## Final Report Summary - ANALYSISDIRAC (The analysis of the Dirac operator: the hypoelliptic Laplacian and its applications)

The ERC project was to bring together various fields of mathematics: geometry, analysis, probability theory, mathematical physics, and dynamical systems, by showing the potential of a new object introduced by the PI, the hypoelliptic Laplacian, that has the potential to interpolate between the classical Laplacian, and the geodesic flow. The classical Laplacian is the most important partial differential operator in mathematical physics: the Laplacian of a function is the sum of the second derivatives of this function. The Laplacian appears in the heat equation, which itself describes the propagation of Brownian motion, which models the erratic motion of particles of negligible mass. The geodesic flow is used to describe the motion of objects with a large mass.

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.

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.