Linear and nonlinear problems in CR manifolds

本项目主要研究一阶线性偏微分方程组和利用cr流形分析结果和方法的非线性问题。所研究的方程组是由局部可积结构产生的，一个很好的模型是cr流形。有潜在用途的工具包括Baouendi-Treves近似方案，来自微局部分析的思想（例如FBI变换），解析盘，Sussmann轨道的几何，以及在半刚性cr -流形上cr -函数的全纯可扩展性理论中发展的技术。本项目研究的潜在应用领域包括偏微分方程和几何。一些非线性偏微分方程在气象学中出现，试图模拟和解释大气现象。同样的方程描述了弹性固体中非线性波振幅的演化。理解非线性波的传播和相互作用是物理学中一个非常重要和具有挑战性的问题。在几何学和物理学中，这些方程出现在对空间表面形状的研究中，这些形状在一类随机表面模型中作为极限形状出现，极限形状使表面张力最小化。该研究活动也有望为研究生带来有趣的问题。

This project focuses on the study of systems of linear first-order partial differential equations and nonlinear problems that exploit the results and methods of analysis on CR-manifolds. The systems of equations under investigation arise from locally integrable structures, a good model for which is a CR-manifold. The tools that are of potential use include the Baouendi-Treves approximation scheme, ideas from microlocal analysis (e.g., the FBI transform), analytic discs, the geometry of Sussmann's orbits, and the techniques developed in the theory of holomorphic extendability of CR-functions on semirigid CR-manifolds. Potential application areas for the research in this project include partial differential equations and geometry. Some of the nonlinear partial differential equations under consideration surface in meteorology, in attempts to model and explain atmospheric phenomena. The same equations describe the evolution of the amplitudes of nonlinear waves in elastic solids. Understanding the propagation and interaction of nonlinear waves is a very important and challenging problem in physics. In geometry and physics, these equations arise in the study of the shapes of surfaces in space that turn up as limit shapes in a class of random surface models, limit shapes that minimize surface tension. The research activity is also expected to generate interesting problems for graduate students.