In this talk, we will discuss various approaches that have emerged over the last decades to tackle different questions related to the analysis of sub-elliptic and sub-Riemannian settings. For instance, the study of hypoellipticity in these contexts has made considerable progress using groupoid methods from operator algebra while spectral properties of certain sub-Laplacians have been analysed...
Zero dispersion problems for nonlinear evolution equations are known to create very strong oscillations so that there are very few examples for which it is possible to describe the limit. In this talk I will discuss the case of the Benjamin-Ono equation on the line, for which the Lax pair structure provides valuable tools leading to a complete answer to this problem.
In 1979, Helffer and Nourrigat made a very broad conjecture about the hypoellipticity of differential operators which are polynomials in a family of vector fields. Their conjecture generalises a vast number of results — eg, the elliptic regularity theorem, Hörmander’s sums-of- squares theorem, and Rockland’s Theorem (proven by Helffer-Nourrigat) on hypoellipticity for left invariant vector...
In this talk I will present some recent results obtained
independently in collaboration with Gabriel Rivière and Chenmin Sun on
the spectral study of sub-elliptic operators. In the particular cases
of the Baouendi-Grushin operator on the torus and certain perturbations
of sub-Riemannian contact Laplacians in dimension three, we will
describe the quantum limits associated with these operators.
In the context of an infinite dimensional analogue of the Weyl pseudodifferential calculus, we have to work with the Fock space and with the Wiener space. This talk aims at giving a characterization, in terms of the Fock space, of a concept (a set of test functions) initially defined on the Wiener space.
The second part is concerned with the explicit computation of the Wick symbol of...
The semiclassical analysis of magnetic Laplacians is closely related to the analysis of hypoelliptic sums of squares. I will present some semiclassical normal forms for the magnetic Laplacian, which provide precise description of its spectrum. I will especially emphasize on the geometry of the underlying phase space. One purpose of this talk is to open discussions to understand better the...
Laplace-Beltrami operators on rank-varying sub-Riemannian structures have been recently gaining interest due to their exotic properties. In this talk we will start from the 0th property of their analysis: self-adjointness. In a large number of cases, and in contrast with the Riemannian case, the sub-Riemannian setting presents large families of operators which are essentially self-adjoint even...
It is a fundamental problem in mathematical physics to derive macroscopic transport equation from the underlying microscopic transport equations. In this talk, we will consider such a problem. To be precise we will consider solutions to a time-dependent Schrödinger equation for a potential localised at the points of a Poisson point process. For these solutions we will present a result stating...
Decoherence is the suppression of interference effects in quantum mechanics due to the coupling of a system to an environment. The evolution in an open quantum system is typically described by the Lindblad equation and I will describe how semiclassical analysis of the Lindblad equation leads to a classical diffusion equation in the Hoermander sum of squares form and how hypoellipticity of...
In this talk we will discuss a class of symmetric pseudo-differential calculi on graded nilpotent Lie groups using the Hörmander symbol classes introduced by V. Fisher and M. Ruzhansky. Among the quantizations generating these calculi, we shall identify a candidate Weyl quantization on general graded nilpotent Lie groups by comparison with the well-know Weyl quantization on Rn. Finally, we...
