SRGI conference: Sub-Riemannian Geometry and Interactions
Paris, September 7--11, 2020

This conference is the final event of the ANR SRGI project. The objective of the conference is to cover all aspects of sub-Riemannian geometry, making a state-of-the-art of recent advances in the field, ranging over metric, geometric and spectral issues, harmonic analysis, control and observability, optimal transport, applications to imaging.

Organization committee: Davide Barilari, Ugo Boscain, Yacine Chitour, Dario Prandi, Luca Rizzi, Emmanuel Trélat.

 Invited speakers:

Location: Sorbonne Université, Faculté Pierre et Marie Curie (4 place Jussieu, Paris), amphi 25.

Schedule2020, September 7--11

 MONDAY TUESDAY WEDNESDAY THURSDAY FRIDAY 09:15 -- 10:00 N. Burq N. Garofalo I. Gallagher M. Léautaud 10:00 -- 10:45 A. Thalmaier K. Habermann T. Rossi Y. Chitour 10:45 -- 11:15 BREAK BREAK BREAK BREAK 11:15 -- 12:00 C. Letrouit Z. Balogh A. Malchiodi Y. Colin de Verdière 12:00 -- 14:00 LUNCH LUNCH LUNCH LUNCH 14:00 -- 14:45 A. Agrachev K. Fässler V. Franceschi L. Sacchelli END 14:45 -- 15:30 A. Belotto N. Savale E. Le Donne G. Tralli 15:30 -- 16:00 BREAK BREAK BREAK BREAK 16:00 -- 16:45 R. Monti F. Baudoin R. Montgomery M. Sigalotti 16:45 -- 17:30 J. Petitot 19:00 banquet 23:00 (Tour Zamansky)

- From tuesday to friday, lunch will be offered to all participants, in the patio of the amphi 25.
- On wednesday evening, banquet will be offered to all participants, at the top of Tour Zamansky (reception room, 24th floor).

Titles and abstracts:

A. Agrachev
Title: TBA
Abstract: TBA

Z. Balogh
Title: TBA
Abstract: TBA

F. Baudoin
Title: H-type sub-Riemannian spaces.
Abstract: We introduce the class of H-type sub-Riemannian manifolds, which is a generalization of the class of H-type groups. A complete classification of those spaces is obtained and a comparison geometry is developed. This is joint work with E. Grong, G. Molino and L. Rizzi.

A. Belotto
Title: TBA
Abstract: TBA

N. Burq
Title: TBA
Abstract: TBA

Y. Chitour
Title: TBA
Abstract: TBA

Y. Colin de Verdière
Title: Essential self-adjointness of differential operators on closed manifolds.
Abstract: A classical result of Gaffney says that a Riemannian laplacian on a complete Riemannian manifold is essentially self-adjoint (ESA). Of course this applies in particular to closed manifolds. It holds true even for sR Laplacians. In this lecture, I will study the case of non subelliptic operators. A basic example for geometers is the Laplacian of a Lorentzian metric. Generically in dimension 2, such a Laplacian is not essentially self-adjoint. I will also discuss the case of higher dimensions which is related to a recent work in collaboration with Laure Saint-Raymond on a spectral problem coming from the study of internal waves. A general conjecture will appear relating ESA to classical completeness of the Hamiltonian vector field of the symbol. This is joint work with Corentin Le Bihan (ENS Lyon).

K. Fässler
Title: Notions of rectifiability in Heisenberg groups.
Abstract: Rectifiable sets are central objects in geometric measure theory that serve as measure-theoretic generalizations of smooth curves and surfaces. In this talk, which is based on collaboration with D. Di Donato and T. Orponen, I will discuss two notions of rectifiability for hypersurfaces in Heisenberg groups proposed in the literature since the early 2000s: an intrinsic one using H-regular hypersurfaces, and a metric one built around Lipschitz images of codimension-1 vertical subgroups. While the relation between these two definitions is not yet fully understood, I will focus on the metric rectifiability of H-regular hypersurfaces in H^n with Hölder continuous horizontal normal (and a slightly stronger result in the first Heisenberg group H^1). The proofs are based on a new criterion for finding bi-Lipschitz maps between "big pieces" of metric spaces.

V. Franceschi
Title: The sub-Finsler isoperimetric problem in the Heisenberg group.
Abstract: We present recent results on optimal shapes in the Heisenberg group H^1 for the isoperimetric problem relative to sub-Finsler structures. Namely, we consider left-invariant perimeter measures associated with a general left-invariant norm $\phi$ on the horizontal distribution. The case where \phi is the standard Euclidean norm corresponds to the sub-Riemannian case, subject of the well-known Pansu’s conjecture on the shape of isoperimetric sets. Assuming the norm \phi to be regular enough, we present a characterization of C^2-smooth isoperimetric sets as the sub-Finsler analogue of Pansu's bubbles. To this purpose we provide a fine description of the characteristic set of regular surfaces that are locally extremal for the isoperimetric problem. In the case where \phi is non-regular, this procedure does not allow to deduce a characterization of isoperimetric sets, and we present some preliminary results. By an approximation procedure, we finally prove a conditional minimality property for the candidate solutions in the general case (including the case where \phi is crystalline). This is based on a joint work with R. Monti, A. Righini and M. Sigalotti.

