Analysis and CR Geometry in Several Complex Variables

在每个微积分课上，一个基本的研究对象是导数：它是一个描述函数变化率的量。例如，位置的导数是速度，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，也是数学在物理学和其他科学、经济学和统计学中的许多应用的基础。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，它们描述了世界：牛顿的运动定律，麦克斯韦的电磁辐射方程，量子力学中的薛定谔方程都是微分方程的例子。这个导数可以直接推广到复值函数，在这种情况下它被称为复导数。然而，为了使一个复值函数的复导数存在，该函数必须满足一个特定的微分方程。换句话说，复可微函数具有额外的结构，这种额外的结构为广泛的学科提供了深远的应用，包括数学、工程、物理、化学和经济学的各个领域。本文研究多变量复可微函数。增加额外的变量会带来重大的新挑战，但对数学和科学的应用是众多多样的，而且正如PI所相信的那样，意义深远。在一些复杂的变量中，理解dbar-Neumann和Kohn laplacian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类别中的问题。与阿肯色大学的Phillip Harrington一起，PI将证明非伪凸和/或无界域上的dbar-Neumann算子和复Green算子的存在性和规律性结果。由于技术原因，这些问题超出了有界伪凸情况的范围和工具，PI将开发新的方法来解决这些问题。在伪凸情况下，可解性是很容易理解的，问题集中在边界类型和解的正则性之间的关系上。在有限型的情况下，PI和亚利桑那大学的Albert Boggess正在合作建立二次子流形上复杂热核的点估计。很少有可以直接计算的例子，二次子流形提供了一个大的这样的类，包括更高余维的例子，所有这些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也对指数平面情况下（切向）Cauchy-Riemann方程解的规则性感兴趣。他正在与新加坡国立大学的Khanh Tran合作，为Bergman和Szego核以及dbar-Neumann和复杂的Green算子建立L^p和Holder边界。

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.