## Noncommutative Geometry Seminar

**Date: ** September 20, 2023

**Time: ** 2:00PM - 3:00PM

**Location: ** BLOC 302

**Speaker: **Shiqi Liu, TAMU

**Title: ***Introduction to the hypoelliptic Laplacian and Bismut’s formula.*

**Abstract: **Invented by Jean-Michel Bismut, the hypoelliptic Laplacian is the centerpiece of a new type of index theory. It provides a remarkable trace formula (Bismut’s formula). In the circle case, it is an application of Possion summation formula. In the compact Lie group case, it becomes Frenkel’s formula. In the symmetric space case, it provides an explicit calculation of Selberg trace formula. In this talk, I will give an overview of the hypoelliptic Laplacian, and briefly explain the analytical proof of Bismut’s formula. Recently, using noncommutative geometry, we developed a series of new techniques in analysis to reduce the difficulty of the proof. This is joint work with N. Higson, E. MacDonald, F. Sukochev, and D. Zanin.