The notion of ampleness captures essential properties of projective spaces over fields. It is natural to ask whether any sufficiently ample strongly minimal set arises from an algebraically closed field. In this talk I will explain the question and present recent results on ample strongly minimal structures.
Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms source, target: Mor → Ob, inversion: Mor → Mor, composition: Mor ×_{Ob} Mor → Mor and identity: Ob → Mor that satisfy certain axioms. Yet this description of the stack X might be misleading. Namely, given a field F which is not algebraically closed, we have a natural functor between the groupoid (Ob(F),Mor(F)) and the groupoid X(F). While this functor is fully faithful, it […]
Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between `stringy' Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using non-archimedean integrals on these moduli spaces, building on work of Denef-Loeser and Batyrev. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô.In my talk I will outline the main arguments of the proofs and discuss the adjustments […]
Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms source, target: Mor &rarr
I will present a work on flattening by blow-ups in the context of Berkovich geometry (inspired by Raynaud and Gruson's paper on the same topic in the scheme-theoretic setting), and explain how it gives rise to the description of the image of an arbitrary analytic map between two compact Berkovich spaces, and why this description is (very likely) related to quantifier elimination in the Lipshitz-Cluckers variant of Lipshitz-Robinson's analytic language. (I plan to spend most of the talk discussing the results rather than their proofs.)
My goal, in this talk, is to explain a new notion of minimality for (characteristic zero) Henselian fields, which generalizes C-minimality, P-minimality and V-minimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, h-minimality, can be defined, analogously to other minimality notions, by asking that 1-types, over algebraically closed sets, are entirely determined by their reduct to some sublanguage - in that case the pure language of valued fields. However, contrary to what happens with other minimality notions, particular care […]
In this talk, we characterize NIP henselian valued fields modulo the theory of their residue field. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.
The famous Mumford-Tate conjecture asserts that, for every prime number l, Hodge cycles are Q_l linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato-Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The […]
Étant donné un anneau de valuation V de corps résiduel F et contenant un corps k, et une extension k' de k, on cherche à construire une extension V' de V contenant k', d'idéal maximal engendré par celui de V, et de corps résiduel composé de F et k'. On y parvient notamment si F ou k' est séparable sur k.
Lipschitz geometry is a branch of singularity theory that studies the metric data of a germ of a complex analytic space.I will discuss a new approach to the study of such metric germs, and in particular of an invariant called Lipschitz inner rate, based on the combinatorics of a space of valuations, the so-called non-archimedean link of the singularity. I will describe completely the inner metric structure of a complex surface germ showing that its inner rates both determine and are determined by global geometric data: the topology of the […]
(travail en commun avec Silvain Rideau-Kikuchi)Les imaginaires (c'est-à-dire les quotients définissables) dans la théorie ACVF des corpsalgébriquement clos non-trivialement valués sont classifiés par les sortes “géométriques”.Ceci est un résultat fondamental dû à Haskell, Hrushovski et Macpherson. En utilisantl'approche via la densité des types définissables/invariants, nous donnons une réductiondes imaginaires dans des corps valués henséliens, sous des hypothèses assez générales,aux sortes géométriques et à des imaginaires de RV avec des sortes pour certains espacesvectoriels de dimension finie sur le corps résiduel.
Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra () to obtain the following quantitative information: in order to prove the for polynomials of degree d, the Intermediate Value Theorem () is requested to hold for real polynomials of degree at most d^2. We also explain that the classical algebraic proof due to Laplace requires for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.
