Hypoelliptic equations and geometry

In linear partial differential equations, hypoellipticity is the condition that if Df=g, with g smooth, then f is necessarily smooth too. The best-known hypoelliptic equations are the elliptic equations, which are characterized by an isotropy property that can be readily checked point-by-point. Various more general point-by-point sufficiency criteria for hypoellipticity have been studied, beginning with famous work of Lars Hormander in the 1960’s. Quite recently these criteria have been used to formulate and prove index theorems for hypoelliptic operators in the spirit of the famous Atiyah-Singer index theorem, and to apply index-theoretic techniques to new problems.  I shall give a survey of some of these developments. Of special interest is Jean-Michel Bismut's hypoelliptic Laplacian, which is a remarkable family of operators interpolating between the Laplace operator on a Riemannian manifold and the geodesic flow on its tangent bundle.