Hypoelliptic Calculus, Noncommutative Geometry and CR Related Geometries

拟议的研究包括3个主要项目。第一个项目的目的是利用非交换几何的框架，以重新制定的CR和接触几何的指数公式。 它也很好地适合与长期计划的Febrilman和斯坦和其他人的目的是有关的次椭圆分析的Kohn-Rossi的CR几何的thebrilliant流形。此外，它与爱泼斯坦-梅尔罗斯-门多萨最近的工作有一些重叠。这个项目有两个自然的后续行动。一个在合作与亨利莫斯科维奇处理一个指数公式严格pseudoconvular域有关subellipticanalysis的dbar-Neuman问题的几何形状的域及其边界。另一个目的是从非对易几何的角度研究具有费曼度量的Lorentz流形，从而为Lorentz流形的一般非对易几何研究迈出了第一步。第二个项目是与乔治Marinescu和处理获得CR类似的全纯莫尔斯不等式Demaily为了得到嵌入定理CR和复杂的流形。最后一项建议发展一个几何适应的亚椭圆演算equigurarCarnot-Carath\'eodory流形，以便能够利用非交换的几何框架，在此设置和解决Yamabe问题接触四元数流形。第一个层次是在数学范围内，但在非交换几何的范围之外，这是PI的主要领域。一方面，提出利用Connes的非对易几何框架来解决与CR、切触和复流形有关的几何问题。另一方面，本文（iii）部分的伪微分工具将为亚椭圆型偏微分方程的研究提供一个强有力的工具。首先，通过洛伦兹几何部分的说明，这个命题旨在为统一量子力学和引力的引人注目的计划做出贡献，因此应该有助于更好地理解宇宙。其次，亚椭圆偏微分方程出现在许多科学领域，如物理学、工程学、金融学、机器人学等。因此，开发亚椭圆偏微分方程的研究工具，将有助于在这些领域取得进展。

The proposed research consists of 3 main items. The first item aims to make use of the noncommutative geometry framework in order toreformulate the index formula in CR and contact geometry. It also fits nicely with the long term program ofFefferman and Stein and others aiming to relate the subelliptic analysis of the Kohn-Rossi to the CR geometry of theunderlying manifold. Moreover, it has some overlap with recent work ofEpstein-Melrose-Mendoza. This project has two natural follow-ups. One in collaboration withHenri Moscovici dealing with an index formula for strictly pseudoconvex domains relating the subellipticanalysis of the dbar-Neuman problem with the geometry of the domain and its boundary. The otherone aims to study Lorentzian manifolds with Fefferman metric from a noncommutative geometric viewpoint, henceis a first step towards a general noncommutative geometric study of Lorentzian manifolds. The second item is a joint project with George Marinescu anddeals with obtaining CR analogues of the holomorphic Morse inequalities of Demaillyin order to get embedding theorems for CR and complex manifolds. The last item proposes toto develop a geometrically adapted hypoelliptic calculus on equiregularCarnot-Carath\'eodory manifolds in order to be able to make use of the noncommutative geometry framework in thissetting 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 ofnoncommutative geometry which is the primary field of the PI. On the one hand, it is proposed to make use ofthe framework of Connes' noncommutative geometry for solvinggeometric problems related to CR, contact and complex manifolds. On the other hand, the pseudodifferential toolsof 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 proposalaims to contribute to the compelling program of unifying quantum mechanics andgravity, hence should contribute to a better understanding of the Universe. Second, hypoelliptic PDE's arise inmany fields of science, for example physics, engineering, finance, androbotics. Therefore developing tools for studying hypoelliptic PDE's should help making progress in these fields.