Hypoelliptic Calculus, Noncommutative Geometry and CR Related Geometries

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

The proposed research consists of 3 main items. The first item aims to make use of the noncommutative geometry framework in order to reformulate the index formula in CR and contact geometry. It also fits nicely with the long term program of Fefferman and Stein and others aiming to relate the subelliptic analysis of the Kohn-Rossi to the CR geometry of the underlying manifold. Moreover, it has some overlap with recent work of Epstein-Melrose-Mendoza. This project has two natural follow-ups. One in collaboration with Henri Moscovici dealing with an index formula for strictly pseudoconvex domains relating the subelliptic analysis of the dbar-Neuman problem with the geometry of the domain and its boundary. The other one aims to study Lorentzian manifolds with Fefferman metric from a noncommutative geometric viewpoint, hence is a first step towards a general noncommutative geometric study of Lorentzian manifolds. The second item is a joint project with George Marinescu and deals with obtaining CR analogues of the holomorphic Morse inequalities of Demailly in order to get embedding theorems for CR and complex manifolds. The last item proposes to to develop a geometrically adapted hypoelliptic calculus on equiregular Carnot-Carath\'eodory manifolds in order to be able to make use of the noncommutative geometry framework in this setting and to solve the Yamabe problem on contact quaternionic manifolds. The broader impact of this proposal is at two levels. The first level is within mathematics but outside the scope of noncommutative geometry which is the primary field of the PI. On the one hand, it is proposed to make use of the framework of Connes' noncommutative geometry for solving geometric problems related to CR, contact and complex manifolds. On the other hand, the pseudodifferential tools of the part (iii) of this proposal will provide a powerful tool for studying hypoelliptic PDE's. The second level is that of other sciences. First, at illustrated by its part on Lorentzian geometry this proposal aims to contribute to the compelling program of unifying quantum mechanics and gravity, hence should contribute to a better understanding of the Universe. Second, hypoelliptic PDE's arise in many fields of science, for example physics, engineering, finance, and robotics. Therefore developing tools for studying hypoelliptic PDE's should help making progress in these fields.