Soit $F(X,Y)$ une forme binaire à coefficients entiers, de discriminant non nul, de degré $\geq 3$.
A quelle condition, nécessaire et suffisante, existe-t-il une forme $G (X,Y)$, non $GL(2, Z)$-équivalente à $F(X,Y)$, telle qu’on ait l’égalité des images $F(Z^2) = G(Z^2)$ ?
La condition trouvée repose sur l’existence d’un élément d’ordre $3$, d’un certain type, dans le groupe d’automorphismes de $F$.
Travail en commun avec Peter Koymans.

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Résumé

Brascamp–Lieb inequalities are multilinear integral inequalities, classically set in Euclidean spaces, in which the functions involved possess symmetries that can be described via annihilation by translation invariant vector fields. With this point of view in mind, in this seminar I will present a general strategy to obtain inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. The proof relies on a monotonicity property of the heat flow. In particular, I will focus on the case of real spheres, where the obtained inequalities are seen to be sharp for certain choices of symmetries.

Le but de cet exposé est d’introduire une classe indice pour les opérateurs longitudinaux G-transversalement elliptiques. Je commencerai par rappeler la définition de l’indice d’un opérateur elliptique G-invariant sur une variété compacte et le théorème de l’indice d’Atiyah-Singer. Ensuite j’introduirai la définition de la classe indice pour un opérateur G-transversalement elliptique introduite par Atiyah et celle pour les familles d’opérateurs G-transversalement elliptiques introduite dans ma thèse. Je discuterai dans le même temps les différents axiomes vérifiés par ces classes indices. Je terminerai avec les derniers résultats obtenus en collaboration avec Moulay Benameur, dans le cadre des feuilletages.

Quantum groups have been defined in the analytical/topological setting by Woronowicz in the 1980s. In this talk, we will give a brief introduction to these objects and see how they can be used for describing quantum symmetries. We will discuss some examples arising from so called « easy » quantum groups or quantum automorphism groups of finite graphs. Some ongoing research project on de Finetti type theorems with Isabelle Baraquin (Metz) will be mentioned as well.

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In this talk I will present the problem of finding extremal potentials for the functional determinant of a one-dimensional Schrödinger operator defined on a bounded interval with Dirichlet boundary conditions. We consider potentials in a fixed $L^q$ space with $qgeq 1$. Functional determinants of Sturm-Liouville operators with smooth potentials or with potentials with prescribed singularities have been widely studied, I will present a short review of these results and will explain how to extend the definition of the functional determinant to potentials in $L^q$. The maximization problem turns out to be equivalent to a problem in optimal control. I will explain how we obtain existence and uniqueness of the maximizers. The results presented in the talk are join work with J-B. Caillau (UCDA, CNRS, Inria, LJAD) and P. Freitas (Lisboa).

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Résumé

For an arbitrary morphism of (super)manifolds, the pull-back is a linear map of the space of functions. In 2014 Th.Voronov have introduced thick morphisms of (super)manifolds which define a generally non-linear pull-back of functions. This construction was introduced as an adequate tool to describe L-infinity morphisms of algebras of functions provided with the structure of a homotopy Poisson algebra. It turns out that if you go down from `heaven to earth’, and consider usual (not super!) manifolds, then we come to constructions which have natural interpretation in classical and quantum mechanics. In particular in this case the geometrical object which defines the thick diffeomorphism becomes an action of classsical mecanics, and pull-back of the thick diffeomorphism with a quadratic action give a spinor representation. I will define a thick morphism and will tell shortly about their application in homotopy Poisson algebras. Then I will discuss the relation of thick morphisms with notions such as the action in classical mechanics and spinors. The talk is based on the work: « Thick morphisms of supermanifolds, quantum mechanics and spinor representation’, J.Geom. and Phys., 2019, article Number: 103540, https://doi.org/10.1016/ j.geomphys.2019.103540, arXiv:1909.00290 Authors: Hovhannes Khudaverdian, Theodore Voronov