SGAS

Séminaire de Géométrie et Analyse Sous-Riemannienne

SGAS ACTIVITY IS  ONLINE STARTING FROM 21/04/2020

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

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.

Past seminars

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.

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).

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.

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.

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.

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.

 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.

Determinantal processes and reproducing kernels 

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.

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.