Analysis and CR Geometry in Several Complex Variables

多复杂变量的分析与CR几何

• 批准号: 1405100
• 负责人: Andrew Raich
• 金额: $ 13.33万
• 依托单位: University of Arkansas
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405100&HistoricalAwards=false
• 关键词: Analysis; CR; Geometry; Several; Complex

A basic object of study in every calculus class is the derivative: it is a quantity that describes the rate of change of a function. For example, the derivative of position is velocity, and in economics, the code word for derivative is marginal. The derivative is a foundational concept in mathematics and the basis for much of the application of mathematics to physics and the other sciences, economics, and statistics. In applications, equations often contain functions and their derivatives. Such equations are called differential equations, and they describe the world: Newton's Laws of Motion, Maxwell's equations for electromagnetic radiation, Schrodinger's equations in quantum mechanics are all examples of differential equations. The derivative has a straight forward generalization to complex valued functions, and it is called the complex derivative in this situation. In order for the complex derivative of a complex valued function to exist, however, the function must satisfy a particular differential equation. In other words, complex differentiable functions have extra structure, and this extra structure provides far-reaching applications to a wide variety of subjects, including every areas of mathematics, engineering, physics, chemistry, and economics. This proposal investigates complex differentiable functions of several variables. Adding in additional variables creates significant new challenges, but the applications to mathematics and science are numerous and varied, and, as the PI believes, profound.In several complex variables, understanding the dbar-Neumann and Kohn Laplacians is a major driver of research questions. The PI and his collaborators will examine questions in the both the pseudoconvex and nonpseudoconvex categories. Together with Phillip Harrington of the University of Arkansas, the PI will prove existence and regularity results for the dbar-Neumann and complex Green operators on nonpseudoconvex and/or unbounded domains. These questions are beyond the scope and tools in the bounded pseudoconvex case for technical reasons, and the PI will develop new approaches to tackle these questions. In the pseudoconvex case, solvability is well-understood, and the questions instead focus on the relationship between the type of the boundary and the regularity of solutions. In the finite type case, the PI and Albert Boggess of the University of Arizona have an ongoing collaboration to establish pointwise estimates for the complex heat kernel on quadric submanifolds. There are few examples which are accessible to direct computation, and quadric submanifolds provide a large such class, including examples of higher codimension, all of which have a formulas that can be exploited. Quadric submanifolds have a very regular curvature structure, and the PI is also interested in the regularity of solutions to the (tangential) Cauchy-Riemann equations in the exponentially flat case. He has an ongoing collaboration with Khanh Tran of the National University of Singapore to establish L^p and Holder bounds for the Bergman and Szego kernels as well as the dbar-Neumann and complex Green operators.

每门微积分课程的一个基本研究对象是导数：它是一个描述函数变化率的量。例如，头寸的导数是速度，而在经济学中，导数的代号是边际。导数是数学中的一个基本概念，也是许多数学应用于物理和其他科学、经济学和统计学的基础。在应用中，方程通常包含函数及其导数。这样的方程被称为微分方程组，它们描述了世界：牛顿运动定律，电磁辐射的麦克斯韦方程，量子力学中的薛定谔方程都是微分方程组的例子。导数是对复值函数的直接推广，在这种情况下称为复数导数。然而，为了使复值函数的复导数存在，该函数必须满足特定的微分方程式。换句话说，复可微函数具有额外的结构，这种额外的结构为包括数学、工程、物理、化学和经济的各个领域的各种学科提供了深远的应用。这项建议研究的是多变量的复可微函数。添加额外的变量会带来重大的新挑战，但在数学和科学中的应用是多种多样的，而且正如PI所认为的那样，是深刻的。在几个复杂的变量中，理解dbar-Neumann和Kohn Laplace是研究问题的主要驱动力。PI和他的合作者将研究伪凸和非伪凸范畴中的问题。与阿肯色大学的Phillip Harrington一起，PI将证明非伪凸和/或无界区域上dbar-Neumann和复Green算子的存在性和正则性结果。由于技术原因，这些问题超出了有界伪凸情形的范围和工具，PI将开发新的方法来解决这些问题。在伪凸的情况下，可解性是很容易理解的，而问题的焦点是边界类型和解的正则性之间的关系。在有限类型的情况下，亚利桑那大学的Pi和Albert Boggess正在合作建立二次子流形上的复热核的逐点估计。可以直接计算的例子很少，二次子流形提供了一大类这样的例子，包括更高余维的例子，所有这些例子都有一个可以利用的公式。二次子流形具有非常规则的曲率结构，PI也对(切向)Cauchy-Riemann方程在指数平坦情况下解的正则性感兴趣。他正在与新加坡国立大学的Khanh Tran合作，为Bergman核、Szego核以及dbar-Neumann和Complex Green算子建立L^p和Holder界。