Université libre de Bruxelles, 28/11/2017
“Einstein metrics, desingularization and non degeneracy.”
I will explain a recent non degeneracy result for desingularizations of Einstein 4-orbifolds. This has two types of applications: first it provides an obstruction to the existence of families of desingularizations; second it gives in principle a recursive procedure to desingularize asymptotically hyperbolic Einstein orbifolds with higher singularities.
“Quasi-Asymptotically Conical Geometries.”
In this talk we introduce the class of quasi-asymptotically conical (QAC) geometries, a less rigid Riemannian formulation of the QALE geometries introduced by Joyce in his study of crepant resolutions of Calabi-Yau orbifolds. Our set-up is in the category of real stratified spaces and Riemannian geometry. Given a QAC manifold, we identify the appropriate weighted Sobolev spaces, for which we prove the finite dimensionality of the null space for generalized Laplacians as well as their Fredholmness. We conclude with new examples of Ricci-flat Kähler metrics which have these type of asymptotic geometries. This talk is based on joint work with Rafe Mazzeo and with Ronan Conlon and Frederic Rochon.
“Filtering the Heegaard Floer contact invariant”
According to the initial value formulation of general relativity, all that is future and all that is past is contained in a glimpse of a spacetime. This correspondence between the physics of the evolving spacetime and the geometry of “initial data” for the Einstein equations is dramatically and famously non-linear. The work of H. Bray, G. Huisken, R. Schoen, S.-T. Yau, and others suggests that the classical question of isoperimetry – How much area is needed to enclose a given amount of volume in initial data for the spacetime? – plays a pivotal role in this correspondence. In my lecture, I will discuss the recent proofs with O. Chodosh and with O. Chodosh, Y. Shi, and H. Yu of several long-standing conjectures in this direction.