Géométrie et Théorie
des Modèles
Année 2021  2022
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 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 nonAr
chimedean fields, aimed to mimic ominimality. 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. RideauKikuchi, and partially with V. CantoralFarfan 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 KrasnerKazhdanDeligne 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 crosssection.
We will also discuss a tentative step towards understanding the
underlying model theory of arithmetic phenomena in this area, by
presenting a modeltheoretic way of seeing the FontaineWintenberger
theorem.
Notes de l'exposé
et vidéo.
Vendredi 18 février 2022. Orateurs
:
14h : MinhChieu 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 ominimal expansions of the
reals and are differentially algebraic, but not Pfaffian.
vidéo.
Vendredi 18 mars 2022, session sur Zoom. Programme :
14h  15h30 : Tamara Servi
(IMJPRG/Fields), Interdefinability and
compatibility in certain ominimal expansions of the real
field.
Let us say that a real function f is ominimal if the expansion (R,f) of the real field by f is ominimal. A function g is definable from f if g is definable in (R,f). Two ominimal functions are compatible if there exists an ominimal expansion M of the real field in which they are both definable. I will discuss the ominimality, 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 ominimal 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 ominimality (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 firstorder 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 realanalytic geometers to the
ominimal 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 welldefined sense [...]. The analysis
of such structures often requires a mixture of modeltheoretic,
analyticgeometric and descriptive settheoretic techniques. An
underlying idea is that firstorder 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.
vidéo
Vendredi 13 mai à l'ENS, salle W. Orateurs
11h : Arthur Forey (EPFL), Complexity of ladic
sheaves
To a complex of ladic sheaves on a quasiprojective 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 ladic
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.
Vidéo.
14h15 : Thomas
Scanlon (UC Berkeley), Skewinvariant curves and algebraic independence
A σvariety over a difference field (K,σ) is a pair
(X,φ) consisting of an algebraic variety X over K and
φ:X → X^{σ} is a regular map from X to its
transform X^{σ} under σ. A subvariety Y ⊆ X is
skewinvariant if φ(Y) ⊆ Y^{σ}. In earlier
work with Alice Medvedev we gave a procedure to describe
skewinvariant varieties of σvarieties of the form
(𝔸^{n},φ) where φ(x_{1},...,x_{n}) =
(P_{1}(x_{1}),...,P_{n}(x_{n})). 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 skewinvariant
curves in the case where P_{2} = P_{1}^{τ} 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 skewconjugate to a monomial or ±
Chebyshev (i.e. P is “nonexceptional”) then
skewinvariant curves in (𝔸^{2},(P,P^{τ})) are
horizontal, vertical, or skewtwists: described by equations
of the form y = α^{σn} ∘ P^{σn1}
∘ ⋅⋅⋅ ∘ P^{σ} ∘ P(x) or x =
β^{σ1}∘ P^{τ σn2}∘
P^{τ σn3}∘
⋅⋅⋅ ∘ P^{τ}(y) where P = α ∘ β and P^{τ} = α^{σn+1}∘ β^{σn} for some integer n.
Vidéo.
16h : Gal Binyamini (Weizmann
Institute), Sharp ominimality:
towards an arithmetically tame
geometry Over the last 15 years a
remarkable link between ominimality and algebraic/arithmetic
geometry has been unfolding following the discovery of
PilaWilkie'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 ominimality 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 ominimal, 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 ominimal
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 AndréOort conjecture by
PilaShankarTsimerman(EsnaultGroechenig).
Vidéo.
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718,
201819,
201920,
202021.
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