Preliminary Schedule
Monday 6
10.00 Welcome coffee
10.30 L.Fanelli
11.30 S.Federico
12:30 Lunch
14.00 U.Boscain
15.00 L.Benedetto
16.00 Coffee Break
16.30 E.Pozzoli
-
20.00 Dinner
Tuesday 7
9.00 D.Prandi
10.00 Coffee Break
10.30 E.Danesi
11.30 D.Cardona
12.30 Lunch
14.00 S.Flynn
ABSTRACTS
Lino Benedetto (ENS-PSL)
Title: Obstruction to smoothing effect and Strichartz estimates on the Engel group
Abstract:
The Engel group is the lowest dimensional nilpotent Lie group of step 3. One can consider its natural subLaplacian and ask about the dispersive properties of the associated Schrödinger equation. These questions have already been considered in the case of nilpotent Lie groups of step 2: dispersive estimates are known and Strichartz inequalities have been proven in particular cases. One observes that such properties are best understood when using the natural Fourier theory of the group under study and are related to its subRiemannian structure.
In this presentation, we will present the natural semiclassical theory adapted to the Engel group and in particular a two-microlocal approach, developped to give more insights on the dispersive nature of its subLaplacian. In particular, we will prove obstruction to smoothing-type estimates, as well as obstruction to some family of Strichartz estimates.
Ugo Boscain (CNRS, Sorbonne Université)
Title: Geometric confinement of a quantum particle on the Grushin plane and
in almost Riemannian manifolds
Abstract:
Two-dimensional almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set Z, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator $\Delta$ cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such counterintuitive phenomenon is called geometric confinement.
For the heat equation an intuitive explanation of this fact can be given in terms of random walk. For the Schoredinger equation an intuitive explanation is more subtle since the evolution of a quantum particle on a manifold can be done in several non-equivalent way. In this talk I will describe the evolution (and the confinement) of a quantum particle described by the curvature Laplacian $-\Delta+cK$ (here $K$ is the Gaussian curvature and c>0 a constant) which originates in coordinate free quantization procedures (as for instance in path-integral or covariant Weyl quantization). - Joint work with Eugenio Pozzoli and Ivan Beschastnyi
Duvan Cardona (Ghent University)
Title: The weak (1,1) boundedness of Fourier integral operators with complex phases
Abstract:
It was proved by Terence Tao that the Fourier integral operators with real-valued phase functions and of order -(n-1)/2 are of weak (1,1) type. In this talk, we present the extension to Tao's estimate when the corresponding Fourier integral operators have complex-valued phase functions. Applications of this result for a family of hyperbolic problems which arise in the calculus of Fourier integral operators due to Melin and Sjöstrand will be presented. Joint work with Michael Ruzhansky.
[1] Cardona, D., Ruzhansky, M. The weak (1,1) boundedness of Fourier integral operators with complex phases, preprint, arXiv:2402.09054.
[2] Tao, T. The weak-type (1,1) of Fourier integral operators of order −(n − 1)/2. J. Aust. Math. Soc. 76(1), 1–21, (2004).
Elena Danesi (Università di Padova)
Title: Strichartz estimates for the Dirac equation on compact manifolds without boundary
Abstract:
The Dirac equation on Rn can be listed within the class of dispersive equations, together with, e.g., the wave and Klein-Gordon equations. In the years a lot of tools have been developed in order to quantify the dispersion of a system. Among these one finds the Strichartz estimates, that are a priori estimates of the solutions in mixed Lebesgue spaces. For the flat case Rn they are known, as they are derived from the ones that hold for the wave and Klein-Gordon equations. However, when passing to a curved spacetime domain, very few results are present in the literature. In this
talk I will firstly introduce the Dirac equation on curved domains. Then, I will discuss the validity of this kind of estimates in the case of Dirac equations on compact Riemannian manifolds without boundary. This is based on a joint work with Federico Cacciafesta (Università di Padova) and Long Meng (CERMICS-École des ponts ParisTech).
Luca Fanelli (BCAM, Bilbao)
Title: A priori estimates for the resolvent of the Heisenberg sublaplacian
Abstract:
The sub-laplacian on the Heisenberg Group $\mathbb H^d$ is a standard example of sub-elliptic operator in a sub-Riemannian geometry. In this talk, we will first introduce some natural inequalities related to the Uncertainty Principle (Hardy, Rellich), then study some recent uniform versions of the same inequalities for the resolvent operator, over the complex plane. As an application, we obtain local smoothing estimates for the associated Schrödinger evolution flow, which is known to show a lack of the usual dispersion, due to the presence of soliton-like solutions.
The results are obtained in collaboration with H. Mizutani (Osaka University), L. Roncal and N. Schiavone (BCAM - Bilbao).
Serena Federico (Università di Bologna)
Title: Unique continuation properties of variable coefficient Schrödinger equations
Abstract:
In this talk we will discuss some unique continuation properties of certain space-variable coefficient Schrödinger equations in the Euclidean setting. We will see that, under suitable natural smallness assumptions on the coefficients, and assuming a certain exponential decay of the solution at two different times, the solution to the equation must be identically zero. (This talk is based on a joint work with Z. Li and X. Yu.)
Steven Flynn (Università di Padova)
Title: A Microlocal Calculus on Filtered Manifolds
Abstract:
A manifold equipped with a bracket generating family of vector fields has a natural filtration given by successive commutators. A pseudodifferential calculus constructed for filtered manifolds offers new methods for studying geometric PDEs involving Hormander sum-of-squares operators. We develop a pseudodifferential calculus on filtered manifolds in a constructive manner to allow for an explicit symbol calculus. The explicit symbolic calculus should led itself to applications including Erogov-type theorems and propagation of singularities.
Dario Prandi (CNRS, Paris Saclay)
Title: Magnetic Hardy inequalities in the Heisenberg group
Abstract:
We introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov-Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian, relying on a novel sharp Hardy inequality for the Folland-Stein operator. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. This is joint work with Biagio Cassano, Valentina Franceschi and David Krejicirik.
Eugenio Pozzoli (CNRS, Université de Rennes)
Title: Small-time controllability of bilinear Schrödinger and wave equations
Abstract:
This talk is devoted to some recent results of small-time global approximate controllability of bilinear Schroedinger and wave equations, posed on boundaryless manifolds such as tori and euclidean spaces. The analysis of such bilinear equations is classical, and several results of global approximate controllability in large times are available. The main novelty of these recent results is thus in the possibility of controlling such equations in arbitrarily small times. We will also discuss the ideas behind the new control strategies, inspired by geometric control techniques of Lie brackets in infinite dimensions.