Institut de Recherche Mathématiques de Rennes
My research lies at the crossroads of analysis and geometry, mainly because I like to consider the phase space geometry and its relationships with Quantum mechanics. While I’m interested in all aspects of semiclassical analysis, I’ve been specializing in integrable systems, classical or quantum, and their singularities. This is a nice door to differential geometry and normal forms.
A quite important class of integrable systems are obtained by two degree of freedom systems with an $S^1$ symmetry. Under additional non-degeneracy conditions, I called them semitoric systems, because they share some rigidity aspects of toric systems and some nice flexibility from general Hamiltonian systems. In the last ten years, they have been studied by various teams and continue to reveal intringuing properties, both at classical and quantum levels.
Recently, I’ve been interested in magnetic fields; the general ideas from semiclassical analysis and (near) integrable systems apply remarkably and give accurate descriptions of quantum dynamics and eigenvalues of magnetic laplacians. While Magnetic hamiltonians do have an (approximate) $S^1$ symmetry, the relationship with semitoric systems remains to be elucidated!
Here are a few keywords that tend to show up again and again in my works.
Microlocal analysis, spectral theory, integrable systems, semitoric systems, inverse spectral theory for integrable systems, symplectic geometry, mathematical physics, Hamiltonian dynamics, Birkhoff normal forms, Morse theory, pseudodifferential operators, Berezin-Toeplitz operators, classical and quantum magnetic fields
Thank you so much to all of you.
While a mathematician seemingly does not need much to get working, we all recognize the importance of the support from our research institutes, and most importantly the invitation programs that allow us to travel around the world and meet our collaborators in person. I find it the most efficient way to advance my research. For this reason I warmly thank the following institutions for their generous invitations.
Please contact me if I was invited to work at your institute and it's not listed below!
Here are some works from the year 2018 on. You can access the complete list of publications here.
For a two degree of freedom quantum integrable system, a new spectral quantity is defined, the quantum rotation number. In the semiclassical limit, the quantum rotation number can be detected on a joint spectrum and is shown to converge to the well-known classical rotation number. The proof requires not only semiclassical analysis (including Bohr-Sommerfeld quantization rules) but also a detailed study on how quantum labels can be assigned to the joint spectrum in a smooth way. This leads to the definition and analysis of asymptotic lattices. The general results are applied to the semitoric case where formulas become particularly natural.
Using a new quantization scheme, we construct approximate semi-classical Bergman projections on weighted $L^2$ spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\mathbb{C}^n$ and for high powers of ample holomorphic line bundles over compact complex manifolds.
We prove that the Schwartz class is stable under the magnetic Schrödinger flow when the magnetic 2-form is non-degenerate and does not oscillate too much at infinity.
We consider a charged particle on a plane, subject to a strong, purely magnetic external field. It is well known that the quantum evolution closely follows the classical dynamics for short periods of time, while for times larger than $\ln \frac{1}{\hbar}$, where $\hbar$ is Planck’s constant, purely quantum phenomena are expected to happen. In this paper we investigate the Schrödinger evolution of generalized coherent states for times of order $1/\hbar$. We prove that, when the initial energy is low, the initial states splits into multiple wavepackets, each one following the average dynamics of the guiding center motion but at its own speed.
We study the Hamiltonian dynamics of a charged particle submitted to a pure magnetic field in a two-dimensional domain. We provide conditions on the magnetic field in a neighbourhood of the boundary to ensure the confinement of the particle. We also prove a formula for the scattering angle in the case of radial magnetic fields.
There is now plenty of evidence to suggest that we all should stop wasting the taxpayer’s money, and instead turn to non-profit editors for handling our beloved manuscripts. If you’re not convinced, please have a look at this post by Tim Gowers, or, if you can read French, at the very complete Frédéric Hélein webpage. For this reason, I have decided to give a lot of my time and energy to the launching of a new journal, free for authors and readers, the Annales Henri Lebesgue. I also serve in the editorial board of Annales Scientifiques de l’École Normale Supérieure which, albeit not totally non-profit, operates now in a very reasonable way.