Topology via the Witten deformation

Hao Zhuang  (Beijing University)

10:00-11:00, July 29-31, 2026   普陀校区 理科大楼A503

Abstract:

The Witten deformation is an effective way to figure out numerical results in the topology of smooth manifolds. Roughly speaking, after adding the Clifford action of a non-degenerate vector field to a Dirac type operator, we obtain an elliptic operator whose eigenvalues are divided into two parts. The first part consists of small eigenvalues and gives us the expected numerical formula of a topological object. The other part consists of sufficiently large eigenvalues and does not contribute to the formula. In this talk, we focus on the Witten deformation in the case of non-degenerate vector fields. We start with Bismut and Zhang’s Witten deformation approach to the Thom-Smale complex associated to a Morse function. Then, we review Zhang’s analytic approaches to the Euler characteristics and the Kervaire semi characteristics. Finally, we present an example of how Bismut and Zhang’s approaches are adapted to the mapping cone situation when a non-degenerate vector field is given.

About the speaker:

庄昊现为北京大学北京国际数学研究中心（BICMR）博士后研究员（ 合作导师为刘小博教授），他于2025年在圣路易斯华盛顿大学获得数 学博士学位（师从唐翔教授）。他的研究兴趣主要集中在数学物理和 几何分析领域，近期核心研究工作包括定义并探讨闭辛流形的辛半特 征（Symplectic semi-characteristic）及其相关的计数公式；此外，他还 在适当李群作用（proper Lie group actions）的框架下，深入研究了 de Rham 外微分的 Witten 形变与拓扑不变量之间的相互作用与内在联系。

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