SÉMINAIRE DE GÉOMÉTRIE SOUS-RIEMANNIENNE - INSTITUT HENRI POINCARÉ, PARIS - 2014/15

The "séminaire de géométrie sous-riemannienne" is a periodic seminar held in Paris since 2011, whose aim is to help connections between the different communities working in sub-Riemannian geometry from different viewpoints.

The seminar takes place at the Institut Henri Poincaré, 11, rue Pierre et Marie Curie, Paris. In the period September 8 - December 15, 2014 it will be hosted as a part of the IHP Trimester on Sub-Riemannian Geometry.

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OTHER INFORMATIONS

The "séminaire de géométrie sous-riemannienne" is a periodic seminar held in Paris since 2011, whose aim is to help connections between the different communities working in sub-Riemannian geometry from different viewpoints.

The seminar takes place at the Institut Henri Poincaré, 11, rue Pierre et Marie Curie, Paris. In the period September 8 - December 15, 2014 it will be hosted as a part of the IHP Trimester on Sub-Riemannian Geometry.

- Frequence Sept-Dec 2014: part of the IHP Trimester.
- Frequence Jan-Apr 2015: pause
- Frequence May-June 2015: one session in May 2015.
- Topics: sub-Riemannian geometry and related fields.
- Some notes of the seminars of the past years are available on this blog.
- Organizers: Davide Barilari, Pierre Pansu.
- Next sessions: May 21 at 10.00 (confirmed), then the seminar resumes in October 2015.

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*NEXT SEMINARS*

*Thursday, May 21, 2015. - Salle 01 IHP*

- 10.00 - 11.00.
**Elisa Paoli**(SISSA) -*Small time asymptotic on the diagonal for Hörmander's type hypoelliptic operators*

*Abstract:*On a smooth manifold we consider differential operators of Hormander's type, defined by k vector fields and possibly a drift vector field. Under the assumption that the fields satisfy weak Hormander's condition of hypoellipticity, we study the small time asymptotic of the fundamental solution of the differential operator, in a stationary point $x_0$ of the drift field. We prove that the order of the asymptotic depends on the controllability of an associated control problem and of its approximating system: if the control problem of the approximating system is controllable at $x_0$, then so is also the original control problem, and in this case we show that the fundamental solution blows up as $t^{-N/2}$. The integer $N$ is a number determined by the Lie algebra at $x_0$ of the fields, which define the hypoelliptic operator. It generalizes the homogeneous dimension, that appears in the asymptotic of the fundamental solution in the sub-Riemannian case without drift.

- 11.15 - 12.15.
**Luca Rizzi**(CMAP, Ecole Polytechnique) -*Intrinsic random walks and sub-Laplacians in sub-Riemannian geometry*

*Abstract:*In Riemannian geometry, the Laplace-Beltrami operator can be defined as the generator of the diffusion obtained as the limit of geodesic random walks. This coincides with the usual divergence of the gradient (the divergence is computed with respect to the Riemannian volume). We discuss how to extend these definitions in sub-Riemannian geometry, where it is not clear what is an intrinsic random walk and an intrinsic volume. Joint work with U. Boscain and R. Neel.

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*PAST SEMINARS*

*Wednesday, September 10, 2014. (IHP, Amphi Darboux)*

- 15.00 - 16.00.
**Andrea Bonfiglioli**(Univ. Bologna) -*Maximum principle and Harnack inequality for hypoelliptic degenerate non-Hormander operators.*

*Abstract:*We consider a class of hypoelliptic second-order partial differential operators L in divergence form on R^N, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for L, along with some Potential Theory results on the characterization of the subharmonic functions with respect to L. The main novelty with respect to the vast existing literature on Harnack inequalities lies in that the involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, or Muckenhoupt-type conditions on the degeneracy of the principal part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity in order to recover a meaningful geometric information on connectivity and maxima propagation, yet in the absence of any maximal rank condition. Finally, the potential-theoretic results are presented under the assumption that L also possesses a global, positive fundamental solution, vanishing at infinity.

*Friday, September 12, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Marcello Seri**(Univ. College London) -*Spectral properties and Aharonov-Bohm effect on Grushin-like structures*

*Abstract:*We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. As for general almost-Riemannian structures (under certain technical hypothesis), the singular set acts as a barrier for the evolution of the heat and of a quantum particle, although geodesics can cross it. This is a consequence of the self-adjointness of the Laplace-Beltrami operator on each connected component of the manifolds without the singular set. We get explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we get a Weyl law with dominant term $E\log E$. We then study the effect of an Aharonov-Bohm non-apophantic magnetic potential that has a drastic effect on the spectral properties.

