We investigate properties of some spherical fonctions defined on hyperbolic groups using boundary representations on the Gromov boundary endowed with the Patterson-Sullivan measure class. We prove sharp decay estimates for spherical functions as well as spectral inequalities associated with boundary representations. This point of view on the boundary allows us to view the so-called property RD (Rapid Decay, also called Haagerup’s inequality) as a particular case of a more general behavior of spherical functions on hyperbolic groups. Then I will explain how these representations are related to the so-called « complementary series ». The problem of the unitarization of such representations will be at the heart of the discussion.
If time permits, I will try to explain the idea of a constructive proof, using a boundary unitary representation, of a result due to de la Harpe and Jolissaint asserting that hyperbolic groups satisfy property RD.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

Lecture 2

Kontsevich’s oriented graphs with decorated edges and outgoing edge ordering at vertices. The Poisson complex, the Poisson cohomology.

 * Universal (with respect to all affine Poisson manifolds) symmetries of the brackets.

 * The Leibniz graphs. Examples of factorizations via the Leibniz graphs.

A classical principle in deformation theory asserts that any formal deformation problem over a field of characteristic zero is classified by a differential graded Lie algebra. Using the Koszul duality between Lie algebras and commutative algebras, Lurie and Pridham have given a more precise description of this principle: they establish an equivalence of categories between dg-Lie algebras and formal moduli problems indexed by Artin commutative dg-algebras. I will describe a variant of this result for deformation problems around schemes over a field of characteristic zero. In this case, there is an equivalence between the homotopy categories of dg-Lie algebroids and formal moduli problems on a derived scheme. This can be viewed as a derived version of the relation between Lie algebroids and formal groupoids.

For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.

The talk is dedicated to flag manifold sigma-models, which are theories of generalized harmonic maps from a Riemann surface to a manifold of flags in $C^N$. These theories feature interesting geometric properties and are in certain cases examples of the so-called ‘integrable’ models. I will review some of these facts.

In this talk we will discuss the Besson-Courtois-Gallot (BCG) theorem in the context of convex projective geometry. The BCG theorem is a rigidity statement relating the volume and entropy of a negatively curved Riemannian manifold, and has many applications including Mostow rigidity. In the world of convex real projective structures, the natural Hilbert geometry on these objects is only Finsler and the geometry is generally not even $C^2$. We discuss our analogous BCG theorem and some applications in the case where the manifold is closed. We will include some ongoing work to extend the result to finite volume. This is based on joint work with Ilesanmi Adeboye and David Constantine.

Séminaire Laurent Schwartz — EDP et applications

Dans le cas de coefficients indépendants du temps, le propagateur de Feynman peut être défini comme une valeur au
bord de la résolvante de l’opérateur d’onde (interprété comme un opérateur auto-adjoint). Un résultat célèbre de Duistermaat et
Hörmander en donne une caractérisation microlocale ainsi qu’une paramétrix qui se généralisent bien au cas des opérateurs de
type principal réel. Toutefois, la question d’existence d’un inverse canonique est longtemps restée ouverte. Le but de cet exposé
sera de présenter quelques résultats récents sur l’inversibilité de l’opérateur de Klein-Gordon sur des espaces-temps asymptotiquement
plats et d’expliquer comment interpréter l’inverse en termes de conditions aux limites globales à l’infini ainsi que dans un
langage plus « spectral». Quelques applications en théorie quantique des champs seront présentées. (travaux en collaboration avec
Christian Gérard et András Vasy)

Séminaire Laurent Schwartz — EDP et applications

Je discuterai d’un résultat récent obtenu avec Joackim Bernier et Benoît Grébert concernant la stabilité de petites solutions
régulières à l’équation de Schrödinger non linéaire (NLS) en dimension 1 sur un tore. Il s’agira de montrer un résultat du
type: « le plus souvent, les solutions de NLS sont stables sur des temps très longs » en précisant l’énoncé et les techniques utilisées.
Celles ci sont basées sur un nouveau type de formes normales dites rationnelles qui permettent de conjuguer le flot de NLS à
une dynamique intégrable sur des ensembles dont on mesure la taille par calculs probabilistes. La possible existence de contre
exemples et l’extension à d’autres situations, seront aussi évoquées et discutées.

Weight structures give certain filtrations of triangulated categories; the definition is a certain cousin of that of  t-structures. Particular cases of weight decompositions give stupid filtrations of complexes and cellular towers for spectra.  I will also mention interesting motivic and Hodge-theoretic examples of weight structures along with methods for constructing them. Weight structures yield weight filtrations and spectral sequences for cohomology as well as certain weight complex functors. In particular,  there exists an exact conservative functor from (relative) constructible Voevodsky motives into complexes of Chow motives, whereas the corresponding weight spectral sequences vastly generalize Deligne’s ones. Lastly, the relations between weight structures and t-structures yield new methods for constructing t-structures and proving their properties.