The Regularity of Cauchy-Riemann Mappings and Solutions of Systems of Nonlinear Partial Differential Equations

该研究项目的目的是研究多复变量偏微分方程组的解。 这项研究的重要数学应用包括建立的解决方案的规律性和解决方案的收敛性，只有泰勒级数系数是已知的。 物理系统的重要应用包括统计力学和气象学中出现的方程。 例如，平面二聚体模型是空间中某些二维表面的统计力学模型，它是整数上简单随机游走的高维推广。这类曲面的极限形状由与本项目中研究的复Burgers方程密切相关的偏微分方程来模拟。同样的方程也适用于模拟某些大气现象行为的准地转方程的研究。该项目的主要部分涉及的几何条件，这意味着一个充分光滑的CR映射之间的两个柯西-黎曼（CR）流形是光滑的流形时，和真实的分析时，流形是真实的分析的理解问题。密切相关的问题，将被调查包括当地和微观局部规则的CR功能抽象CR流形。可以使用的工具包括一个新的家庭的傅立叶-Bros-Iagolnitzer变换，解析盘，和方法开发的全纯扩展理论的CR功能的超曲面。

The aim of this research project is to study the solutions of systems of partial differential equations that arise in several complex variables. Important mathematical applications of this research include establishing the regularity of solutions and the convergence of solutions for which only Taylor series coefficients are known. Important applications to physical systems include equations that arise in statistical mechanics and meteorology. For example, the planar dimer model is a statistical mechanical model of certain two-dimensional surfaces in space that is a higher dimensional generalization of the simple random walk on integers. The limit shape of such surfaces is modeled by a partial differential equation closely related to the complex Burgers' equation studied in this project. The same equation is also relevant in the study of the quasi-geostrophic equations that model the behavior of certain atmospheric phenomena. The main part of the project involves the problem of understanding geometric conditions that imply that a sufficiently smooth CR mapping between two Cauchy-Riemann (CR) manifolds is smooth when the manifolds are smooth, and real analytic when the manifolds are real analytic. Closely related problems that will be investigated include the local and microlocal regularity of CR functions on abstract CR manifolds. The tools that may be used include a new family of Fourier-Bros-Iagolnitzer transforms, analytic discs, and the methods developed in the theory of holomorphic extendability of CR functions on hypersurfaces.