Hypoelliptic Calculus, Noncommutative Geometry and CR Related Geometries

建议的研究包括3个主要项目。第一项旨在利用非对易几何框架来重新表述CR和接触几何中的指数公式。它也很好地符合Fefferman和Stein等人的长期计划，目的是将Kohn-Rossi的亚椭圆分析与底层流形的CR几何联系起来。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。与Henri Moscovici合作研究了严格伪凸域的一个指数公式，该公式将dbar-Neuman问题的次椭圆分析与域的几何及其边界联系起来。另一个目的是从非交换几何的角度研究具有Fefferman度量的洛伦兹流形，这是通向一般非交换洛伦兹流形几何研究的第一步。第二个项目是与George Marinescu的一个联合项目，它涉及到获得Demaillin的全纯Morse不等式的CR类似，从而得到CR和复流形的嵌入定理。最后一项建议在等价正则Carnot-Carath‘eodory流形上建立几何适应的亚椭圆演算，以便能够在这种情况下利用非对易几何框架来解决接触四元数流形上的Yamabe问题。这一建议的更广泛的影响是在两个层面上。第一个层次是在数学范围内，但不在非对易几何的范围内，这是PI的主要领域。一方面，它被建议利用Connes的非对易几何框架来解决与CR、接触和复流形有关的几何问题。另一方面，本文(III)部分的拟微分工具将为研究亚椭圆偏微分方程组提供有力的工具。第二个层次是其他科学的层次。首先，通过对洛伦兹几何的说明，这一提议旨在为统一量子力学和引力的令人信服的计划做出贡献，因此应该有助于更好地理解宇宙。其次，亚椭圆型偏微分方程组出现在许多科学领域，例如物理、工程、金融和机器人。因此，开发研究亚椭圆型偏微分方程组的工具应该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.