- 11.15 - 12.15.
**Eugene Malkovich**(Univ. Novosibirsk) -*Splitted geometrical flows and dynamics of contact forms on S^3*

*Abstract:*Classical Eguchi-Hanson metric with SU(2)-holonomy can be found via special flow defined on 2 contact forms on S^3. Also Fubini-Study metric can be obtained using weighted flow of this type. This flows can be regarded as splitted flows on Hopf fibration and presumably have connection with an anisotropic dilations.

*Monday, September 15, 2014. (IHP, Amphi Darboux)*

- 14.30 - 15.00.
**Giulio Tralli**(Univ. Bologna) -*Scale-invariant Harnack inequality of Cordes-Landis type for some horizontally elliptic operators*

*Abstract:*We consider second order linear degenerate-elliptic operators which are elliptic with respect to horizontal directions generating a stratified algebra. An axiomatic approach developed by Di Fazio, Gutiérrez, and Lanconelli allows us to study the so-called "double-ball property" and "critical density property" in order to get an invariant Harnack inequality for such operators. We will show the validity of the double-ball property in any Carnot group of step two and, under some Cordes-Landis type conditions, the critical density in H-type groups.

- 15.15 - 15.45.
**Isidro Munive**(SISSA, Trieste) -*The Dirichlet and Neumann problems for Hormander type operators*

*Abstract:*We study the solvability of the Dirichlet and Neumann problems for sub-Laplacians associated with a system of Hormander vector fields. The study of these problems in the sub-Riemannian setting differs substantially from the ordinary Laplacian due to the presence of the so-called characteristic points on the boundary.

*Wednesday, September 17, 2014. (IHP, Amphi Darboux)*

- 15.00 - 16.00.
**Alexander Zuyev**(Univ. Donetsk) -*Stabilization of a Class of Non-Holonomic Systems by Using Fast Oscillating Controls*

*Abstract:*This talk is devoted to the stabilization problem of nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.

- 16.30 - 17.30.
**Victoria Grushkovskaya**(Univ. Donetsk) -*Steering of Nonholonomic Systems by using Second-Order Lie Brackets*

*Abstract:*The talk is devoted to the steering problem of nonholonomic systems without drift. The class of systems under consideration satisfies the controllability rank condition with the Lie Brackets up to the second order. We use trigonometric polynomials to steer a system from the initial to the final point at a given time. The Volterra series expansion of solutions is applied for computing coefficients of control functions. By using this technique, the steering problem is reduced to solving a system of algebraic equations. It is proved that such an algebraic system locally admits at least one real solution. The control design scheme is illustrated by several examples.

*Friday, September 19, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Marcos M. Diniz**(ICEN, UFPA) -*Gauss-Bonnet theorem in Heisenberg space $\mathbb H^1$*

- 11.15 - 12.15.
**Xiaoping Yang**(Nanjing Univ.) -*Some properties for solutions to some elliptic equations in the Heisenberg group*

*Abstract:*In this talk, we will discuss some properties such as growth, nodal sets and singular sets of solutions to some elliptic equations in the Heisenberg group. We will show growth of H-harmonic functions in the Heisenberg group. The measure estimates for nodal sets of some H-harmonic functions are proved. The geometric structure of horizontal singular sets of H-harmonic functions in the Heisenberg group are investigated. This is a joint work with H-R. Liu and L. Tian

*Tuesday, September 23, 2014. (IHP, Amphi Darboux)*

- 15.00 - 16.00.
**Alessia Kogoj**(Univ. Bologna) -*L^p Liouville Theorems for Invariant Evolution Equations*

*Abstract:*Some L^p-Liouville theorems for several classes of evolution equations will be presented. The involved operators are left invariant with respect to Lie group composition laws in R^{n+1}. Results for both solutions and sub-solutions will be given.

*Friday, September 27, 2014. (IHP, Amphi Darboux)*

- 15.00 - 16.00.
**Yves Colin de Verdière**(IF Grenoble) -*Quantum ergodicity for sub-Riemannian Laplacians*

*Abstract:*This is joint work in progress with Luc Hillairet (Orléans) & Emmanuel Trélat (Paris 6). A. Shnirelman proved in 1974 the following Theorem: "let $(X,g)$ be a closed Riemannian manifold with ergodic geodesic flow and $(\phi_n,\lambda_n)$ an eigen-decomposition of the Laplace operator. Then there exists a density $1$ sub-sequence $\lambda _{n_j}$ of the sequence $\lambda _n $ so that the sequence of probability measures $|\phi_{n_j}|^2 dx_g $ ($dx_g$ being the Riemannian measure) converges weakly to the measure $dx_g$." I plan to describe extensions of this Theorem to the case of contact 3D sR Laplacians and to discuss possible extensions to other sR geometries.

