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Nonuniqueness for the Navier–Stokes equations and model equations / Singularities in Fluid Mechanics

ENS Salle W

JG: In this talk, I will discuss fundamental properties of the solutions to the incompressible Navier–Stokes equations in three dimensions. After reviewing the classical local well-posedness results, I will explain how numerical simulations suggest local ill-posedness at the borderline of the known results. I will discuss a plausible scenario of non-uniqueness from smooth initial data through finite-time blow-up. Finally, I will describe how this scenario is actually happening in a model equation sharing the same fundamental properties as the Navier–Stokes equations.KM: Singularities of the Navier-Stokes equations occur when some derivative […]

Après-midi de théorie de groupes

Salle W (DMA)

14.00-14.45 Stefan Witzel (Ecole Polytehchnique)15.00-15.45 Katie Vokes (IHES)15.45-16.15 coffee break16.15-16.45 Adrien Le Boudec (ENS Lyon)Stefan Witzel, Arithmetic approximate groups and their finiteness propertiesI will talk about approximate groups, a geometric generalization of groups. Approximate groups were discovered independently in various contexts and I will describe how they arise very naturally in the context of arithmetic groups. I will then explain how to extend topological finiteness properties of groups to approximate groups. This allows to make a connection betweenarithmetic groups in positive characteristic and arithmetic approximate groups in characteristic zero. The […]

Après-midi de théorie de groupes

IHP salle 1

14.00-14.45 Katrin Tent (Munster), Burnside groups of relatively small odd exponent15.00-15.45 Arindam Biswas (Vienne), On minimal complements in groups16.15-16.45 Laurent Bartholdi (Institut d'études avancées, ENS Lyon), Dimension series and homotopy groups of spheres

An abstract elementary class framework for fields with commuting automorphisms

We take a look at structures that consist of a field together with finitely many distinguished field automorphisms required to commute. The theory of fields with one distinguished automorphism has a model companion known as ACFA, which Z. Chatzidakis and E. Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. This leads us to investigate them within a non-elementary framework. One way […]

H-structures

Sophie Germain salle 2015

A complete theory T is called geometric if the algebraic closure has the exchange property in all models of T and the theory eliminates the quantifier exists infinity. In such theories there is a rudimentary notion of independence given by algebraic independence. Examples of geometric theories include SU-rank one theories and dense o-minimal theories.An expansion of a model M of T by a unary predicate H is called dense-codense if for every finite dimensional subset A of M and every non algebraic type p(x) over A, there is a realization […]

Sous-groupes qui pavent génériquement et géométrie des involutions

Sophie Germain salle 2015

(En collaboration avec Joshua Wiscons)L'exposé mélange théorie des modèles, théorie des groupes, et algèbre géométrique. On y parlera de groupes de rang de Morley fini, mais il suffit de savoir naïvement ce qu'est une dimension à valeurs entières, sans devoir maîtriser les finesses de la conjecture de Cherlin-Zilber.Un groupe abstrait porte peu d'information de nature géométrique, même au sens des géométries d'incidence, et c'est toujours remarquable si cela se produit.Le pur groupe SO(3,R), par exemple, permet de redéfinir l'espace projectif réel. PGL(2,C) permet presque la même chose : il définit […]

An abstract elementary class framework for fields with commuting automorphisms

We take a look at structures that consist of a field together with finitely many distinguished field automorphisms required to commute. The theory of fields with one distinguished automorphism has a model companion known as ACFA, which Z. Chatzidakis and E. Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. This leads us to investigate them within a non-elementary framework. One way […]

Permutations et polymères aléatoires

ENS (amphithéâtre Galois sous la bibliothèque de mathématique)

 On s’intéressera dans l’exposé au problème suivant, appelé problème d'Ulam: si on prend une permutation s de {1,…,n} au hasard, uniformément parmi toutes les permutations possibles, quelle est la longueur de la plus longue sous-suite croissante de s(1), s(2),…, s(n) ?Ce problème d’apparence simple est en réalité très riche, et on verra qu’il est relié à certaines modèles de physique statistique, dont un modèle de polymère aléatoire.

Quantifier elimination in algebraically closed valued fields in the analytic language: a geometric approach

ENS Salle W

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.)