Frédéric Hélein

Séminaire de géométrie et physique mathématique


organisé par Serguei Barannikov, Daniel Bennequin, Christian Brouder,
Frédéric Hélein et Volodya Roubtsov



Bâtiment Sophie Germain, Paris 13ème
(voir le plan d'accès)

Salle 1016
Année 2017-2018



Vendredi 15 juin 2018, 14 h
salle 1016 :

François Gay-Balmaz

Ecole Normale Supérieure

Variational principles and Dirac structures in nonequilibrium thermodynamics

Résumé : We present a Lagrangian variational formulation for nonequilibrium thermodynamics. This formulation extends the Hamilton principle of classical mechanics to include irreversible processes in both discrete and continuum systems. The irreversibility is encoded into a nonlinear nonholonomic constraint given by the expression of entropy production associated to the irreversible processes involved. The introduction of the concept of thermodynamic displacement allows the definition of a corresponding variational constraint. We show that the evolution equations for nonequilibrium thermodynamics admit an intrinsic formulation in terms of Dirac structures. Finally, we illustrate our theory with both finite and infinite dimensional examples, including mechanical systems with friction, chemical reactions, electric circuits, reacting fluids, and moist atmospheric modelling.
This is a joint work with Hiroaki Yoshimura.

Prochaines séances :

Séances précédentes :

Vendredi 25 mai 2018, 14 h
salle 1016 :

Alice-Barbara Tumpach

Université de Lille 1

Bruhat-Poisson structure of the restricted Grassmannian and the KdV hierarchy

Résumé : In the first part of this talk, we explore the notion of Poisson structure in the Banach context. We show that the Leibniz rule for a Poisson bracket on a Banach manifold does not imply the existence of a Poisson tensor and we give examples of Poisson brackets (called queer) without Poisson tensor. The existence of a Hamiltonian vector field on a Banach manifold endowed with a Poisson tensor is also not guaranteed. We propose a definition of a (generalized) Poisson structure on a Banach manifold which includes all weak symplectic Banach manifolds.

The second part of the talk is devoted to the study of particular examples of generalized Poisson manifold, namely Banach Poisson-Lie groups related to the Korteweg-de-Vries hierarchy. We construct a generalized Banach Poisson-Lie group structure on the unitary restricted group, as well as on a Banach Lie group consisting of (a class of) upper triangular bounded operators. We show that the restricted Grassmannian inherite a Bruhat-Poisson structure from the unitary restricted group, and that the action of the triangular Banach Lie group on it by ``dressing transformations'' is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian.

Vendredi 18 mai 2018, 14 h 30

Attention, la séance commencera 30 minutes après l'horaire habituelle !
(cela afin de permettre à ceux qui le désirent de suivre le groupe de travail sur Batalin-Vilkovisky qui aura lieu à l'IHP le matin à 11 h)

Salle 1016 :

Marie Kerjean

IRIF, Université Paris Diderot

Smooth and classical models of Linear Logic

Résumé : Linear logic (LL) is a refinement of Intuitionstic Logic involving an involutive linear negation. While proofs of Linear Logic are traditionally interpreted by discrete models inherited from domains, the development of Differential Linear Logic asks for more elaborate models involving smooth functions. In a joint work with Y. Dabrowski, we constructed several smooth models of LL, therefore developing a theory of smooth functions on (some notion of) reflexive spaces. In particular, we isolate a completeness condition, called k-quasi-completeness, and an associated notion stable by duality called k-reflexivity, allowing for a * autonomous product. We adapt Meise's definition of Smooth maps into a first model of Differential Linear Logic, made of k-reflexive spaces. We also build two new models of LL with conveniently smooth maps, on categories made respectively of Mackey-complete Schwartz spaces and Mackey-complete Nuclear Spaces (with extra reflexivity conditions).

I will start this talk by an introduction to Linear Logic and its semantics. I will in particular draw links with the semantics of Quantum mechanics, following work by John Baez. I will then illustrate the semantics of Linear Logic by detailing the models we constructed with Y. Dabrowski.

Vendredi 13 avril 2018, 14 h
salle 1016 :

Jérémy Attard

CPT, Université d'Aix-Marseille

Tractor and Twistor spaces and connection from conformal Cartan geometry

Résumé : Tractors and twistors are closely related geometric structures and naturally provide conformally covariant calculus on (pseudo-)Riemannian manifolds. These structures are usually constructed by a bottom-up procedure from prolongations of defining equations. We propose here a top-down construction, starting from conformal Cartan bundle and connection, then applying to this framework the dressing field method of gauge symmetry reduction. This construction allows to recover the same objects as in the usual one, but with a deeper insight into their geometric nature. In particular, tractors and twistors and their associated connections exhibit a behaviour of a non standard kind with respect to the (residual) Weyl transformation, which means that they implement the gauge principle but are of a different geometric nature than the usual differential objects.

Vendredi 15 décembre 2017, 14 h
salle 1016 :

Jordan François

Université de Lorraine

Tracteurs conformes, une nouvelle approche

Résumé : Le fibré des tracteurs muni de la connexion tracteur est aux variétés conformes ce que le fibré tangent muni de la connexion de Levi-Civita est aux variétés riemanniennes. Habituellement il est construit par prolongation d'une équation différentielle nommée "Almost Einstien", ce qui réclame des calculs patients. Nous proposerons une approche constructive alternative et économique apportant de nouveaux résultats.

Vendredi 24 novembre 2017, 14 h
salle 1016 :

Alberto Cattaneo

Université de Zürich

Geometrical construction of reduced phase spaces

Résumé : The reduced phase space of a field theory is the space of its possible initial conditions endowed with a natural symplectic structure. An alternative to Dirac’s method, relying on natural geometric aspects of variational problems, was introduced by Kijowski and Tulczijev. This method also has the advantage of having a natural generalization in the BV context. In this talk, I will explain the method and describe some examples, focusing in particular on the tetradic version of general relativity in four dimensions.

Vendredi 10 novembre 2017, 14 h
salle 1016 :

Andrzej Zuk

IMJ-PRG et Université Paris 7 - Paris Diderot

Groupes associés à l'équation KdV

Vendredi 20 octobre 2017, 14 h
salle 1016 :

Serguei Barannikov

IMJ-PRG et Université Paris 7 - Paris Diderot

Quantum master equation on cyclic cochains, Feynman transforms and cohomological field theories.

Résumé : The purpose of the talk is description of the constructions of cohomology classes of the compactified moduli spaces of curves via summation over stable ribbon graphs, based on the speaker's earlier work "Modular operads and Batalin-Vilkovisky geometry" (IMRN, vol 2007). The constructions are related with solutions to the quantum master equation on cyclic cochains, and with A-infinity algebras with scalar product. The constructions are based on the identification of the cell decomposition of the compactified moduli spaces of curves with the Feynman transform of modular operad of symmetric groups. The Feynman transform controls various properties of the constructed cohomology classes.



Année précédente (2016-2017)
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