Géométrie et Théorie des Modèles

Année 2020 - 2021

Organisateurs : Zoé Chatzidakis, Raf Cluckers.
Pour recevoir le programme par e-mail, 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 quelques-uns des exposés sont disponibles.

Vendredi 16 octobre, Zoom.

9h - 10h20 : Dmitry Novikov (Weizmann Institute), Complex Cellular Structures

Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes.
The Gromov-Yomdin Lemma (later generalized by Pila and Wilkie) proves that the maps could be chosen of C^r-smooth norm at most one, and the number of such maps is uniformly bounded for finite-dimensional 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 Yomdin-Gromov Lemma with polynomial bounds on the complexity, thus proving a long-standing 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 long-term 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 Manin-Mumford conjecture, the Andre-Oort conjecture, Galois-orbit 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 étale-open topology
Fix an abstract field K. For each K-variety V, we will define an “étale-open” 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 pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties.
The étale-open topology is mostly of interest when K is large (also known as ample). On non-large fields, the étale-open topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some well-known facts about large fields, and classify the model-theoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with Chieu-Minh 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 pro-definable and we propose a model-theoretic 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.


Programme des séances passées : 2006-07, 2007-08, 2008-09, 2009-10, 2010-11, 2011-12, 2012-13, 2013-14, 2014-15, 2015-16, 2016-17, 2017-18, 2018-19, 2019-20.
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