Abstract:

 In this talk, I will introduce recent progress on Arveson's curvature invariant. We complete Arveson's framework for an operator-theoretic version of the Gauss-Bonnet-Chern formula by giving a necessary and sufficient condition for the formula to hold. 
Furthermore, for regular unitarily invariant complete Nevanlinna-Pick reproducing kernel Hilbert spaces on the unit ball in $\mathbb C^d$, we establish Arveson's version of the Gauss–Bonnet–Chern formula holds unconditionally when these Hilbert spaces are regarded as Hilbert modules over their multiplier algebras.  Moreover, we solve the finite defect problem in the same framework. As a special case, we prove that the only nonzero finite-rank submodule of the Dirichlet module is the whole Dirichlet module itself. This phenomenon is completely different from what happens for the Drury-Arveson module, the Hardy module, and the weighted Bergman modules. This talk is based on joint work with my advisor, Penghui Wang, and my junior colleague, Ruoyu Zhang.