SRGI
conference: SubRiemannian Geometry and Interactions
Paris, September 711, 2020
This conference is the final event of the ANR SRGI
project. The objective of the conference is to cover all
aspects of subRiemannian geometry, making a stateoftheart 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.
REGISTRATION: mandatory, please contact Emmanuel Trélat.
Invited speakers:


Location: Sorbonne Université, Faculté Pierre et Marie Curie
(4 place Jussieu, Paris), amphi 25.
Schedule: 2020, September 711

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: Htype subRiemannian spaces.
Abstract: We introduce the class of
Htype subRiemannian manifolds, which is a generalization of the
class of Htype 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 selfadjointness of differential operators
on closed manifolds.
Abstract: A classical result of Gaffney
says that a Riemannian laplacian on a complete Riemannian manifold
is essentially selfadjoint (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 selfadjoint. I will also discuss
the case of higher dimensions which is related to a recent work in
collaboration with Laure SaintRaymond 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
measuretheoretic 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 Hregular
hypersurfaces, and a metric one built around Lipschitz images of
codimension1 vertical subgroups. While the relation between these
two definitions is not yet fully understood, I will focus on the
metric rectifiability of Hregular 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 biLipschitz maps between "big pieces"
of metric spaces.
V. Franceschi
Title: The subFinsler 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 subFinsler structures. Namely, we consider
leftinvariant perimeter measures associated with a general
leftinvariant norm $\phi$ on the horizontal distribution. The
case where \phi is the standard Euclidean norm corresponds to the
subRiemannian case, subject of the wellknown Pansu’s conjecture
on the shape of isoperimetric sets. Assuming the norm \phi to be
regular enough, we present a characterization of C^2smooth
isoperimetric sets as the subFinsler 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
nonregular, 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 nondispersive
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
pseudodifferential 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 threedimensional
contact subRiemannian manifolds.
Abstract: We are concerned with
stochastic processes on surfaces in threedimensional contact
subRiemannian manifolds. By considering the Riemannian
approximations to the subRiemannian 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
LaplaceBeltrami 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
subRiemannian 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 highfrequency eigenfunctions of
the LaplaceBeltrami operator heavily depends on the properties of
the geodesic flow: if it is ergodic, nearly all eigenfunctions
become equidistributed in the highfrequency 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 subRiemannian geometry. Given m\in\N, we
consider the subLaplacian \sum_{j=1}^m
\partial_{x_j}^2+(\partial_{y_j}x_j\partial_{z_j})^2, which is
the natural subLaplacian 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 highfrequency 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 subRiemannian
geometry.
Abstract: We consider threedimensional
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 Sobolevtype 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
endpoint map.
Abstract: We will present a third order
study of the endpoint map in subRiemannian 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 subRiemannian manifold. This is a joint work
with F. Baorotto and F. Palmurella.
J. Petitot
Title: The primary visual cortex as a subRiemannian engine.
Abstract: Since the 1990s, methods of
"in vivo optical imaging based on activitydependent 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
wellknown 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 subRiemannian geometry of the 1jet
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 subRiemannian 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
subRiemannian structures.
Abstract: It is a longstanding problem
in subRiemannian geometry whether lengthminimizing 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 lengthminimizers in rank 2 subRiemannian
structures. As a consequence of such a result, all
lengthminimizers for rank 2 subRiemannian 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
steptwo Carnot groups.
Abstract: We discuss the subRiemannian
counterpart of a famous result by BourgainBrezisMironescu and
Dávila concerning the limiting behaviour of a bynow well
established nonlocal notion of sperimeter 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
suitablydefined nonlocal horizontal sperimeter 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.