Final Report Summary - ANALYSISDIRAC (The analysis of the Dirac operator: the hypoelliptic Laplacian and its applications)
The theory of the hypoelliptic Laplacian predicts two things. First, it is possible to interpolate between the Laplacian and the geodesic flow, and this interpolation preserves fundamental quantities, connected with the spectrum of the Laplacian, that characterises propagation of waves in the considered geometric bodies. In certain cases, the full spectrum is preserved. On flat spaces, the hypoelliptic Laplacian is known as a Fokker-Planck operator. The PI discovered that such a known object could precisely produce otherwise unknown interpolation properties. From a dynamical point of view, the interpolation parameter corresponds to a mass, which varies from zero (for Brownian motion) to infinity (for the geodesic flow), the interpolation of dynamical systems being completely explicit.
The mathematics of the program consisted in putting together the geometric, algebraic, analytic and probabilistic intuitions coming from the above model, and to derive all the possible consequences of the above theory, by associating it to index theory, a cornerstone of analysis and topology. Among the outcomes of the project, there have been solutions to various problems connected with representation theory, complex geometry, and dynamical systems.