Institut Henri Poincaré, Paris - 2013/14

The seminar takes place at the Institut Henri Poincaré, 11, rue Pierre et Marie Curie, Paris.


Wednesday, May 14, 2014. (Salle 421)

  • 15.30 - 16.30. Jean-Baptiste Caillau (Univ. Bourgogne) - Approximation of controlled mechanical systems by Riemannian metrics and generalizations.

    Abstract: Given periodic controlled vector fields and some cost, the long-time asymptotics of the minimizing curves are considered using averaging. For the Newtonian potential (two-body problem) and the $L^2$ cost, a metric on the space of ellipses is obtained; depending on whether one or two controls are used, the metric is either Riemannian or almost-Riemannian. For minimum time, a Finsler metric is obtained. Joint work with B. Bonnard (Dijon) and J.-B. Pomet (Nice).

  • 16.40 - 17.40. Nicola Gigli (IMJ, UPMC) - A flow tangent to the Ricci flow via heat kernel and mass transport

    Abstract: I will discuss a relation between the short time behavior of the heat flow, the geometry of optimal transport and the Ricci flow. I will also point out how this relation can be used to define an evolution of metrics on non-smooth metric measure spaces with Ricci curvature bounded from below. Several basic questions about this evolution are still open, and I will present the most significant ones. The content of the talk comes from a collaboration with C. Mantegazza.


Wednesday, April 9, 2014. (Salle 01)

  • 15.30 - 16.30. Frédéric Jean (ENSTA ParisTech) - Volumes de Hausdorff en géométrie sous-riemannienne

    Abstract: Dans un espace métrique, on définit le volume de Hausdorff d'un ensemble E comme sa mesure de Hausdorff de dimension D , où D est la dimension de Hausdorff de E. Le but de cet exposé est de présenter les résultats connus sur les volumes de Hausdorff dans une variété sous-riemannienne, où la dimension de Hausdorff diffère de la dimension topologique. Je rappellerai d'abord les premiers résultats de Mitchell sur la dimension de Hausdorff et ceux plus récents de Agrachev, Boscain et Barilari sur la régularité des volume de Hausdorff dans le cas équirégulier. J'exposerai ensuite les résultats que j'ai obtenus avec Roberta Ghezzi sur la décomposition de Radon-Nykodim du volume de Hausdorff par rapport à un volume lisse dans des variétés non équirégulières.

  • 16.40 - 17.40. Frédéric Paulin (Univ. Paris Sud) - Comptage et équidistribution dans le groupe de Heisenberg

    Abstract: Nous montrons un résultat d'équidistribution dans le groupe de Heisenberg des points rationnels sur un corps quadratique imaginaire, ainsi qu'un résultat de comptage de chaines de Cartan arithmétiques dans le groupe de Heisenberg quand leur diamètre de Cygan (ou de Koranyi) tend vers zéro. Nous développons pour cela la relation entre la géométrie hyperbolique complexe et les applications arithmétiques de comptage et d'équidistribution, qui proviennent des actions de groupes arithmétiques sur les espaces hyperboliques complexes, en utilisant la géométrie sous-riemannienne de leur sphere à l'infini. (En commun avec Jouni Parkkonen)

Wednesday, February 19, 2014. (Salle 421)

  • 15.30 - 16.30. François Vigneron (Univ. Paris Est) - Analyse Multifractale sur le Groupe de Heisenberg.

    Abstract: L'analyse multifractale consiste à étudier la taille (au sens de Hausdorff) des ensembles de points où une fonction possède une régularité ponctuelle prescrite. On réalise cette tache dans le contexte géométrique anisotrope du groupe de Heisenberg en utilisant des ondelettes. On obtient ainsi des bornes supérieures sur le spectre multifractal des fonctions dans les espaces de Holder et de Besov et on vérifie leur caractère génériquement optimal.

  • 16.30 - 17.30. Zoltan Balogh (Bern University) - Horizontal convexity in the Heisenberg group

    Abstract: In this talk I will discuss aspects of convexity in the sub-Riemannian context of Heisenberg groups. Various aspects of horizontal convexity will be compared to the corresponding notions of Eulidean convexity. Aleksandrov-type comparison and maximum principles will be proven for continuous functions on Heisenberg groups. This is a joint work with Andrea Calogero and Alexandru Kristaly.

Wednesday, January 22, 2014. (Salle 421)

  • 15.30 - 16.30. Antonio Lerario (Univ. Lyon 1) - The topology of loop spaces in Carnot groups.

