Loop Spaces and the hypoelliptic Laplacian

Abstract

The purpose of this paper is to introduce some ideas that motivated the construction of the hypoelliptic Laplacian as a deformation of Hodge theory, which interpolates between Hodge theory and the geodesic flow. Results obtained with Lebeau on the analysis of the hypoelliptic Laplacian are also presented.

The motion of a random string

“…To some extent, we can probably learn more from the proof of Theorem 1.1 than from the result itself, although a more systematic study of the properties of the limiting process, especially in the small noise limit, could be very interesting, especially in view of [Wit82,Bis08]. We will assume in the sequel that the reader has at least some familiarity with the theory of regularity structures as developed in [Hai14] and surveyed for example in [Hai15,CW15].…”

Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau, and Shen)

“…We hope this article can be used as an invitation to the original papers [6,7,8,12,14,16] and to several surveys on this topic [9,10,11,13,15,17] and [47].…”

The Mathematical Work of V. K. Patodi

“…As for the most significant recent advance of the (local) index theory, we mention the groundbreaking theory of hypoelliptic Laplacians developed by Bismut and his collaborators in [20], [21], [32], [23] and [25], of which two surveys are given in [22] and [24]. The motivation of this theory comes in part from an attempt to generalize the methods and results developed in [34] to the loop space of the underlying smooth manifold.…”