Time：10:00-12:00, Saturday, August 3, 2024
Venue：E4-233
Host：Jintian Zhu, ITS
Speaker：Junrong Yan, BICMR
Title: Higher Torsions and Witten-type Deformation
Abstract：This talk will present a version of the higher Cheeger-Müller/Bismut-Zhang theorem. Bismut and Goette have derived a version of this theorem assuming the existence of a fiberwise Morse function that satisfies the fiberwise Thom-Smale transversality condition (TS condition). To fully generalize the theorem, our aim is to remove these assumptions from Bismut and Goette's paper. Unlike fiberwise Morse functions, fiberwise generalized Morse functions (GMFs) always exist. Thus, we extend Bismut and Goette's setup by considering a fibration $ M \to S $ with a unitarily flat complex bundle $ F \to M $ and a fiberwise GMF $ f $, while retaining the TS condition.
The critical point set of a generalized Morse function consists of both non-degenerate critical points and birth-death points. This talk will focus on addressing the difficulties posed by the analysis near birth-death points using Witten-type deformation. Removing the TS condition involves higher algebra constructions by Igusa-Klein and Goette, which will be the focus of our future research and will be outlined if time permits.
This is joint work with Martin Puchol.