*Tuesday, October 7, 2014. (IHP, Amphi Darboux)*

- 15.00 - 16.00.
**Nizar Touzi**(CMAP, Ecole Polytechnique) -*Viscosity solutions of path-dependent parabolic PDEs*

*Abstract:*Path-dependent PDEs provide a local characterization for some non-Markovian phenomena. Typical examples are the HJB equation for stochastic control of non-Markov controlled systems, and the HJBI equation for the corresponding stochastic differential games. A notion of classical solutions was introduced by Dupire 2009. Similar to the Markov case, classical regularity of potential solutions is rather exceptional. We therefore introduce a notion of viscosity solutions which by-passes the main difficulty due to the fact that the underlying path space is not locally compact. The key-idea is to replace the pointwise tangency in the definition of test function, by the tangency in mean under appropriate probability measures. In particular, when restricted to the Markov case, our definition induces a larger set of test functions. We provide existence and uniqueness results for a general class of nonlinear equations. In the special case of semilinear equations, our proof of the comparison result relies on the notion of punctual differentiation, which is similar to the corresponding concept introduced by Caffarelli-Cabre in the standard Crandall-Lions viscosity solutions, and we prove that semimartingales are punctually differentiable Leb$\otimes\dbP-$a.e. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions.

*Tuesday, October 14, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Abdol-Reza Mansouri**(Queens University) -*Heat kernel asymptotics on Lie groups*

*Abstract:*We consider the problem of computing heat kernel small-time asymptotics for hypoelliptic Laplacians associated to left-invariant sub-Riemannian structures on unimodular Lie groups of type I. In this setting, the heat kernel can be expressed using the non-commutative Fourier transform associated with the group (as shown by Agrachev et al.), and we show how such an expression for the heat kernel can be used in order to compute its small-time asymptotics. We present the examples of the Heisenberg, Cartan, and Engel groups, as well as possible links with the cut/conjugate locus of the sub-Riemannian structure.

*Monday, October 20, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Valerii Berestovskii**(Sobolev Institute of Mathematics, Omsk) -*Sub-Riemannian limits of geodesic orbit Riemannian manifolds*

*Abstract:*A connected Riemannian manifold M is called geodesic orbit (go) if every its geodesic is an orbit of some one-parameter group of isometries of M. Evidenly, any such space is homogeneous. I shall discuss different subclasses of go Riemannian manifolds, in particular weakly symmetric Riemannian manifolds of A.Selberg, a classification of last class in compact case. Other examples include total spaces of Hopf fibrations, Heisenberg groups and groups of Heisenberg types by A.Kaplan. I proved that arbitrary two points of every sub-Riemannian limit P of go Riemannian manifolds can be joined by a shortest arc which is a part of some geodesic, an orbit of some one-parameter subgroup of isometries of P.

*Tuesday, October 21, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Jean-Paul Gauthier**(Univ. Toulon) -*More about Goursat subriemannian metrics and nonholonomic motion planning for the car with n trailers.*

*Monday, October 27, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Jean-Michel Bismut**(Univ. Paris Sud) -*Hypoelliptic Laplacian, spectral theory and the Langevin process*

*Abstract:*Click here to read the abstract

*Wednesday, October 29, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Piermarco Cannarsa**(Univ. Roma Tor Vergata) -*Singular dynamics for Hamilton-Jacobi equations*

*Abstract:*In dynamic programming, the set of points at which the value function, V, of an optimal control problem fails to be differentiable - in short, the singular set of V - is usually regarded as a region to keep away from. Indeed, the uniqueness of optimal trajectories is generally lost on such set and numerical schemes become less reliable. Such a viewpoint, however, could be partly reversed thinking of all the data that can be compressed at singular points. This talk will be focussed on singularities of solutions to Hamilton-Jacobi equations in connection with optimal control problems, and the dynamics that describes their propagation. We will be particularly interested in the study of the invariance of singular sets under such dynamics for two examples of solutions to eikonal-type equations: the euclidean (and riemannian) distance function from the boundary of a bounded domain, which has interesting applications to homotopy, and the solution of a Cauchy problem which is given by the Hopf-Lax formula.

- 11.15 - 12.15.
**Dmitry Faifman**(Univ. Tel Aviv) -*The quotient Finsler structure, and Schaffer's dual sphere conjecture.*

*Abstract:*In the 70's, Schaffer investigated the geometry of spheres in Banach spaces. He conjectured that the length of the shortest closed geodesic of the unit sphere is invariant under duality. This was proved by him in dimension 2, and then in full (in fact, greater) generality by Alvarez-Paiva in '06. In this talk, I will explain how an attempt to generalize the conjecture to higher grassmannians leads naturally to the notion of quotient Finsler structure, and prove the corresponding duality result.

