Institut de recherche mathématique avancée

Résumé : The theory of curvature-dimension conditions for metric measure spaces, such as the Riemannian Curvature-Dimension condition RCD(K,N), has emerged as a powerful tool in differential geometry and geometric analysis. I will give a short introduction to this theory, and I will highlight one of its key features: the geometric and functional-analytic stability under measured Gromov-Hausdorff convergence. I will then present a nonlinear analogue of almost splitting maps into Euclidean space, as harmonic maps into a flat torus. Existence of such maps implies Gromov-Hausdorff closeness to a flat torus in any dimension. Combining these results with a Bochner-type inequality by Stern yields a new Gromov-Hausdorff stability theorem for flat 3-tori with almost nonnegative Scalar curvature. This is partly a joint project with Shouhei Honda, Ilaria Mondello, Raquel Perales and Chiara Rigoni.

Résumé : L'espace des modules d'une surface hyperbolique est un espace probabilisé pour la mesure de Weil-Petersson. Dans cet exposé, nous verrons dans un premier temps certains résultats de Mirzakhani sur l'intégration de variables aléatoires dont la définition fait intervenir les longueurs de géodésiques simples. Nous déterminerons ensuite l'expression de la longueur d'une géodésique ayant des auto-intersections en coordonnées de Fenchel-Nielsen, et nous observerons comment cette expression nous permet de généraliser les résultats probabilistes de Mirzakhani à des courbes remplissant un pantalon.

Résumé : The McKay correspondence establishes a strong relationship between the classical minimal resolution of a Kleinian singularity and its standard orbifold, or non-commutative, resolution. In this talk I will introduce a new class of non-commutative resolutions of a Kleinian singularity, and show how the McKay correspondence can be extended to them. I will also show how Hilbert schemes of points on these resolutions are Nakajima quiver varieties for various generic stability parameters. If time permits, I will introduce a process called cornering, and describe how it relates to partial resolutions of the Kleinian singularity and Nakajima quiver varieties for certain non-generic stability parameters.

Résumé : The spectral theory of Toeplitz operators originates in Szegő’s classical work on the eigenvalue distribution of Toeplitz matrices and was later extended by Boutet de Monvel and Guillemin to general complex manifolds. This talk has two objectives. First, we present a generalization of these results in the framework of arbitrary Bernstein-Markov measures on complex manifolds. Second, we investigate the asymptotic behavior of the smallest eigenvalue of Toeplitz operators, a problem that — unexpectedly — turns out to be closely related to Mabuchi geometry originally introduced in connection with constant scalar curvature Kähler metrics. Our aim is to show how techniques from Kähler geometry and pluripotential theory emerge organically in the asymptotic analysis of Toeplitz operators.