Géométrie et Théorie des Modèles

Année 2021 - 2022

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 21 janvier 2022, de 14h à 17h10, en Zoom. Programme :

14h : Floris Vermeulen (KU Leuven), Hensel minimality and counting in valued fields
Hensel minimality is a new axiomatic framework for doing tame geometry in non-Ar chimedean fields, aimed to mimic o-minimality. It is designed to be broadly applicable while having strong consequences. We will give a general overview of the theory of Hensel minimality. Afterwards, we discuss arithmetic applications to counting rational points on definable sets in valued fields.
This is partially joint work with R. Cluckers, I. Halupczok and S. Rideau-Kikuchi, and partially with V. Cantoral-Farfan and K. Huu Nguyen.

Notes de l'exposé et vidéo.

15h40 : Konstantinos Kartas (Oxford), Decidability via the tilting correspondence
We discuss new decidability and undecidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. The reduction uses extensively the theory of perfectoid fields and also the earlier Krasner-Kazhdan-Deligne principle. Our main results will be:
(1) A relative decidability theorem for perfectoid fields. Using this, we obtain decidability of certain tame fields of mixed characteristic.
(2) An undecidability result for the asymptotic theory of all finite extensions of ℚ_p (fixed p) with cross-section.
We will also discuss a tentative step towards understanding the underlying model theory of arithmetic phenomena in this area, by presenting a model-theoretic way of seeing the Fontaine-Wintenberger theorem.

Notes de l'exposé et vidéo.

Vendredi 18 février 2022. Orateurs :

14h : Minh-Chieu Tran (Notre Dame U.), The Kemperman inverse problem
Let G be a connected locally compact group with a left Haar measure μ, and let A,B ⊆ G be nonempty and compact. Assume further that G is unimodular, i.e., μ is also the right Haar measure; this holds, e.g., when G is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that

μ(AB) ≥ min {μ(A)+μ(B), μ(G)} .

The Kemperman inverse problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the connections to model theory. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang).

Notes de l'exposé et vidéo.

15h45 : James Freitag (UI Chicago), Not Pfaffian
This talk describes the connection between /strong minimality/ of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being /Pfaffian/. The talk will not assume the audience knows these notions previously, and will attempt to motivate why each of them are important notions in various areas. The connection we give, combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian.


Vendredi 18 mars 2022, session sur Zoom. Programme :

14h - 15h30 : Tamara Servi (IMJ-PRG/Fields), Interdefinability and compatibility in certain o-minimal expansions of the real field.

Let us say that a real function f is o-minimal if the expansion (R,f) of the real field by f is o-minimal. A function g is definable from f if g is definable in (R,f). Two o-minimal functions are compatible if there exists an o-minimal expansion M of the real field in which they are both definable. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler's Gamma and Riemann's Zeta, restricted to the reals. If time allows it, I will present a general technique for establishing whether a function is definable or not in a given o-minimal expansion of the reals. Joint work with J.-P. Rolin and P. Speissegger.

Notes de l'exposé et vidéo

15h45 - 17h15 : Philipp Hieronymi (Bonn/Fields), Tameness beyond o-minimality (in expansions of the real ordered additive group)

In his influential paper “Tameness in expansions of the real field” from the early 2000s, Chris Miller wrote:
“ What might it mean for a first-order expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and real-analytic geometers to the o-minimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some well-defined sense [...]. The analysis of such structures often requires a mixture of model-theoretic, analytic-geometric and descriptive set-theoretic techniques. An underlying idea is that first-order definability, in combination with the field structure, can be used as a tool for determining how complicated is a given set of real numbers.”
Much progress has been made since then, and in this talk I will discuss an updated account of this research program. I will consider this program in the larger generality of expansions of the real ordered additive group (rather than just in expansions of the real field as envisioned by Miller). In particular, I will mention in this context recent joint work with Erik Walsberg, in which we produce an interesting tetrachotomy for such expansions.


Vendredi 13 mai à l'ENS, salle W. Orateurs

11h : Arthur Forey (EPFL), Complexity of l-adic sheaves
To a complex of l-adic sheaves on a quasi-projective variety one associate an integer, its complexity. The main result on the complexity is that it is continuous with tensor product, pullback and pushforward, providing effective version of the constructibility theorems in l-adic cohomology. Another key feature is that the complexity bounds the dimensions of the cohomology groups of the complex. This can be used to prove equidistribution results for exponential sums over finite fields. This is due to Will Sawin, written up in collaboration with Javier Fresán and Emmanuel Kowalski.


14h15 : Thomas Scanlon (UC Berkeley), Skew-invariant curves and algebraic independence
A σ-variety over a difference field (K,σ) is a pair (X,φ) consisting of an algebraic variety X over K and φ:X → Xσ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Yσ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties of the form (𝔸n,φ) where φ(x1,...,xn) = (P1(x1),...,Pn(xn)). The most important case, from which the others may be deduced, is that of n = 2. In the present work we give a sharper description of the skew-invariant curves in the case where P2 = P1τ for some other automorphism of K which commutes with σ. Specifically, if P in K[x] is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or ± Chebyshev (i.e. P is “nonexceptional”) then skew-invariant curves in (𝔸2,(P,Pτ)) are horizontal, vertical, or skew-twists: described by equations of the form y = ασn ∘ Pσn-1 ∘ ⋅⋅⋅ ∘ Pσ ∘ P(x) or x = βσ-1∘ Pτ σ-n-2∘ Pτ σ-n-3∘ ⋅⋅⋅ ∘ Pτ(y) where P = α ∘ β and Pτ = ασn+1∘ βσn for some integer n.


16h : Gal Binyamini (Weizmann Institute), Sharp o-minimality: towards an arithmetically tame geometry
Over the last 15 years a remarkable link between o-minimality and algebraic/arithmetic geometry has been unfolding following the discovery of Pila-Wilkie's counting theorem and its applications around unlikely intersections, functional transcendence etc. While the counting theorem is nearly optimal in general, Wilkie has conjectured a much sharper form in the structure R_exp. There is a folklore expectation that such sharper bounds should hold in structures “coming from geometry”, but for lack of a general formalism explicit conjectures have been made only for specific structures.
I will describe a refinement of the standard o-minimality theory aimed at capturing the finer “arithmetic tameness” that we expect to see in structures coming from geometry. After presenting the general framework I will discuss my result with Vorobjov showing that the restricted Pfaffian structure is sharply o-minimal, and how this was used in our recent work with Novikov and Zack to prove Wilkie's conjecture for the restricted Pfaffian structure and for Wilkie's original case of R_exp. I will also discuss some conjectures on the construction of larger sharply o-minimal structures, and some partial results in this direction. Finally I will explain the crucial role played by these results in my recent work with Schmidt and Yafaev on Galois orbit lower bounds for CM points in general Shimura varieties, and subsequently in the recent resolution of general André-Oort conjecture by Pila-Shankar-Tsimerman-(Esnault-Groechenig).


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, 2020-21.
Retour à la page principale.