Partial differential equations in conformal and CR geometry

本计画将研究共形几何与CR几何中的高阶Q曲率方程。这些方程与共形几何中的Gauss-Bonnet积分和CR几何中的Szego核中的对数项密切相关。主要研究者以前的工作已经确立了这些方程在帮助理解问题中空间几何方面的重要性。目前的项目采取进一步的步骤，探索这些Q-曲率方程的分析发展和基本问题有关的积极性的基本运营商。在CR几何的情况下，研究试图将Q曲率方程与CR结构的质量概念联系起来。同样，潜在操作者的积极性是调查的焦点。该项目的主要研究人员率先使用涉及大量微分的自然偏微分方程作为分析工具来研究四维空间的几何。该提案的科学部分是这一发展的自然延续，重点是验证物理学家创造的时空宇宙的某些几何图像。 该项目的很大一部分将涉及培训下一代数学家作为数学研究人员和教师。主要研究人员还将继续他们过去的记录，指导一些成功的女性数学家。

This project will study the higher order Q-curvature equations in conformal geometry and CR geometry. These equations are closely related to the Gauss-Bonnet integral in the case of conformal geometry and to the logarithm term in the Szego kernel in the case of CR geometry. The previous work of the principal investigators has established the significance of these equations in helping to understand the geometry of the spaces in questions. The current project takes further steps to explore the analytic development of these Q-curvature equations and basic questions relating to the positivity of the underlying operators. In the case of CR geometry, the research seeks to relate the Q-curvature equation to the notion of mass for a CR structure. Again the positivity of the underlying operator is the focus of the investigations. A successful outcome would lead to a strong characterization of the standard sphere in complex space.The principal investigators of this project have pioneered the use of a natural partial differential equation involving a larger number of differentiations as analytic tools to study the geometry of four-dimensional spaces. The scientific part of the proposal is a natural continuation of this development, where the focus is on verifying certain conjectural pictures of the space-time universe created by physicists. A good portion of the project will involve the training of the next generation of mathematicians as researchers and teachers of mathematics. The principal investigators will also continue their past record of mentoring a number of successful women mathematicians.