Research in Several Complex Variables

这个项目结合了两个非常强大的数学思想--复数和微积分。复数包括负一的平方根之类的量。微积分研究物理量或几何量在空间或时间上的变化。这些思想的结合(称为复分析)导致了关于光滑变化的复数(称为全纯函数)的深远而美丽的结果。人们可以用这些想法来模拟各种自然现象，如电吸引或液体运动。这些考虑在数学的其他部分也有令人惊讶的结果，如素数的性质，高维空间的几何，以及用于描述许多物理过程的方程(称为偏微分方程)的研究，如热传导和波的传播。本研究项目研究了在高维空间中，当一个函数逼近定义该函数的区域的边界时，全纯函数的行为。这位研究人员还将继续成功地指导本科生在相关主题上的研究。在强伪凸域上，多变量全纯函数的行为是众所周知的。在这个项目中，目标是研究这种结果的扩展到新的和更一般的域类型。正在研究的问题包括非齐次柯西-黎曼方程的解的估计，理解当没有这样的估计时会发生什么，以及全纯函数的边界行为。要研究的区域包括分段光滑域(特别是积域)、复流形中的Levi-Flat Stein域和非伪凸域，既有无“洞”。这项研究既包括对一般问题的调查，也包括对特定例子的仔细研究，这些例子可能会显示与这些领域相关的意外现象。在这个项目中用来研究这些问题的工具包括先验估计、积分公式和基于鞘理论的代数方法，以及来自数学的其他部分的见解，如偏微分方程组、泛函分析和微分几何。

This project combines two very powerful ideas of mathematics -- that of complex numbers and that of the calculus. Complex numbers include quantities such as the square root of negative one. The calculus studies how physical or geometric quantities vary in space or time. The combination of these ideas (called complex analysis) leads to far-reaching and beautiful results about smoothly varying complex quantities (called holomorphic functions). One may use these ideas to model various natural phenomena such as electrical attraction or the motion of liquids. These considerations also have surprising consequences in other parts of mathematics, such as the properties of prime numbers, the geometry of higher dimensional spaces, and the study of equations (called partial differential equations) used to describe many physical processes such as heat conduction and the propagation of waves. This research project studies the behavior of holomorphic functions as one approaches the boundary of the region in higher dimensional space where the function is defined. The investigator will also continue his successful mentoring of undergraduate student research in related topics.The behavior of holomorphic functions of several variables is well-understood on strongly pseudoconvex domains. In this project, the goal is to study the extension of such results to new and more general types of domains. Among the questions under study are estimates for the solutions of the solutions of the inhomogeneous Cauchy-Riemann equations, understanding what happens when there are no such estimates, and the boundary behavior of holomorphic functions. The domains to be studied include piecewise smooth domains (in particular product domains), Levi-flat Stein domains in complex manifolds, and non-pseudoconvex domains, both with and without "holes." The research will involve both the investigation of general questions and the careful study of particular examples, which can exhibit unexpected phenomena associated to these domains. Tools employed in this project to study these problems include a priori estimates, integral formulas, and algebraic methods based on sheaf theory, as well as insights from other parts of mathematics such as partial differential equations, functional analysis, and differential geometry.