Noncommutative Geometry Seminar

Date: October 6, 2021

Time: 2:00PM - 3:00PM

Location: Zoom

Speaker: Jesus Sanchez Jr, Penn State

Title: The Geometry of Mehler's Kernel

Abstract: Mehler's Kernel made its first appearance in Index Theory through the work of Ezra Getzler in his computation of the index of a Dirac Operator. The appearance of Mehler's Kernel in this approach is through the introduction of a symbol calculus which refines the usual symbol calculus of differential operators and smoothing operators. In a different approach to computing the index of a Dirac Operator, Nicole Berline and Michele Vergne study heat flow on the principal Spin bundle and show that the corresponding Index density arises naturally by studying the local geometry in this setting. What we will show is that we can extend the insight of Berline and Vergne to fully recover Mehler's Kernel and give geometric insight into the curvature terms appearing within the kernel. This will give a more unified treatment of these two seemingly different proofs of the local index theorem for Dirac Operators.