*Friday, October 31, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Alessio Figalli**(Univ. Texas Austin) -*Quantitative stability estimates for the Brunn-Minkovsky inequality*

*Abstract:*Given a Borel A in R^n of positive measure, one can consider its semisum S=(A+A)/2. It is clear that S contains A, and it is not difficult to prove that they have the same measure if and only if A is equal to his convex hull minus a set of measure zero. We now wonder whether this statement is stable: if the measure of S is close to the one of A, is A close to his convex hull? More generally, one may consider the semisum of two different sets A and B, in which case our question corresponds to proving a stability result for the Brunn-Minkowski inequality. When n=1, one can approximate a set with finite unions of intervals to translate the problem to the integers Z. In this discrete setting the question becomes a well-studied problem in additive combinatorics, usually known as Freiman's Theorem. In this talk I will review some results in the one-dimensional discrete setting and describe how to answer to the problem in arbitrary dimension.

*Monday, November 3, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Jose Miguel Veloso**(UFPA) -*Variation of (spherical) Hausdorff measure of non horizontal submanifolds in Carnot groups of constant metric factor*

*Abstract:*We discuss first and second variation of the (spherical) Hausdorff measure for non-horizontal submanifolds in Carnot groups of constant metric factor. The condition for a critical point is a tensor equation $H+L=0$, where $H$ is analogous to the mean curvature and $L$ is a mean Lie bracket. We also write the second variation formula.

*Tuesday, November 4, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Adrian Nachman**(University of Toronto) -*Imaging an Anisotropic Conductivity in a Known Conformal Class and Minimal Surfaces*

*Abstract:*This talk will give an overview of electric conductivity imaging from interior data obtainable using Magnetic Resonance Imagers, and the beautiful underlying Riemannian structure. We show that an anisotropic conductivity in a known conformal class can be determined from measurement of one current using gemetric measure theory methods. Further, we show that the associated equipotential surfaces are area minimizing with respect to a Riemannian metric obtained entirely from the physical data. This is joint work with Nicholas Hoell, Robert Jerrard and Amir Moradifam. The experimental results are joint work with Weijing Ma, Nahla Elsaid, Michael Joy and Tim DeMonte.

*Friday, November 21, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Andrey Sarychev**(Univ. Firenze) -*On Lie "Rank" Criteria for Approximate Ensemble Controllability*

- 11.15 - 12.15.
**Ilya Kossovskiy**(Wien University) -*On Poincaré's problème local*

*Abstract:*In this talk, we describe a result giving a complete solution to the old question of Poincare on the possible dimensions of the automorphism group of a real-analytic hypersurface in two-dimensional complex space. As the main tool, we introduce the so-called CR (Cauchy-Riemann manifolds) - DS (Dynamical Systems) technique. This technique suggests to replace a real hypersurface with certain degeneracies of the CR-structure by an appropriate dynamical system, and then study mappings and symmetries of the initial real hypersurface accordingly. It turns out that symmetries of the singular differential equation, associated with the initial real hypersurface, are much easier to study than that of the real hypersurface, and in this way we obtain the solution for the question of Poincare. This work is joint with Rasul Shafikov.

*Monday, December 1, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Corey Shanbrom**(California State Univ.) -*Periodic Orbits in the Heisenberg-Kepler Problem*

*Abstract:*Abstract: The Kepler problem is among the oldest and most fundamental problems in mechanics. It has been studied in curved spaces, such as the sphere and hyperbolic plane. Here, we formulate the problem on the Heisenberg group, the simplest sub-Riemannian manifold. We take the sub-Riemannian Hamiltonian as our kinetic energy, and our potential is the fundamental solution to the Heisenberg sub-Laplacian. The resulting dynamical system is known to contain a fundamental integrable subsystem. We will discuss the use of variational methods in proving the existence of periodic orbits with k-fold rotational symmetry for any odd integer k greater than one, and show approximations for k=3.

*Thursday, December 4, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Alexey Bufetov**(CNRS Aix Marseille & Steklov Institute) -*Quasi-symmetries of determinantal point processs.*

*Monday, December 8, 2014. (IHP, Amphi Darboux)*

- 10.00 - 11.00.
**Elisha Falbel**(Univ. Paris 6) -*Locally homogeneous contact structures on three manifolds*

*Abstract:*We discuss locally homogeneous geometries associated to contact structures on three manifolds. Examples include Cauchy-Riemann geometry and flag structures.

*Tuesday, December 9, 2014. (IHP, Amphi Darboux)*

- 11.15 - 12.15.
**Witold Respondek**(INSA Rouen) -*Mechanical control systems and their linearization*

OTHER INFORMATIONS

- Links to the wepage of the last years:

Séminaire de Géométrie Sous-Riemannienne - 2013/2014

Séminaire de Géométrie Sous-Riemannienne - 2012/2013

Séminaire de Géométrie Sous-Riemannienne - 2011/2012 - (organized by Enrico Le Donne) - Blog of the SR geometry seminar in SISSA - 2013/2014 - (organized by Luca Rizzi)