Ph.D Thesis Defense: "Index theory for Toeplitz Operators on Algebraic Spaces"
Abstract: This dissertation is about the abstract Toeplitz operators obtained by compressing the multishifts of the usual Hilbert spaces of analytic functions onto co-invariant subspaces generated by polynomial functions. These operators were introduced by Arveson in regard to his multivariate dilation theory for spherical contractions. The main technical issue here is essential normality, addressed in Arveson's conjecture. If this conjecture holds true then the fundamental tuple of Toeplitz operators associated to a polynomial ideal $I$ can be thought as noncommutative coordinate functions on the variety defined by $I$ intersected with the boundary of the unit ball. This interpretation suggests operator-theoretic techniques to study certain algebraic spaces. More specifically, we are interested in Douglas' index problem.
In the special case of monomial ideals we give a new proof for Arveson's essential normality conjecture, also answer Douglas' index problem. Our main construction is a certain resolution (in the sense of homological algebra) of Hilbert modules.
Thinking of the fundamental tuple of Toeplitz operators as noncommutative coordinate functions, we start applying them to study the isolated singularities of algebraic hypersurfaces. The main extra operator-theoretic ingredient here is a unitary operator, the holonomy of a certain Gauss-Manin connection induced by the monodromy of the singularity. We want to understand how this unitary operator interacts with the Toeplitz operators. This study could lead to an analytic way for detecting exotic smooth structures on odd-dimensional spheres.
Host: Xiang Tang