Catégorie : Petit amphithéâtre MIM, technopôle Metz

Pseudodifferential operators of Toeplitz type

Joerg Seiler (Turin)

 

There are many examples of calculi/algebras of pseudodifferential operators that have been designed to analyze different sorts of elliptic partial differential operators, in particular to characterize their Fredholm property and regularity properties of solutions of associated pde’s in suitable function spaces, using a parametrix construction within the algebra. This ranges from  pseudodifferential operators on smooth closed manifolds (where ellipticity of an operator is characterized by the invertibility of its homogeneous principal symbol) to operator algebras for singular manifolds like manifolds with conical points, edges, and corners (where ellipticity is characterized by a hierarchy of principal symbols associated with the stratification of the manifold). Also boundary value problems can be treated in such a way. L. Boutet de Monvel developed a calculus for smooth manifolds with boundary which allows to treat classical boundary conditions like Dirichlet or Neumann conditions. Ellipticity in this calculus corresponds to the classical Shapiro-Lopatinskij ellipticity. This calculus has been extended by Schulze to also cover so-called global projection conditions, for example spectral boundary conditions for Dirac operators.

It will be discussed how parts of Schulze’s construction can be obtained in a general framework of  so-called operators of Toeplitz type associated with a given algebra of pseudodifferential operators and that a corresponding approach also applies to complexes of operators. Fredholm property in this context means finite dimension of all associated cohomology spaces. For smooth manifolds with boundary it turns out that every complex of differential operators, which is fibre-wise exact on the level of homogeneous principal symbols, can be complemented with boundary conditions (i.e., a complex-isomorphism to a complex of operators on the boundary) in such a way that the resulting mapping cone is a Fredholm complex. There is a topological obstruction which decides whether these boundary conditions can be chosen from the usual Boutet de Monvel calculus or when they must involve global projection conditions. This extends and makes precise results due to A. Dynin. Parts of this talk are joint work with B.-W. Schulze.

Orbit method and unipotent representations

Chengbo Zhu (National University of Singapore)

 

matzhucb

 

Abstract: A fundamental problem in representation theory is to determine the unitary dual of a given Lie group G, namely the set of equivalent classes of irreducible unitary representations of G. A principal idea, originated in a famous paper of A. A. Kirillov in 1962, is that there is a close connection between irreducible unitary representations of G and the orbits of G on the dual of its Lie algebra. This is known as the orbit method (or the philosophy of coadjoint orbits).

In this talk, I will describe basic ideas of the orbit method as well as a recent development on the problem of unipotent representations, which is to associate unitary representations to nilpotent coadjoint orbits and which is the hardest part of the orbit method. We solve this problem for real classical groups, by profitably combining analytic ideas of R. Howe on theta lifting and algebro-geometric ideas of D. A. Vogan, Jr. on associate varieties. This is joint work with J.-J. Ma and B. Sun.

The talk is aimed at a general audience of mathematicians and graduate students.

Almost homogeneous Schrödinger operators

Jan Derezinsky

(Université de Varsovie)

Abstract: First I will describe a certain natural holomorphic family of closed operators with interesting spectral properties. These operators can be fully analyzed using just trigonometric functions. Then I will discuss one- dimensional Schrödinger operators with inverse square potential and general boundary conditions, which I studied recently with S.Richard. Even though their description involves Bessel and Gamma functions, they turn out to be equivalent to the previous family.

Some operators that I will describe are homogeneous – they get multiplied by a constant after a change of the scale. In general, their homogeneity is weakly broken-scaling and induces a simple but nontrivial ow in the parameter space. One can say (with some exaggeration) that they can be viewed as « toy models of the renormalization group ».

Based on

• J.D. Laurent Bruneau and Vladimir Georgescu: Homogeneous Schrödinger operators on half-line, Annales Henri Poincaré 12 (2011), 547-590 ;

• J.D., Serge Richard: On Schrödinger operators with inverse square potentials on the half-line, Annales Henri Poincaré 18 (2017) 869-928;

• J.D.: Homogeneous rank one perturbations, to appear in Annales Henri Poincaré

Inégalités de Strichartz

Gilles Lebeau (Université de Nice)

505px-Gilles_Lebeau

 

Résumé de l’exposé. Dans l’article « Restriction of Fourier Transform to Quadratic Surfaces and Decay of Solutions of Wave Equations. Duke Math. Journal, 44, 1977 », R. Strichartz a introduit les inégalités qui portent son nom, pour résoudre certaines équations d’ondes non linéaires. Elles sont devenues un outil fondamental pour l’étude du problème de Cauchy pour les équations d’évolutions dispersives non linéaires (ondes, Schrödinger,…) et en analyse harmonique pour l’étude des estimations Lp des projecteurs spectraux. Nous présenterons ces inégalités, ainsi que des résultats récents (en collaboration avec R. Lascar, O. Ivanovici et F. Planchon) dans des domaines bornés, et certains problèmes ouverts.

De l’approche à l’équilibre thermodynamique : quels mécanismes dynamiques ?

Stéphane de Bièvre
(Université de Lille)

IMG_2705

Que les systèmes macroscopiques isolés tendent vers un état d’équilibre thermodynamique est une loi de base de la thermodynamique. Expliquer comment et pourquoi ceci se passe en termes de la dynamique sous-jacente des constituents de ces systèmes reste un problème difficile et largement ouvert et activement étudié. Après avoir posé le problème, je passerai en revue quelques résultats récents sur des systèmes modèle simples.

Institut Elie Cartan de Lorraine