High frequency analysis: from operator algebras to PDEs

Liste des Contributions

7. Analysis in sub-elliptic and sub-Riemannian settings: challenges and recent approaches.

Véronique Fischer (University of Bath)

28/08/2023 14:30

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...

2. The zero dispersion limit for the Benjamin-Ono equation on the line

Patrick Gérard (Université Paris-Saclay)

28/08/2023 15:45

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.

14. Hypoellipticity, pseudodifferential operators, tangent groupoids, and the Helffer-Nourrigat conjecture

Robert Yuncken (Université de Lorraine)

29/08/2023 09:00

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...

4. Quantum limits of some perturbed sub-Laplacians

Victor Arnaiz (université de Nantes)

29/08/2023 10:15

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.

1. Wick symbol of evolution operators for operators acting on the Fock space or on the Wiener space

Lisette Jager (Université de Reims Champagne-Ardennes)

29/08/2023 14:30

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...

11. Semiclassical normal forms for magnetic Laplacians

Léo Morin (University of Aarhus)

29/08/2023 15:45

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...

15. Magnetic Hardy inequalities in the Heisenberg group

David Krejcirik

30/08/2023 09:00

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...

13. A panorama of singular sub-Laplacians and their spectra

Marcello Seri (University of Groningen)

30/08/2023 10:15

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...

16. Spectral summability for 1D oscillators and Fourier Analysis in Carnot groups

Davide Barilari

30/08/2023 14:00

In this talk, I will address some questions concerning spectral properties of the sublaplacian $-\Delta_{G}$ on Carnot groups. The attention will focus on the Engel group, which is the main example of a Carnot group of step~3.

Thanks to Fourier analysis on the Engel group in terms of a frequency set, we give fine estimates on the convolution kernel satisfying $F(-\Delta_{G})u=u\star...

3. Maximal multiplicity of Laplacian eigenvalues in negatively curved surfaces

Cyril Letrouit

30/08/2023 15:00

I will present a joint work with Simon Machado (IAS) in which we make progress on a longstanding conjecture by Colin de Verdière: we obtain a sublinear bound on the maximal multiplicity of first Laplacian eigenvalues for negatively curved surfaces. For the proof, we take inspiration from arguments developed recently in the context of graphs of bounded degree.

5. Fredholm operators on graded Lie groups

Ewert Eske

31/08/2023 09:00

A graded Lie algebra has a decomposition which is compatible with the
Lie bracket. This allows to define a differential calculus on the
corresponding group G in which an element of the Lie algebra can
have order higher than one when viewed as a left-invariant differential
operator.
This notion of order is implemented in the pseudodifferential calculi by
Fischer--Ruzhansky (for graded Lie...

10. Schrödinger evolution in a low-density random potential: convergence to solutions of the linear Boltzmann equation

Soeren Mikkelsen (University of Bath)

31/08/2023 10:15

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...

12. Decoherence time scales and the Hoermander condition

Roman Schubert (University of Bristol)

31/08/2023 15:00

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...

6. Weyl calculus on graded groups

Serena Federico (University of Bologna)

01/09/2023 09:00

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...

8. Field $C^*$-algebra and spectral analysis of quantum many channel Hamiltonians.

Vladimir Georgescu

01/09/2023 10:15

This talk concerns the field $C^*$-algebra associated to a symplectic space (in a representation, this is the $C^*$-algebra generated by the field operators) and the spectral theory of the self-adjoint operators affiliated to it. The field algebra is graded by the semilattice of finite dimensional subspaces of the symplectic space and this fact has deep consequences in the spectral analysis of...