    Abstract: Carnot groups are tangent spaces to subriemannian manifolds. The horizontal loop space P(x,y) (i.e. the space of admissible paths joining two points x and y on a step-two Carnot group) has the homotopy type of an infinite dimensional sphere S. I will discuss the Morse-Bott theory of the function giving the "energy" of each path in P(x,y); critical points of this function are admissible geodesics between x and y. As the energy increases, all geodesics are considered and the full path space is obtained. A very special phenomenon occurs when x is the origin and y is a "vertical" point. I will show how to count geodesics in this situation (the number of critical manifolds with energy bounded by s is bounded by a polynomials of degree c in s, where c is the corank of the horizontal distribution). Surprisingly enough, Morse-Bott inequalities are not sharp in this case and the sum of the Betti numbers of the space of paths with energy less than s increases at most as a polynomial of degree c-1. For c=2 these results are exact and the leading order can be analytically computed, using a "topological" coarea formula. (joint work with A. A. Agrachev and A. Gentile)

Wednesday, November 20, 2013. (Salle 05)

  • 15.30 - 16.30. Isidro Munive (SISSA) - The heat equation in sub-Riemannian spaces: Curvature-dimension inequalities, Li-Yau inequalities and volume growth.

    Abstract: In this talk we study parabolic Harnack inequalities and sharp volume and distance estimates in sub-Riemannian manifolds. The approach that we follow is based on a generalization of the curvature-dimension inequality from Riemannian geometry and new entropy functional inequalities for the heat semigroup recently discover by Baudoin and Garofalo.

Wednesday, November 6, 2013. (Salle 421)

  • 15.30 - 16.30. Davide Barilari (IMJ, Université Paris Diderot) - On the heat diffusion for generic Riemannian and sub-Riemannian structures

    Abstract: In this talk I will discuss the small-time heat kernel asymptotics for generic low-dimensional Riemannian and sub-Riemannian manifolds. I will first recall some classical and more recent results about the small time heat kernel asymptotics, both off-diagonal and at the cut locus, showing how the asymptotic of p_t(x,y) behave depending on whether (and how much) y is conjugate to x. Then I will discuss how these techniques let us to identify the possible asymptotics for the heat kernel at the cut locus for a generic Riemannian manifolds of dimension less or equal than 5. This is a consequence of the fact that, among the stable singularities of Lagrangian maps appearing in the classification of Arnold, only two of them (namely A_3 and A_5) can appear as ``optimal'', i.e. along minimizing geodesics. Similar techniques apply to characterize the heat kernel expansion for a 3D contact sub-Riemannian structure, where the cut/conjugate locus of a point is adjacent to the point itself, due to the explicit form of the local singularity of the exponential map described by Agrachev, Gauthier et al.

    The seminar is followed by a short talk:

  • 16.40 - 17.00. Andrei Agrachev (SISSA) - Some models of constant geodesic curvature in sub-Riemannian geometry

Tuesday, October 22, 2013. (Salle 421)

  • 15.30 - 16.30. Stéphane Menozzi (Université de Evry) - Estimées de densités pour certains operateurs strictment hypoelliptiques

    Abstract: Pour un operateur degenere verifiant une condition de Hormander faible nous presenterons des techniques de controle stochastique, transformee de Fleming, permettant d'obtenir des bornes de type Aronson multi-echelles sur la solution fondamentale

  • 16.40 - 17.40. Luca Rizzi (SISSA, Trieste) - Comparison Theorems in sub-Riemannian geometry

    Abstract: The typical Riemannian comparison theorem is a result in which a local curvature-type bound (e.g. Ric > k) implies a global comparison between some property on the actual manifold (e.g. its diameter) and the same property on a constant curvature model. The generalization of these results to the sub-Riemannian setting is not straightforward, the main difficulty being the lack of a proper theory of Jacobi fields, an analytic definition of curvature and, a fortiori, constant curvature models. In this talk, we propose a theory of Jacobi fields valid for any sub-Riemannian manifold, in which the Riemannian sectional curvature is generalized by the curvature introduced by Agrachev and his students. This allows to extend a wide range of comparison theorems to the sub-Riemannian setting. In particular, we discuss the sub-Riemannian Bonnet-Myers theorem and the generalized Measure Contraction Property for sub-Riemannian manifolds with bounded canonical Ricci curvatures. In this setting, the models with constant curvature are represented by Linear-Quadratic optimal control problems with constant potential. (This is a joint project with D. Barilari.)

Wednesday, October 2, 2013. (Salle 421)

  • 15.30 - 16.30. Dario Prandi (CMAP, Ecole Polytechnique) - Complexity of control-affine motion planning

    Abstract: In this talk we study the complexity of the motion planning problem for control-affine systems. Such complexities are already defined and rather well-understood in the particular case of sub-Riemannian (or driftless) systems. Our aim is to generalize these notions and results to systems with a drift. Accordingly, we present various definitions of complexity, as functions of the curve that is approximated, and of the precision of the approximation. Due to the lack of time-rescaling invariance of these systems, we consider geometric and parametrized curves separately. Then, we give some asymptotic estimates for these quantities and discuss the difficulties that arise due to the presence of a drift. This is a joint work with F. Jean.

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