School of Mathematical Sciences Academic Lecture [2025] No.087

(High-level University Construction Series Report No.1109)

Lecture Title: Arveson's version of Gauss-Bonnet-Chern formula for Hilbert modules

Lecturer: Professor Penghui Wang (Shandong University)

Lecture Time: September 19, 2025, 14:00–15:00

Venue: Room 1420, Huiwen Building Office Area

Abstract:
In this talk, I will introduce the recent development on Arveson’s curvature invariant. We complete Arveson’s framework on the operator-theoretic version of the Gauss-Bonnet-Chern formula by giving a necessary and sufficient condition for the formula to hold. Such a formula can be generalized to the infinite-many-variables case. The talk is based on joint work with Ruoyu Zhang and Zeyou Zhu.

Lecturer’s Biography:
Professor Penghui Wang, School of Mathematics, Shandong University, Distinguished Taishan Scholar of Shandong Province. His main research areas include geometric analysis of Hilbert modules and eigenvalue problems of differential operators. 

1) He solved the polydisc version of the Arveson–Douglas conjecture, providing a complete characterization of the essential normality of homogeneous and quasi-homogeneous Hardy quotient modules on the polydisc. He also gave an operator-theoretic characterization of distinguished subvarieties in the bidisc, and further studied the K-homology groups of distinguished subvarieties using the extension theory of operator algebras.

2) He established a multivariable operator-theoretic version of the Gauss-Bonnet-Chern formula. The Gauss-Bonnet-Chern formula is one of the most fundamental formulas in modern mathematics. Operator theory experts hope to understand curvature from the perspective of operator theory and operator algebras. The former Deputy Editor-in-Chief of Duke Mathematical Journal, based on invariant subspaces of the symmetric Fock space, proposed an operator-theoretic version of curvature and proved the corresponding multivariable operator-theoretic Gauss-Bonnet-Chern formula for homogeneous submodules. By introducing local algebraic properties of submodules, Professor Wang and collaborators completely characterized the necessary and sufficient conditions for this formula to hold.

His major results have been published in Crelle’s Journal, Advances in Mathematics, and other journals.

All teachers and students are welcome to attend!

Inviter: Yong Han

School of Mathematical Sciences

September 16, 2025