SÉMINAIRE DE GÉOMÉTRIE SOUS-RIEMANNIENNE - INSTITUT HENRI POINCARÉ, PARIS - 2015/16

The "séminaire de géométrie sous-riemannienne" is a periodic seminar held in Paris since 2011, whose aim is to help connections between the different communities working in sub-Riemannian geometry from different viewpoints.

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

Starting from October 2015, it will be hosted as a part of the activities of the ANR project SRGI - Sub-Riemannian Geometry and Interactions.

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OTHER INFORMATIONS

The "séminaire de géométrie sous-riemannienne" is a periodic seminar held in Paris since 2011, whose aim is to help connections between the different communities working in sub-Riemannian geometry from different viewpoints.

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

Starting from October 2015, it will be hosted as a part of the activities of the ANR project SRGI - Sub-Riemannian Geometry and Interactions.

- Frequence: October 2015 - June 2016, one session per month.
- Topics: sub-Riemannian geometry, hypoelliptic operators and related fields.
- Some notes of the seminars of the past years are available on this blog.
- Organizers: Davide Barilari, Pierre Pansu.
- Fall 2015 : 21 Oct - 18 Nov - 9 Dec.
- Spring 2016 : Feb 3 - Feb 17 - Mar 16 - May 4 - Jun 15.

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*NEXT SEMINARS*

*Wednesday, June 15, 2016. - Salle 01 IHP*

- 15.00 - 16.00
**Mario Sigalotti**(INRIA, CMAP Ecole Polytechnique) -*The Whitney Extension Theorem for Curves in Carnot Groups*

*Abstract:*We study the validity of the C^1 Whitney extension theorem for maps from a closed subset of the real line into a Carnot group. We characterize Carnot groups for which such a Whitney Extension Theorem holds, in terms of a notion which is similar to the non-rigidity of curves defined by Bryant and Hsu. We provide several examples of Carnot groups for which the Whitney Extension Theorem does or does not hold. The talk is based on a joint work with Nicolas Juillet.

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*PAST SEMINARS*

*Wednesday, May 4, 2016. - Salle 01 IHP*

- 15.00 - 16.00
**Antonio Lerario**(SISSA, Trieste) -*The homotopy of singular spaces and a subriemannian Minimax principle*

*Abstract:*Given a function J on a topological space X (e.g. the Energy functional on the space of paths between two points on a manifold), the classical calculus of variations allows to predict the existence of critical points of J (i.e. geodesics) using topological methods (for example, Morse theory tells that geodesics are exactly the obstruction for the deformation of one Lebesgue set to another). In the subriemannian framework the space X under consideration (the H^1 space of horizontal curves between two points) can be very singular, due to the possible existence of the so-called abnormal curves. In this talk I will discuss recent results introducing some variational techniques on this singular space (for example, I will present a subriemannian version of the classical Minimax principle). (This is joint work with A. A. Agrachev and F. Boarotto)

*Wednesday, March 16, 2016. - Salle 01 IHP*

- 15.00 - 16.00
**Davide Barilari**(IMJ-PRG, Paris Diderot) -*Volume geodesic distortion and Ricci curvature in sub-Riemannian geometry*

*Abstract:*We study the variation of a smooth volume form under the sub-Riemannian exponential map. We introduce a new invariant describing the interaction between the volume and the dynamic and we show how this invariant, together with curvature-like invariants of the sub-Riemannian structure, appear in the expansion of the volume. This generalizes the well-known expansion of the Riemannian volume in terms of the Ricci curvature. We discuss possible conjectures/applications to measure contraction properties and small time heak kernel expansion. [Joint with A. Agrachev and E. Paoli]

*Wednesday, February 17, 2016. - Salle 01 IHP*

- 15.00 - 15.50
**Yacine Chitour**(Orsay) -*Horizontal holonomy**Abstract:*In this talk, we present the concept of horizontal holonomy (HH) which is associated with a triple $(M,\Delta,\nabla)$, where $M$ is a smooth manifold equipped with a linear connection $\nabla$ and $\Delta$ is a smooth completetely controllable distribution on $M$. Horizontal holonomy $H^{\Delta}^{\nabla}$ of $(M,\Delta,\nabla)$ refers to the $\nabla$-parallel transport along loops of $M$ tangent to $\Delta$. We prove that $H^{\Delta}^{\nabla}$ is a Lie subgroup of the holonomy $H^{\nabla}$ of $(M,\nabla)$ and we provide Ambrose-Singer and Ozeki type of results around regular points of $\Delta$. These results are obtained by defining adapted distributional differential derivatives throguh the concept of selector. We finally present several open question relative to the selectors and to the possible links between horizontal holonomy and sub-riemannian geometry. Joint work with E. Grong, F. Jean and P. Kokkonen.

- 16.00 - 16.50
**Erlend Grong**(University of Luxembourg) -*Asymptotic expansion of holonomy on metric spaces.*

*Abstract:*Let P be a principal bundle over a metric space M. Relative to the metric on M and a chosen connection on P, we want to look at the first terms in an asymptotic expansion of the holonomy of short loops. We are considering the case when M is either a Riemannian or a sub-Riemannian manifolds, while the principal bundle is arbitrary.

