Institut de recherche mathématique avancée

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.

Résumé : TBA