Résumés des exposés
Martin Bordemann
Title: Combinatorics of iterated covariant derivatives for Lie-Rinehart algebras
In differential geometry covariant derivatives, i.e. connections (mostly in the tangent bundle) are ubiquitary, and some of their iterations (e.g. Laplacians, bidifferential star-products on Fedosov manifolds). We give an explicit structure of a Rinehart bialgebra on the free algebra generated by the A-module of a Lie Rinehart algebra equipped with a connection in itself by means of iterated covariant derivatives.
There is a morphism of Lie Rinehart algebras from the module of primitive elements of this free algebra to the initial Lie Rinehart algebra involving a recursively computable sequence of terms involving derivatives of torsion and curvature. We hope (but did not make it yet) to thus get information on the multiplication of the universal enveloping algebra of the initial Lie Rinehart algebra.
Thomas Strobl
Title: Particles moving across singular foliations and a generalization to Strings constrained by Dirac structures
There is a traditional definition of a singular foliation on a manifold M as an appropriate subdivision of M into leaves of possibly different dimensions and an algebraic one in terms of an appropriately defined submodule of vector fields on M, generating such leaves. While the algebraic notion is more restrictive, it also contains more information and is mathematically richer to study. There are likewise two definitions for the case of singular Riemannian foliations (SRFs) when applied to the algebraic setting on a Riemannian manifold (M,g): the traditional one where one requires that geodesics initially orthogonal to a leaf remain orthogonal and an algebraic one introduced recently by Kotov and the speaker. We prove that algebraic SRF implies traditional SRF but that, in general, the opposite direction does not hold true.
We show that the algebraic version of an SRF is precisely the one needed for the movement of point particles on (M,g) whose velocities are constrained to be orthogonal to the leaves of a given singular foliation. We also consider a Dirac geometric generalization of such SRFs (joint work with Severa) which arises when replacing point particles by strings: extended one-dimensional objects moving on a Riemannian manifold (M,g) equipped with a closed 3-form H and constrained by a Dirac structure.
Olivier Peltre
Champs aléatoires de Gibbs et homologie des algorithmes à passage de messages
Le problème de décrire la statistique d'un grand nombre de variables en interaction x1, x2,... est apparu en physique pour donner des fondements microscopiques à la thermodynamique et se présente aujourd'hui dans de nombreux champs d'application. On attend notamment des statistiques en haute dimension de produire des modèles tractables et raisonnables en intelligence artificielle et en biologie: cet exposé cherchera à décrire la continuité que l'on peut retracer de la modélisation des atomes d'un cristal à celle des neurones dans un réseau.
Les phénomènes collectifs ne peuvent pas être calculés sur la variable jointe globale x, mais leur influence sur des sous-ensembles de variables a = {i, j, ...} peut-être estimée et c'est une étape cruciale de l'apprentissage bayésien qui guide la mise à jour des paramètres. S'appuyant sur la structure locale des interactions décrite par un hypergraphe X = {a, b, c, ...}, les algorithmes à passage de messages fournissent une méthode asynchrone et parallélisable pour estimer les distributions marginales p(x_a) pour a dans X de la loi globale p(x). Un champ de Gibbs étant une distribution p(x) qui se factorise comme un produit de fonctions locales f_a(x_a), nous introduisons un complexe de chaînes dont les classes d'homologie décrivent l'ensemble des factorisations équivalentes d'un champ de Gibbs. Le passage de message consiste alors à explorer l'une de ces classes jusqu'à trouver une collection de facteurs (g_a), définissant par produits locaux des croyances (q_a) qui satisfont une condition cohomologique de consistance exprimant que q_b est bien la marginale de q_a dès que b est inclus dans a.
Leonid Ryvkin
Title: The higher holonomies of a singular foliation (j.w. Camille Laurent-Gengoux).
We use the universal Lie $\infty$-algebroid of a singular foliation (recently discovered by Camille Laurent-Gengoux, Sylvain Lavau and Thomas Strobl) to define the homotopy groups of a singular foliation. We introduce the higher holonomies of a singular leaf $L$ of a singular foliation as a sequence of group morphisms from $\pi_n(L)$ to the $\pi_{n-1}$ of the universal Lie $\infty$-algebroid of the transverse foliation of $L$.
Vladimir Salnikov
Title: Graded and generalized geometry for mechanics.
In this contribution we will describe some objects of the generalized geometry that appear naturally in the qualitative analysis of mechanical systems. In particular we will discuss the Dirac structures within the framework of the systems with constraints and eventually of port-Hamiltonian systems. From the mathematical point of view, Dirac structures generalize simultaneously symplectic and Poisson structures. As for mechanics, the idea is to design numerical methods that preserve these structures and thus guarantee good physical behaviour in simulations. Then, we will present a framework which is even more general - the one of differential graded manifolds (also called Q-manifolds), and discuss some possible ways of using them for the ``structure preserving integrators'' in mechanics.
Christian Blohmann
Title: Hamiltonian Lie algebroids
The description of hamiltonian actions in terms of their action Lie groupoids and action Lie algebroids naturally leads to the notion of hamiltonian Lie algebroids, which generalize hamiltonian actions but share some of their good properties. I will motivate this notion, give basic examples, and outline applications to symplectic geometry, Poisson geometry, equivariant cohomology, and classical field theory. This is work in progress with Alan Weinstein.