I. Gallagher
Title: Strichartz estimates and local dispersion on the Heisenberg group.
Abstract: The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation, and for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. We shall nevertheless show that Strichartz estimates do exist, by use of Fourier restriction methods. Using a representation of the Schrödinger kernel, we shall also prove local dispersive estimates. This is a joint work with Hajer Bahouri and Davide Barilari.

N. Garofalo
Title: Feeling the heat in a group of Heisenberg type.
Abstract: The main character of this talk Is the heat flow in a group of Heisenberg type and some modifications of the latter. These tools are used to provide a unified treatment of the very different extension problems for two pseudo-differential operators arising in analysis and conformal CR geometry. One of the main objectives is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equations and semigroup methods. When s=1 our results recapture the famous fundamental solution found by Folland and generalised by Kaplan. This is recent joint work with Giulio Tralli.

K. Habermann
Title: Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds.
Abstract: We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. By considering the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in SU(2) and SL(2,R) equipped with the standard sub-Riemannian contact structures as model cases for this setting. This is joint work with Davide Barilari, Ugo Boscain and Daniele Cannarsa.

E. Le Donne
Title: TBA
Abstract: TBA

M. Léautaud
Title: TBA
Abstract: TBA

C. Letrouit
Title: Quantum limits of products of Heisenberg manifolds.
Abstract: In Riemannian geometry, the distribution on the manifold of high-frequency eigenfunctions of the Laplace-Beltrami operator heavily depends on the properties of the geodesic flow: if it is ergodic, nearly all eigenfunctions become equidistributed in the high-frequency limit, whereas eigenfunctions of completely integrable systems, due to the high multiplicity of some eigenvalues, may present more complicated patterns. In this talk, we deal with the same problem in the more general framework of sub-Riemannian geometry. Given m\in\N, we consider the sub-Laplacian \sum_{j=1}^m \partial_{x_j}^2+(\partial_{y_j}-x_j\partial_{z_j})^2, which is the natural sub-Laplacian on products of compact quotients of the 3D Heisenberg group. The associated geodesic flow is completely integrable, and the study of the Quantum Limits, which characterize possible limits of high-frequency eigenfunctions, reveals a very rich structure, in which an infinite number of flows comes into play.

A. Malchiodi
Title: On a geometric Sobolev quotient in sub-Riemannian geometry.
Abstract: We consider three-dimensional CR manifolds, which are modelled on the Heisenberg group, introduce a natural concept of “mass” and prove its positivity under the condition that the scalar curvature is positive and in relation to their (holomorphic) embeddability properties. We apply this result to discuss extremality of Sobolev-type quotients, giving some counterexamples in cases of lack of embeddability, and discussing their relevance to the CR Yamabe problem. This is joint work with J.H.Cheng and P.Yang.

R. Montgomery
Title: TBA
Abstract: TBA

R. Monti
Title: Third order open mapping theorems and analysis of the end-point map.
Abstract: We will present a third order study of the end-point map in sub-Riemannian geometry. We first discuss some new open mapping theorems of the third order in a general setting and then we use them to find necessary conditions involving brackets of length three satisfied by minimzing strictly singular curves in a sub-Riemannian manifold. This is a joint work with F. Baorotto and F. Palmurella.

J. Petitot
Title: The primary visual cortex as a sub-Riemannian engine.
Abstract: Since the 1990s, methods of "in vivo optical imaging based on activity-dependent intrinsic signals" have made possible to visualize the extremely special connectivity of the primary areas of the visual cortex, that is to say their “functional architectures.” Cortical visual neurons function as wavelets detecting local geometric cues (position, contrast, orientation, etc) encoded in the optical signal processed by the retina. But it is the intracortical architectures that explain how these local cues can be integrated so as to generate the global geometry of the images perceived, with all the well-known phenomena studied since Gestalt theory (illusory contours, etc). Neurogeometry is based on the discovery that the functional architecture of V1 (the first visual area) implements the contact structure and the sub-Riemannian geometry of the 1-jet space of plane curves. From there, the illusory contours can be interpreted as geodesics of the polarized Heisenberg group or of the SE(2) group, which specifies previous models of David Mumford using the theory of elastica. These sub-Riemannian models have many applications, in particular for inpainting algorithms.

T. Rossi
Title: TBA
Abstract: TBA

L. Sacchelli
Title: TBA
Abstract: TBA

N. Savale
Title: TBA
Abstract: TBA

M. Sigalotti
Title: On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures.
Abstract: It is a longstanding problem in sub-Riemannian geometry whether length-minimizing curves are smooth. It is known that normal extremals are smooth, but the case of abnormal minimizers is still open. We present an improvement of the existing partial results that guarantees the C^1 regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of such a result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class $C^{1}$. This is a joint work with Davide Barilari, Yacine Chitour, Frédéric Jean and Dario Prandi.

A. Thalmaier
Title: TBA
Abstract: TBA

G. Tralli
Title: Nonlocal approximation of the horizontal perimeter in step-two Carnot groups.
Abstract: We discuss the sub-Riemannian counterpart of a famous result by Bourgain-Brezis-Mironescu and Dávila concerning the limiting behaviour of a by-now well established nonlocal notion of s-perimeter as the fractional order of differentiation s approaches the local case (s=1/2). We show, in the context of Carnot groups of step two, that a suitably-defined nonlocal horizontal s-perimeter does converge to the horizontal perimeter up to a dimensionless constant. The relevant notion of nonlocal horizontal perimeter is introduced with the aid of the heat kernel associated with the horizontal Laplacian. The talk is based on a joint work with N. Garofalo.