abstracts

Magnetic Laplacians on hyperbolic surfaces

On a closed hyperbolic surface, we investigate semiclassical defect measures associated with the magnetic Laplacian in the presence of a constant magnetic field. Depending on the energy level where the eigenfunctions concentrate, three distinct dynamical regimes emerge. In the low-energy regime, we show that any invariant measure of the magnetic flow in phase space can be obtained as a semiclassical measure. At the critical energy level, we establish Quantum Unique Ergodicity, together with a quantitative rate of convergence of eigenfunctions to the Liouville measure. In the high-energy regime, we prove a Shnirelman-type result: a density-one subsequence of eigenfunctions becomes equidistributed with respect to the Liouville measure. Joint work with Thibault Lefeuvre

Dynamics and spectrum of non-self-adjoint Berezin--Toeplitz operators and Mabuchi space

I will present recent advances and perspectives on the study of non-self-adjoint Berezin--Toeplitz quantization, and in particular, the role played by the geometry of Mabuchi space. The technical core of these results is semiclsasical analysis in real-analytic regularity, as developed by Sjöstrand.

Fluctuations pour les processus déterminantaux sur les variétés Kähleriennes

Je présenterai une extension en dimension complexe supérieure d'un résultat dû à Rider et Virag, qui établissent un théorème central limite pour les fluctuations d'un gaz de Coulomb dans un potentiel extérieur quadratique sur le plan complexe, dans le cadre général des processus déterminantaux sur les variétés Kähleriennes introduit par Berman.

Equidistribution of random real algebraic hypersurfaces

I will first describe a model of random algebraic hypersurface in the real projective space, which is defined by considering the zero set of a so-called Kostlan random polynomial of degree d. Then, I will present results showing that this hypersurface equidistributes in the ambient space as d \to +\infty. More generally, these results hold when Kostlan polynomials are replaced with random real sections of the d-th tensor power of an ample line bundle L over a real projective manifold. In this general setting, our results rely on large degree asymptotics for the Bergman kernel of L^d. This talk is based on joint works with M. Ancona, L. Gass, M. Puchol and M. Stecconi.

Spectral Asymptotics of Semi-Classical Toeplitz Operators on CR Manifolds

The study of Toeplitz operators is a classical subject in several complex variables, deeply intertwined with microlocal analysis. This talk will begin with a brief review of the foundational microlocal frameworks developed by Melin–Sjöstrand, Boutet de Monvel–Sjöstrand, and Boutet de Monvel–Guillemin in the context of CR manifolds. Following this, I will present recent joint work with H. Herrmann, C.-Y. Hsiao, and G. Marinescu. We develop a semi-classical approach for the spectral theory of first-order elliptic self-adjoint Toeplitz operators on compact, strictly pseudoconvex, and embeddable CR manifolds. The primary benefit of our method is that it yields the embeddability of CR manifolds into complex Euclidean spaces not only on the differential geometric level, but also on a precise metric level. If time permits, I will conclude by discussing an extension of this work from my thesis to Levi non-degenerate CR manifolds.