structures and an overview of Johnson's work.
In the first talk, we define dprank (which is a notion of rank in NIP theories) and give examples of dpfinite structures. In particular, we discuss the dprank of ordered abelian groups and use them to construct multitude of examples of dpfinite fields. We also prove that every dpfinite field is perfect and sketch a proof that any valued field of dprank 1 is henselian.
In the second talk, we give an overview of Johnson's proof that every
infinite dpfinite field is either algebraically closed, real closed or
admits a nontrivial henselian valuation. Crucially, this relies on the notion of a Wtopology, a natural generalization of topologies arising from valuations, and the construction of a definable Wtopology on a
sufficiently saturated unstable dpfinite field.
Vidéos :
Halevi,
Jahnke. Notes
du 2e exposé.
Vendredi 26 mars, 15 h  18h
15h : Jason Bell (U. of
Waterloo), Effective isotrivial MordellLang
in positive characteristic.
The MordellLang conjecture (now a theorem, proved by Faltings, Vojta,
McQuillan,...) asserts that if G is a semiabelian variety G defined
over an algebraically closed field of characteristic zero, X is a
subvariety of G, and Γ is a finite rank subgroup of G, then
X ∩ Γ is a finite union of cosets of Γ. In positive
characteristic, the naive translation of this theorem does not hold,
however Hrushovski, using model theoretic techniques, showed that in
some sense all counterexamples arise from semiabelian varieties
defined over finite fields (the isotrivial case). This was later
refined by Moosa and Scanlon, who showed in the isotrivial case that
the intersection of a subvariety of a semiabelian variety G with a
finitely generated subgroup Γ of G that is invariant under the
Frobenius endomorphism F: G → G is a finite union of sets of the
form S+A, where A is a subgroup of Γ and S is a sum of orbits
under the map F. We show how how one can use finitestate automata
to give a concrete description of these intersections Γ ∩ X
in the isotrivial setting, by constructing a finite machine that
identifies all points in the intersection. In particular, this
allows us to give decision procedures for answering questions such
as: is X ∩ Γ empty? finite? does it contain a coset of an
infinite subgroup? In addition, we are able to read off coarse
asymptotic estimates for the number of points of height ≤ H in the
intersection from the machine. This is joint work with Dragos
Ghioca and Rahim Moosa.
Vidé
et Notes.
16h30 : Rémi
Jaoui (U. of Notre Dame), Linearization procedures in the semiminimal analysis of algebraic differential equations It is wellknown that certain algebraic differential equations restrain in an essential way the algebraic relations that their solutions share. For example, the solutions of the first equation of Painlevé y'' = 6y^2 + t are “new” transcendental functions of order two which whenever distinct are algebraically independent (together with their derivatives).
I will first describe an account of such phenomena using the language of
geometric stability theory in a differentially closed field. I will then
explain how linearization procedures and geometric stability theory fit
together to study such transcendence results in practice.
Vidéo
et Notes.
Vendredi 23 avril 2021. 15h à
18h. Orateurs :
15h : Gabriel Conant
(Cambridge), VCdimension in model
theory, discrete geometry, and
combinatorics
In statistical learning theory, the notion of VCdimension was developed by Vapnik and Chervonenkis in the context of approximating probabilities of
events by the relative frequency of random test points. This
notion has been widely used in combinatorics and computer science,
and is also directly connected to model theory through the study
of NIP theories. This talk will start with an overview of
VCdimension, with examples motivated by discrete geometry and
additive combinatorics. I will then present several model
theoretic applications of VCdimension. The selection of topics
will focus on the use of finitely approximable Keisler measures to
analyze the structure of algebraic and combinatorial objects with
bounded VCdimension.
Vidéo et Notes.
16h30 : Artem Chernikov
(UCLA), Recognizing groups and fields in
Erdős geometry and model theory
Assume that Q is a relation on R^s of arity s definable in an ominimal expansion of R. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of Q with finite n × ... × n grids, for growing n, can only occur if Q is closely connected to a certain algebraic structure.
