Essential normality of Bergman modules on egg domains

During 2005-2006, Arveson and Douglas formulated a challenging conjecture in multivariable operator theory regarding the essential normality of compressed shifts in the usual Hilbert spaces of analytic functions, say, Bergman spaces on strongly pseudoconvex domains. (Essential normality means normality modulo compact operators.) In this talk, after stating this conjecture, I will report on a joint work with Xiang Tang about the essential normality of Bergman spaces on several classes of egg domains. These egg domains are generalizations of the unit ball and are weakly pseudoconvex in general. If time permits, I will say a few words about a resulting K-homology index theorem and discuss p-essential normality (that is normality modulo p-summable operators).