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.

VISIO: The conference, which will take place in Sorbonne Université (campus Jussieu, amphi 25), will be entirely filmed and broadcast live on YouTube:
You just have to click on the above link, for the corresponding day, or copy-paste it to your internet browser. Enjoy the talks. If you want to ask a question during a talk, send an e-mail to SRGI.conference@gmail.com or use the YouTube chat. Both will be constantly checked by some people present in the amphi, who will raise the question for you at the end of the talk.
All talks will also be recorded for later viewing on YouTube (all links can be found below).

 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 G. Molino 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 (cancelled)
* by visio

- From Tuesday to Friday, lunch trays will be offered to all participants, in two successives services (to avoid too much affluence): in the room 15-16 309, Seminar Room of Laboratoire Jacques-Louis Lions (LJLL).
- Unfortunately, the banquet, initially planned on wednesday, had to be cancelled (decision of the university, for sanitary reasons). Coffee breaks are also not allowed to take place near the amphi 25, but coffee can anyway be offered at the Coffee Room of LJLL (in front of the seminar room).

Titles and abstracts:

Title: Asymptotic homology in sub-Riemannian geometry: two cases study.
Abstract: Given a sub-Riemannian manifold M, it is well known that embedding of the horizontal loop space into the whole loop space is a homotopy equivalence. We know however that horizontal loop spaces have interesting singularities and extremely rich local and global structure even if M is contractible like in the case of Carnot groups. In principle, one can recover hidden structural complexity of the horizontal loop spaces calculating homology of some natural filtrations of the space. I am going to show two examples of such calculations.

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.

Title:
Strong Sard Conjecture for analytic sub-Riemannian structures in dimension 3.
Abstract: Given a totally nonholonomic distribution of rank two \Delta on a three-dimensional manifold M, it is natural to investigate the size of the set of points X^x that can be reached by singular horizontal paths starting from a same point x of M. In this setting, the Sard conjecture states that X^x should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. I will present our work in collaboration with A. Figalli, L. Rifford and A. Parusinski, where we show that the (strong version of the) conjecture holds in the analytic category in dimension 3. Our methods rely on resolution of singularities of surfaces, foliations and metrics; regularity analysis of Poincare transition maps; and on a simplectic argument.

Title:
Control for the Grushin Schrödinger equation.
Abstract: We consider the two dimensional Grushin Schrödinger equation posed on a finite cylinder \Omega = (-1,1)_x\times \T_y with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time T_*>0 depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at x=0, we also show that the observability fails for any T\leq T_*.

Title: Weyl laws for singular Riemannian manifolds.
Abstract: In this talk, I will present results on the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, i.e., where the main geometric invariants are unbounded, with (possibly) infinite volume. Under suitable assumptions on the curvature blow-up, it is shown how the singularity influences the Weyl's asymptotics and the localization of the eigenfunctions for large frequencies. We also have a result of converse type, i.e., given any non-decreasing slowly varying function V (in the sense of Karamata), there exists a singular Riemannian structure with discrete spectrum such that   N(\lambda)\sim_{\lambda\to\infty} \frac{\omega_n}{(2\pi)^n}\lambda^{n/2}V(\lambda).    A key tool in our arguments is a new quantitative estimate for the remainder of th heat trace on Riemannian manifolds.   (Joint work with D. Prandi and L. Rizzi)

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).

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.

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.

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.

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.

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.

Title:
Constant-normal sets and horizontally polynomial functions.
Abstract: In the setting of Carnot groups, we consider sets and functions that have some sort of horizontal monotonicity. In particular, we focus on constant-normal sets, which are sets that are invariant with respect to a horizontal half-space and monotone in the other directions, and we also focus on functions that are linearly increasing (or more generally polynomial) in every  horizontal direction. The aim of the talk is twofold: we shall see nice properties of such objects, but also some pathologies. The work is based on several collaborations with G. Antonelli, C. Bellettini, S. Don, T. Moisala,  D. Morbidelli, S. Rigot, and D. Vittone.

Title:
Quantitative unique continuation for hypoelliptic operators.
Abstract: In this talk we consider quantitative unique continuation issues for equations involving a hypoelliptic operator (a sum of squares of vector fields). We give in particular an estimate of the minimal mass left by eigenfunctions of such an operator on subdomains, in the high-frequency limit. We also deduce applications to approximate controllability and stabilization. This is a joint work with Camille Laurent.

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.

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.

Title:
Sub-Laplacian comparison theorems on H-type foliations.
Abstract: We will present comparison theorems for the sub-Laplacian of the distance function achieved on the class of H-type foliations introduced jointly with F. Baudoin, E. Grong, and L. Rizzi, as well as explore the history and utility of such comparison theorems in the sub-Riemannian setting.

Title:
Geodesics in jet space.
Abstract: The space J^k=J^k(\R,\R) admits a canonical rank 2 distribution of Goursat type. Its sR geodesics have a simple and beautiful characterization in terms of degree k polynomials of the independent variable x first described by Anzaldo-Meneses and Monroy-Perez. Among these geodesics are candidate metric lines: geodesics defined on all of the real line which minimize between any two points. These special geodesics are always asymptotic to singular lines -- abnormal geodesics of J^k, with the asymptotic singular line for s tending to -\infty different from the asymptotic singular line for s tending to +\infty. We can flip the process around: fix \epsilon-tubes about two distinct similarly oriented singular lines. Move along the tubes oppositely to infinity, by letting s tending to \pm\infy and joining the points by minimizers. Does the resulting minimizer converge? Surprisingly to us: typically no! Only for specially arranged tubes will the resulting minimizers converge to a metric line. The concrete description of the metric lines allows us to see this specialness" by dimension count, but does not explain it.

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.

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.

Title:
Heat content asymptotics for sub-Riemannian manifolds.
Abstract: We study the small-time asymptotics of the heat content of smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. By adapting to the sub-Riemannian case a technique due to Savo, we establish the existence of the full asymptotic series. We compute explicitly the coefficients up to order five, in terms of sub-Riemannian invariants of the domain and its boundary. Furthermore, we prove that every coefficient can be obtained as the limit of the corresponding one for a suitable Riemannian extension. As a byproduct of our fifth-order analysis, we prove that the higher order coefficients in the expansion can blow-up in presence of characteristic points. This is a joint work with Luca Rizzi.

Title:
Localization and uniformity of asymptotics for sub-Riemannian heat kernels.
Abstract: Molchanov’s method provides a systematic approach to determining the small-time asymptotics of the heat kernel on a sub-Riemannian manifold away from any abnormal minimizers. The expansion is closely connected to the structure of the minimizing geodesics between two points. This method allows to derive a complete asymptotic expansion of the heat kernel from a sufficiently explicit normal form of the exponential map at the minimal geodesics between two points, at least in principle. We are, however, also able to exhibit metrics for which the exponential map is arbitrarily degenerate. Beyond classical applications of complete expansions and universal bounds, the method extends to log-derivatives of the heat kernel and their characterization of the cut locus. This is joint work with R.W. Neel.

Title:
Bochner Laplacian and Bergman kernel expansion of semi-positive line bundles on a Riemann surface.
Abstract: We generalize the results of Montgomery for the Bochner Laplacian on high tensor powers of a line bundle. When specialized to Riemann surfaces, this leads to the Bergman kernel expansion and geometric quantization results for semi-positive line bundles whose curvature vanishes at finite order. The proof exploits the relation of the Bochner Laplacian on tensor powers with the sub-Riemannian (sR) Laplacian. This is a joint work with G. Marinescu.