*Wednesday, February 3, 2016. - Salle 01 IHP*

- 15.00 - 16.00
**Mauricio Godoy Molina**(Universidad de la Frontera) -*Tanaka prolongation of pseudo H-type algebras*

*Abstract:*The old problem of describing infinitesimal symmetries of distributions still presents many interesting questions. A way of encoding these symmetries for the special situation of graded nilpotent Lie algebras was developed by N. Tanaka in the 70's. This technique consists of extending or "prolonging" the algebra to one containing the original algebra in a natural manner, but no longer nilpotent. When this prolongation is of finite dimension, then the algebra we started with is called "rigid", and otherwise it is said to be of "infinite type". The goal of this talk is to extend a result by Ottazzi and Warhurst in 2011, to show that a certain class of 2-step nilpotent Lie algebras (the pseudo $H$-type algebras) are rigid if and only if their center has dimension greater or equal than three. This is a joint work with B. Kruglikov (Tromsø), I. Markina and A. Vasiliev (Bergen)

*Wednesday, December 9, 2015. - Salle 01 IHP*

*Journée "Propriétés spectrales des opérateurs hypoelliptiques"*

*[Poster]*

- 10.30 - 11.30
**Asma Hassannezhad**(Bonn) -*Bounds on eigenvalues of the sub-Laplacian*

*Abstract:*We study eigenvalue problems on regular sub-Riemannian manifolds. Many known examples such as the Heisenberg group, CR, Sasakian and contact Manifolds can be considered as regular sub-Riemannian manifolds. There is a natural hypoelliptic operator on sub-Riemannian manifolds called the sub-Laplacian. We obtain upper bounds for its eigenvalues which are asymptotically optimal. These bounds can be compared to classical results by Korevaar and Buser in Riemannian geometry. We will see examples on which upper bounds are independent of the geometry of sub-Riemannian manifolds. This talk is joint work with Gerasim Kokarev.

- 11.30 - 12.30
**Michel Bonnefont**(Bordeaux) -*Measure doubling property under a curvature-dimension criterion in sub-elliptic geometry*

*Abstract:*In this talk, I will present a curvature-dimension criterion in sub-elliptic geometry. It was introduced and studied by F. Baudoin and N. Garofalo. It generalizes the one of Bakry-Emery in Riemannian geometry. In the non-negative curvature case, using heat semi-groups methods, we show that the measure doubling property holds. We also set gaussian estimates for the heat kernel and the Poincaré inequality on balls. This is a joint work with F. Baudoin and N. Garofalo.

- 14.15 - 15.15
**Luca Rizzi**(CMAP, Ecole Polytechnique) -*A sub-Riemannian Santalò formula with applications to isoperimetric inequality and spectral gap of hypoelliptic operators*

*Abstract:*We prove a sub-Riemannian version of the classical Santalo' formula: a result in integral geometry that describes the intrinsic Liouville measure on the unit cotangent bundle in terms of the geodesic flow. As an application, we prove a universal (i.e. curvature independent) lower bound for the first Dirichlet eigenvalue on a compact domain M with Lipschitz boundary. All our results are sharp for the sub-Riemannian structures on the hemispheres of the complex and quaternionic Hopf fibrations, where the sub-Laplacian is the standard hypoelliptic operator of CR and quaternionic contact geometries. Further applications include (p)-Hardy-type and isoperimetric-type inequalities.

*Wednesday, November 18, 2015. - Salle 314 IHP*

- 15.00 - 16.00
**Isabelle Gallagher**(IMJ-PRG, Paris Diderot) -*Estimations de dispersion pour l'opérateur de Schrödinger sur des groupes de Lie stratifiés*

*Wednesday, October 21, 2015. - Salle 314 IHP*

- 15.00 - 15.50
**Ugo Boscain**(CMAP, Ecole Polytechnique) -*Random walks in sub-Riemannian geometry via volume sampling*

*Abstract: We wish to relate some basic constructions of stochastic analysis to differential geometry, via random walk approximations. The motivation is to explore how one can pass from geodesics to diffusion on sub-Riemannian manifolds, where geodesics are relatively well understood, while there is no completely canonical notion of sub-Laplacian on a general sub-Riemannian manifold. However, even in the Riemannian case, this random walk approach illuminates the geometric significance of Ito and Stratonovich stochastic differential equations.*

- 16.00 - 16.50
**Emmanuel Trélat**(LJLL, UPMC) -*Asymptotique spectrale pour des Laplaciens SR: mesure de Weyl dans des cas équiréguliers ou singuliers.*

OTHER INFORMATIONS

- Links to the wepage of the last years:

Séminaire de Géométrie Sous-Riemannienne - 2014/2015

Séminaire de Géométrie Sous-Riemannienne - 2013/2014

Séminaire de Géométrie Sous-Riemannienne - 2012/2013

Séminaire de Géométrie Sous-Riemannienne - 2011/2012 - (organized by Enrico Le Donne) - Blog of the "Geometry and Control" seminar in SISSA