15 novembre 2017

Amine Marrakchi (LMO - Equipe TopoDyn)
Espaces mesurés non commutatifs

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Lieu : Petit amphi du bâtiment 425

Résumé : Espaces mesurés non-commutatifs
La théorie des algèbres de von Neumann fut fondée en 1930 par John von Neumann qui cherchait à formaliser mathématiquement la toute jeune théorie de la mécanique quantique. Son idée est de remplacer l’algèbre des observables sur un espace classique, qui est nécessairement commutative, par une algèbre d’opérateurs possédant des propriétés très similaires mais non nécessairement commutative. Pour le physicien, la non-commutativité encode alors les propriétés quantiques du système décrit par l’algèbre. Mais pour le mathématicien, elle est aussi la source de quantité de phénomènes surprenants et de liens forts et féconds avec la théorie ergodique. Je donnerai un petit aperçu historique de cette théorie et présenterai quelques résultats récents.
Noncommutative measure theory
Von Neumann algebra were introduced in 1930 by John von Neumann in order to give a mathematical formulation to the young theory of quantum mechanics. His original idea was to replace the algebra of observables on a classical space, which is necessarily commutative, by a noncommutative operator algebra with similar properties. In the physicist’s viewpoint, noncommutativity encodes quantum properties, while for the mathematician it is a source for numerous inspiring phenomena and is deeply linked with ergodic theory. I will give a historical account and present some recent results of this theory.

Espaces mesurés non commutatifs  Version PDF

Luis Garcia (IHES)
Superconnections and special cycles on Shimura varieties and generalizations

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Lieu : Salle 113-115

Résumé : Shimura varieties are of central importance to number theory and their geometry and arithmetic is a topic of intensive research. They come with a very rich collection of subvarieties known as special cycles. In the 1980’s Kudla and Millson gave very general theorems describing the span of these special cycles in cohomology in terms of automorphic forms. Their proofs involve producing explicit differential forms that are Poincare dual to the special cycles. I will describe a different approach to defining these differential forms that uses the formalism of superconnections and takes advantage of theorems proved by Bismut, Gillet and Soule. This allows for simplified proofs of several of the main theorems of Kudla-Millson and also to extend this construction to arithmetic quotients of period domains. I will also discuss work in progress on other applications of this construction, such as constructing Green forms for these special subvarieties (joint work with S. Sankaran) or extending Kudla-Millson modularity theorems to certain variations of Hodge structure not parametrized by Shimura varieties.
The talk will start with a general introduction to special cycles and no number theory background will be assumed.

Superconnections and special cycles on Shimura varieties and generalizations  Version PDF

Jan Dereziński (Warsaw university)
Almost homogeneous Schroedinger operator

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Lieu : salle 228, bâtiment 440

Résumé : 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 1-dimensional Schroedinger operators with a 1/x^2 potential with 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 induces a simple but nontrivial flow 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 Poincare 12 (2011), 547-590
J.D., Serge Richard : On Schrödinger operators with inverse square potentials on the half-line, Annales Henri Poincare 18 (2017) 869-928
J.D. : Homogeneous rank one perturbations, to appear in Annales Henri Poincare

Almost homogeneous Schroedinger operator  Version PDF