Mathematical Sciences: Geometric Function Theory in Several Complex Variables

提案：DMS-9622695 PI: Rosay。Rosay计划研究函数理论中的几个主题；更具体地说，在一些复杂变量和偏微分方程中。他目前正在与E.L. Stout合作研究超功能意义上的边界值的基本问题，试图建立一个非常具体的局部本征理论。他对柯西问题中的唯一性问题非常感兴趣，特别是对路易算子及其微扰。这当然与抽象CR结构的研究有关。关于C^n的自同构和多项式凸性的几个问题也将被研究，特别是与F. Forstneric合作。他目前正在研究与嵌入问题相关的分离（可约）分析集问题。复数变量领域在数学的许多领域都有应用：近似理论、傅立叶分析，以及最重要的偏微分方程。偏微分方程的后一个领域在物理学和工程学中具有中心重要性。事实上，偏微分方程是研究物理和工程中许多过程的天然工具；物理学中的预测和理解以及工程中的设计往往是基于偏微分方程的。因此，对这一领域有深刻的了解是非常重要的。研究偏微分方程的方法有很多，但最重要的方法之一是通过几个复变量。因此，培养几个复杂变量是很重要的。Rosay提出的研究将澄清几个复杂变量中的几个问题，从而推进偏微分方程。一些复杂变量的应用并不像某些数学领域的应用那样直接和直接。然而，数学的历史充满了这样的例子：数学结果的实用价值在其基本发展多年后才得到承认。事实上，最好的数学往往就是这种情况——数学避免深奥和琐碎，而是基于现实，坚持提出并试图回答最基本和最基本的问题。正是由于这个原因，数学必须被视为一项长期投资。

ABSTRACT Proposal: DMS-9622695 PI: Rosay. Rosay plans to work on several topics in function theory; more specifically, in several complex variables and in partial differential equations. He is presently working with E.L. Stout on the basic question of boundary values in the sense of hyperfunctions, trying to develop a very concrete local intrinsic theory. He is extremely interested in questions related to uniqueness in the Cauchy problem, especially for the Lewy operator and its perturbations. This has of course a link with the study of abstract CR structures. Several questions concerning automorphisms of C^n, and polynomial convexity will also be studied, in particular in collaboration with F. Forstneric. He is presently working on the problem of separating (reducible) analytic sets, in connection with embeddings problems. The field of several complex variables has applications to many areas of mathematics: approximation theory, Fourier analysis, and, most importantly, partial differential equations. This latter area of partial differential equations is of central importance in physics and engineering. In fact, partial differential equations are the natural tool for the study of many processes in physics and engineering; prediction and understanding in physics and design in engineering are often based on partial differential equations. It is thus of great importance to have incisive knowledge of this area. There are many approaches to the study of partial differential equations, but one of the most important approaches is via several complex variables. The cultivation of several complex variables is thus important. The research proposed by Rosay will clarify several issues in several complex variables, and thereby advance partial differential equations. The applications of several complex variables are not as direct and immediate as those of some areas of mathematics. The history of mathematics, however, is replete with examples of mathematical results whose utilitarian valu e was recognized only many years after their basic development. In fact, this is often the case with the best mathematics--mathematics that avoids the esoteric and the frivolous, but is grounded in reality and persistently asks and attempts to answer the most basic and fundamental questions. It is for this reason that mathematics must be regarded as a long term investment.