SÉMINAIRE DE GÉOMÉTRIE SOUS-RIEMANNIENNE - INSTITUT HENRI POINCARÉ, PARIS - 2017/18

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.

Also during 2016/17, it is hosted as a part of the activities of the ANR project SRGI - Sub-Riemannian Geometry and Interactions

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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.

Also during 2016/17, it is hosted as a part of the activities of the ANR project SRGI - Sub-Riemannian Geometry and Interactions

- Frequence: October 2017 - June 2018, one session per month.
- Topics: sub-Riemannian geometry, hypoelliptic operators and related fields.
- Organizers: Davide Barilari, Pierre Pansu.
- Fall 2017 : Oct 4 - Nov 8 - Dec 6 (red text = canceled)
- Spring 2018 : Jan 17 - Feb 14 - Mar 14 - Apr 11 - May 23 - Jun 20
- Some notes of the seminars of the past years are available on this blog.
- The
**room**of the seminars are either**01**(ground floor) or**201**(2nd floor), please check!

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*NEXT SEMINARS*

*Wednesday, June 20, 2018 - Salle 201 IHP*

- 15.00 - 15.40
**Mathieu Kohli**(CMAP) -*Geodesic curvature of horizontal curves in 3D contact sub-Riemannian spaces*

*Abstract:*The distance separating two points of a curve is always smaller than the time it takes to go from one of these points to the other by following the curve at unit speed. The curve is a geodesic if and only if these quantities coincide when the two points are close enough. From the difference between these quantities we can extract a relevant notion of geodesic curvature in 3d contact sub-Riemannian spaces such as the Heisenberg group, SL(2) and SU(2).

- 15.50 - 16.30
**Ludovic Sacchelli**(CMAP) -*Géométrie de la caustique sous-riemannienne des variétés de contact de dimension 5*

*Abstract:*TBA

********************* **********************PAST SEMINARS*

*Wednesday, October 4, 2017. - Salle 201 IHP*- 15.00 - 16.00
**Dario Prandi**(L2S, Supelec) -*Quantum confinement and spectral theory of (sub-)Laplacians*

*Abstract:*

*Wednesday, December 6, 2017. - Salle 01 IHP*- 15.00 - 16.00
**Alexey Remizov**(CNRS, CMAP, Ecole Polytechnique) -*Singularities of geodesic flows on 2-surfaces with pseudo-Riemannian metrics*

Abstract: We present a series of recent results about singularities of geodesic flows on 2-dimensional surfaces with smooth signature varying metric (such metrics are often called pseudo-Riemannian). Generically, a surface with a pseudo-Riemannian metric contains a curve that consists of points where the metric degenerates. These points are singular points of the geodesic flow generated by this metric. The behavior of geodesics at degenerate points differs from those in Riemannian metrics. For instance, geodesics cannot pass through a degenerate point in arbitrary tangent directions, but only in certain directions said to be admissible. (It is worst observing that the same phenomenon appears in singular metrics of different types, for instance, in almost-Riemannian structures.) The number of admissible directions for generic pseudo-Riemannian metrics at almost all degenerate points is 1 or 3 (at some isolated points it could be 2). The main technical tool for this study is the theory of local normal forms for vector fields with non-isolated singular points, a far generalization of the theory started by H. Poincaré and H. Dulac.

*Wednesday, January 17, 2018 - Salle 01 IHP*- 15.00 - 16.00
**Erlend Grong**(Orsay) -*Sub-Laplacians on forms and cohomology.*

Abstract: For any sub-Riemannian structure, there is a unique corresponding second order operator on functions which is symmetric with respect to some given volume form. We will call this operator the sub-Laplacian. We discuss the challenges, results and applications that comes from attempting to generalize this operator to differential forms. We will mainly be concerned with operators which has a Weitzenböck decomposition, that is, can be constructed from the Hessian of an affine connection and a zero order differential operator. As one applications, we get results related to cohomology of totally geodesic foliations.

*Wednesday, February 14, 2018 - Salle 314 IHP*- 15.00 - 16.00
**Davide Barilari**(Paris Diderot) -*What is sub-Riemannian scalar curvature? A result and some open questions**Abstract:*In this talk I will discuss some possible approaches to define a notion of scalar curvature in sub-Riemannian geometry. A natural one is to try to extract this invariant from the asymptotics of the volume of small balls, similarly to what happens in Riemannian geometry. In the sub-Riemannian setting this computation is not trivial for different reasons: balls are not smooth and not geodesically homogeneous, geodesics are parametrized on a non-compact set, the cut locus from a point is adjacent to the point itself etc. We will compute this asymptotics in the 3D contact case, and then discuss some difficulties arising when trying to consider higher dimensional situations. [Joint work with A. Lerario and I. Beschastnyi]

*Wednesday, March 14, 2018. - Salle 01 IHP*- 15.00 - 16.00
**Nikhil Savale**(Cologne) -*Magnetic Laplacian via sub-Riemannian geometry*

*Abstract:*We prove asymptotic and localization results for the first eigenvalue/eigenfunction of the Bochner(magnetic) Laplacian acting on high tensor powers of a line bundle. The proof exploits the relation of the magnetic and sub-Riemannian Laplacians as well as the sR heat kernel expansion. This generalizes known results of R. Montgomery from the Martinet case.

*Wednesday, April 11, 2018 - Salle 201 IHP*- 15.00 - 16.00
**Erika Battaglia**(Padova) -*Harnack inequality and fundamental solution for a class of sub-elliptic operators.*

*Abstract:*Click here

*Wednesday, May 23, 2018 - Salle 201 IHP*- 15.00 - 16.00
**Antonio Lerario**(SISSA) -*Visible and invisible singular curves*

*Abstract:*In this talk I will discuss the interplay between singular curves of a totally nonholonomic distribution and the homotopy of the path space X of all horizontal curves. At a global level the homotopy of X doesn't see the existence of singular curves (X is homotopy equivalent to the standard path space, with no restriction on the velocity of the curves). If we introduce on X a subriemannian energy functional J, a similar thing happens for the generic distribution of rank at least 3: even if there is in general no gradient flow for J (as X might be singular!) still it is possible do deform cycles in X and let their energy decrease, as long as there are no normal geodesics in the energy range of the deformation (no matter the existence of singular curves in this energy range!). When the rank of the distribution is 2, the situation can be different and singular curves can be actual obstacles to the deformation of cycles (for example Sussmann's theorem can be seen as a special case of this phenomenon). (joint work with A. A. Agrachev and F. Boarotto)

OTHER INFORMATIONS- Links to the wepage of the last years:

Séminaire de Géométrie sous-riemannienne - 2016/2017

Séminaire de Géométrie sous-riemannienne - 2015/2016

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

- 15.00 - 16.00