On the one hand, if the projection of Q onto any s1 coordinates is finitetoone but Q has maximal size intersections with some grids (of size >n^(s1  ε)), then Q restricted to some open set is, up to coordinatewise homeomorphisms, of the form x_1+...+x_s=0. This is a special case of the recent generalization of the ElekesSzabó theorem to any arity and dimension in which general abelian Lie groups arise (joint work with Kobi Peterzil and Sergei Starchenko).
On the other hand, if Q omits a finite complete spartite hypergraph but can intersect finite grids in more that than n^(s1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and ChieuMinh Tran).
I will explain how these results are connected to the modeltheoretic
trichotomy principle and discuss variants for higher dimensions, and
for stable structures with distal expansions.
Vidéo
et Notes.
Vendredi 21 mai 2021, 9h  12h. Orateurs
:
9h  10h20 : Will Johnson (Fudan
U.), Curveexcluding
fields
Let T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅.
We can use this theory to construct a field K with an interesting combination of properties. On the modeltheoretic side, the theory of K is complete, decidable, modelcomplete, and algebraically bounded, and K is a “geometric structure” in the sense of Hrushovski and Pillay. Additionally, some classificationtheoretic properties might hold in K. On the fieldtheoretic side, K is nonlargethere is a smooth curve C such that C(K) is finite and nonempty. This is unusual; the vast majority of modeltheoretically tractable fields are large or finite. On the other hand, K is “virtually large”it has a finite extension which is large. In fact, every proper algebraic extension of K is pseudo algebraically closed (PAC). The absolute Galois group of K is an ωfree profinite group. This negatively answers a question of Junker and Koenigsmann (is every modelcomplete infinite field large?) and a question of Macintyre (does every modelcomplete field have a small Galois group?).
This is based on joint work with Erik Walsberg and Vincent Ye.
Vidéo
et Notes.
10h30  11h50 : Silvain Rideau
(IMJPRG), PseudoTclosed fields,
approximations and NTP2
Joint work with Samaria Montenegro
The striking resemblance between the behaviour of pseudoalgebraically closed, pseudo real closed and pseudo padically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudoTclosed fields, where T is an enriched theory of fields. These fields verify a “localglobal” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.
The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudoTclosed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with n independent valuations has burden n and, in particular, is NTP2.
Vidéo
et Notes.
Vendredi 18 juin. Orateurs
:
15h : Pierre Simon (Berkeley), Monadically NIP ordered graphs and bounded twinwidth
An open problem in theoretical computer science asks to characterize
tameness for hereditary classes of finite structures. The notion of
bounded twinwidth was proposed and studied recently by Bonnet, Geniet,
Kim, Thommasé and Watrignant. Classes of graphs of bounded
twinwidth have many desirable properties. In particular, they are
monadically NIP (remain NIP after naming arbitrary unary
predicates). In joint work with Szymon Torunczyk we show the converse
for classes of ordered graphs. We then obtain a very clear dichotomy
between tame (slow growth, monadically NIP, algorithmically simple ...)
and wild hereditary classes of ordered graphs. Those results were also
obtained by Bonnet, Giocanti, Ossona de Mendez and Thomassé. In
this talk, I will focus on the model theoretic input.
Vidéo
et Notes.
16h40 : André Belotto da
Silva (AixMarseille), Real perspectives on monomialization.
I will discuss recent work in collaboration with Edward Bierstone on transformation of a mapping to monomial form (with respect to local coordinates) by simple modifications of the source and target. Our techniques apply in a uniform way to the algebraic and analytic categories, as well as to classes of infinitely differentiable real functions that are quasianalytic or definable in an ominimal structure. Our results in the real cases are best possible. The talk will focus on real phenomena and on an application to quantifier elimination of certain ominimal polynomially bounded structure.
Vidéo
et Notes.
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718,
201819,
201920.
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