Algèbre et géométrie

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.

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 → X^σ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Y^σ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties 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 […]

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 […]

Tilting perfectoid fields, developed by Scholze, allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for […]

In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, Casale, Freitag, Nagloo), the following picture has been obtained regarding the transcendence properties of their solutions: - (Strong minimality): outside of an exceptional set of parameters, the corresponding differential equations are strongly minimal, - (Geometric triviality): algebraic independence of several solutions is controlled by pairwise algebraic independence outside of this exceptional set of parameters, - (Multidimensionality): the differential equations defined by generic independent parameters are orthogonal. Are […]