Third Year Candidacy Requirement: "Morse Theory and Witten's Perturbation"

Speaker: Hao Zhuang, Washington University in Saint Louis

Abstract: In 1982, Edward Witten proposed a perturbation technique on the de Rham complex. Using this insightful technique, he gave an analytic proof of the well known Morse inequalities, which were proved through topological ways. This perturbation technique has been seen powerful in proving the Poincaré-Hopftheorem and the isomorphism between the Thom-Smale cohomology and the de Rham cohomology.

In this talk, I will introduce John Roe's rigorous version of Witten’s original idea. First, I will give a quick review of the Morse inequalities and the Hodge theorem. After that,functional calculus is used to rewrite the number of critical points of a certain index as the limit of the trace of an operator. This operator is generated by a Schwartz function and the perturbed Laplacian. It is worthwhile to mention that the Schwartz function is the key point to deal with the tail terms in Witten’s original paper and make his idea rigorous in math. Finally, I will introduce some ofmy ideas about how to generalize Witten’s work to the $G$-invariant de Rhamcomplex, including some evident difficulties.

Host: Xiang Tang