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.
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).
Location: Sorbonne Université, Faculté Pierre et Marie Curie
(4 place Jussieu, Paris), amphi 25.
Schedule: 2020, 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:
A. AgrachevslidesYouTube link 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.
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. BelottoslidesYouTube
link
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.
N. BurqslidesYouTube link 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_*.
Y. ChitourslidesYouTube
link 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)
Y. Colin de Verdière slidesYouTube
link 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 slidesYouTube
link 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. FranceschislidesYouTube link
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. GallagherslidesYouTube link 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. GarofaloslidesYouTube link 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. HabermannslidesYouTube link 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 DonneslidesYouTube link 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.
M. LéautaudslidesYouTube link 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.
C. LetrouitslidesYouTube link 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. MalchiodislidesYouTube link 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.
G. MolinoslidesYouTube link 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.
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.
R. MontislidesYouTube
link 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. PetitotslidesYouTube
link 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. RossislidesYouTube
link 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.
L. SacchellislidesYouTube link 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.
N. SavaleslidesYouTube link 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.
M. SigalottislidesYouTube link 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. ThalmaierslidesYouTube link Title:
Sub-Riemannian Brownian motion, functional inequalities on path
space and horizontal Ricci curvature.
Abstract: Given the general setting of a
sub-Riemannian manifold with a bracket-generating subbundle of
horizontal directions, we describe recent work related to the
concept of "horizontal Ricci curvature". Our approach relies on a
study of sub-Riemannian Brownian motion and stochastic analysis on
the path space over the sub-Riemannian manifold. Motivated by the
work of Aaron Naber (2015) on characterizing bounded Ricci
curvature for Riemannian manifolds, we show that certain
functional inequalities and gradient estimates on path space are
equivalent to boundedness of the horizontal Ricci curvature
tensor. The adaptation of methods from Riemannian geometry to the
sub-Riemannian setting turns out to be a delicate task. This is a
joint work with Li-Juan Cheng and Erlend Grong.
G. TrallislidesYouTube link 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.