报告人：Junrong Yan （University of California at Santa Barbara）
时间：12月12日，周六，早上10-11点
地点：腾讯会议，会议 ID：688 419 891

Abstract: Asymptotic expansions of heat kernel of Schr\"odinger-type operators, as well as its trace on noncompact spaces, are rarely explored, and even for cases as simple as $\mathbb{C}^n$ with (quasi-homogeneous) polynomials potentials, it's already very complicated to compute. Motivated by path integral formulation of the heat kernel, we introduced parabolic distance, which also appeared in Li-Yau's famous work on Harnack inequality. With the help of parabolic distance, we could derive a nice point-wise asymptotic expansion of the heat kernel with a strong remainder estimate. In particular, we get an asymptotic expansion of the heat kernel of Witten Laplacian $\Box_{Tf}$ induced by $d+Tdf\wedge$, where $T>0$ is the deformation parameter. When the deformation parameter of Witten deformation and time parameter are coupled, we derive an asymptotic expansion of trace of heat kernel for small-time t, and obtain a local index theorem. Also, we invented a new rescaling technique to write down the local index density explicitly.