Due to the current health emergency, the activity of the Séminaire de géométrie et analyse sous-riemannienne is proposed online from 21/04/2020 until a later date.
The "Séminaire de Géométrie et Analyse Sous-riemannienne" is a periodic seminar held in Paris since 2011, whose aim is to help connections between researchers working on different aspects of the analysis and geometry of sub-Riemannian manifolds including
Carnot-Carathéodory geometry
hypoelliptic diffusion
nonholonomic mechanics and geometric control
spectral theory of degenerate elliptic operators
anisotropic shape optimization
The "Séminaire de Géométrie et Analyse Sous-riemannienne" is a part of the activities of the ANR project SRGI - Sub-Riemannian Geometry and Interactions
Organizers: Davide Barilari, Ugo Boscain, Valentina Franceschi, Mario Sigalotti
If you want to subscribe to our mailing list, please inform us by sending an e-mail to the address franceschiv AT ljll.math.upmc DOT fr
For the previous editions of this seminar, please visit this page.
16/06/2020 - 15h - Andrea Pinamonti - Online Meeting
Γ-convergence and H-convergence for functionals and operators depending on vector fields.
Abstract: Given a family of locally Lipschitz vector fields $X(x)=(X1(x),\dots,Xm(x))$ on $\mathbb{R}^n$, $m\leq n$, we study functionals depending on $X$. We will discuss some results of $\Gamma$- convergence and we will apply them to study the $\mathbb{H}$-convergence of linear differential operators in divergence form modeled on $X$. The talk is based on joint works with Alberto Maione and Francesco Serra Cassano.
19/05/2020 - 16h30 - Álvaro del Pino Gómez - Online Meeting
Submanifolds tangent to bracket-generating distributions
Abstract: Whenever we study a geometric structure, it is natural to look at submanifolds that interact meaningfully with it. In the case of distributions, tangent submanifolds have been central objects of study. A classic example is Chow's theorem: It states that curves tangent to bracket-generating distributions present extremely flexible behaviours, allowing us to connect any two points in the ambient manifold. Another example is the theory of legendrians (i.e. maximal submanifolds tangent to a contact structure), which has been one of the driving topics shaping the field of Contact Topology (and which additionally has deep connections with Singularity Theory).
In this talk I will review some of the classic ideas involved in the study of legendrian knots. This will serve as motivation for me to discuss knots tangent to other bracket-generating distributions. In particular, I will present a classification result for knots tangent to Engel structures (joint with R. Casals) and some on-going work regarding the general case (joint with F.J. Martínez-Aguinaga). If time allows, I will also comment briefly on recent progress regarding the classification of higher-dimensional submanifolds tangent to the canonical distribution in jet space (joint with L. Toussaint).
04/05/2020 - 16h30 - Davide Vittone - Zoom Meeting
Some new results about the Sard problem in Carnot groups
Abstract: The Sard problem in sub-Riemannian geometry concerns the negligibility of the set spanned by singular (a.k.a. abnormal) curves emanating from a fixed basepoint. We show that singular curves in sub-Riemannian Carnot groups can be obtained by concatenating trajectories of suitable dynamical systems; as an applications, we positively answer the Sard problem in some classes of Carnot groups. Time permitting, we will discuss the construction of some exotic singular curves. The talk is based on a joint work with F. Boarotto.
21/04/2020 - 16h30 - Frédéric Jean - Zoom Meeting
Geodesic equivalence of sub-Riemannian metrics and a sub-Riemannian Weyl theorem.
Abstract: H. Weyl in 1921 demonstrated that, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one (in dimension greater than one). In this talk we investigate the analogous property for sub-Riemannian metrics, called the Weyl projective rigidity. We first review our previous work on projective equivalence of SR metrics (metrics having the same geodesics up to a reparametrization). Then we show that Weyl projective rigidity is related to the minimal order property of the complex abnormal extremals. This a joint work with Sofya Maslovskaya and Igor Zelenko.
25/02/2020 - 14h - Eero Hakavuori - Salle 15-16-309 : barre 15-16, 3ème étage, porte 09
Infinite geodesics in Carnot groups
Abstract: Infinite geodesics in Carnot groups naturally arise as tangents to sub-Riemannian geodesics. This permits a study of the geodesic regularity problem through study of the family of all infinite geodesics. In this talk, I will present some examples of interesting behavior and cover the known results on rigidity of infinite minimizers. This talk is based on joint work with Enrico Le Donne.
21/01/2020 - 14h - François Vigneron - Salle 15-16-309 : barre 15-16, 3ème étage, porte 09
A simple proof of the Hardy inequality on Carnot groups.
Abstract: We give an elementary proof of the classical Hardy inequality on any Carnot group, using only integration by parts and a fine analysis of the commutator structure, which was not deemed possible until now. If we have enough time, we might also discuss the conditions under which this technique can be generalized to deal with hypoelliptic families of vector fields, which, in this case, leads to an open problem regarding the symbol properties of the gauge norm.
17/12/2019 - 15h - Davide Barilari - Salle 15-16-309 : barre 15-16, 3ème étage, porte 09
Bakry-Émery curvature and model spaces in sub-Riemannian geometry
Abstract: To obtain comparison theorems (volume, laplacian, etc.) on Riemannian manifolds with an external measure, one should replace bounds on Ricci tensor with bounds on the so-called Bakry-Emery Ricci tensor, a scalar quantity containing information about both the curvature of the metric and the measure.
In this talk I will discuss about a notion of sub-Riemannian Bakry-Émery curvature and the corresponding comparison theorems. The model spaces for comparison are variational problems coming from optimal control theory.
17/12/2019 - 14h - Bufetov Aleksandr Igorevich - Salle 15-16-309 : barre 15-16, 3ème étage, porte 09
Determinantal processes and reproducing kernels
19/11/2019 - 14h - Pierre Pansu - Salle 15-16-411 : barre 15-16, 4ème étage, porte 11
L^1 cohomology of Heisenberg groups.
Abstract: Although the Laplacian cannot be conveniently inverted on L^1, the Laplacian on closed differential forms has a good inverse. This works in Heisenberg group as well, providing a result of interest in geometric group theory.
22/10/2019 - 14h - Emmanuel Trélat - Salle 15-16-309 (barre 15-16, 3ème étage, porte 09)
Spectral analysis of sub-Riemannian Laplacians and Weyl measure
Abstract: In a series of works on sub-Riemannian geometry with Yves Colin de Verdière and Luc Hillairet, we study spectral properties of sub-Riemannian Laplacians, which are hypoelliptic operators. The main objective is to obtain quantum ergodicity results, what we have achieved in the 3D contact case. In the general case we study the small-time asymptotics of sub-Riemannian heat kernels. We prove that they are given by the nilpotentized heat kernel. In the equiregular case, we infer the local and microlocal Weyl law, putting in light the Weyl measure in sR geometry. This measure coincides with the Popp measure in low dimension but differs from it in general. We prove that spectral concentration occurs on the shief generated by Lie brackets of length r-1, where r is the degree of nonholonomy. In the singular case, like Martinet or Grushin, the situation is more involved but we obtain small-time asymptotic expansions of the heat kernel and the Weyl law in some cases. Finally, we give the Weyl law in the general singular case, under the assumption that the singular set is stratifiable.