January 13, 2023
TITLE: The generalized Kahler Calabi-Yau problem
ABSTRACT: In recent years generalized Kahler geometry has emerged as a natural extension of Kahler geometry with applications to complex, Poisson, and symplectic geometry, as well as mathematical physics. In this talk I will describe an extension of the Calabi-Yau problem to this setting. I will give a nearly complete picture of the existence and uniqueness of the relevant Calabi-Yau geometries using the generalized Kahler-Ricci flow, and explain a consequence for symplectomorphism groups on hyperKahler manifolds. This is joint work with V. Apostolov, X. Fu, and Y. Ustinovskiy.
January 14, 2022
TITLE: Singular sets of generalized Einstein metrics
ABSTRACT: Various considerations in geometry and physics lead to natural generalizations of Einstein metrics which are coupled to differential forms. In this talk I will describe recent joint work with X. Fu and A. Naber on the singularity formation of sequences of such structures. In particular we show that the limit spaces are regular outside of a set of codimension 4, and satisfy certain sharp integral estimates, leading to geometric applications.