Geometric Function Theory in Several Complex Variables

多复变量的几何函数论

• 批准号: 9877194
• 负责人: Jean-Pierre Rosay
• 金额: $ 7.88万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9877194&HistoricalAwards=false
• 关键词: Geometric; Function; Theory; Several; Complex

Proposal: DMS-9877194Principal Investigator: Jean-Pierre RosayAbstract: The first goal of Professor Rosay is to continue his study, in collaboration with E.L. Stout, of the theory of boundary values (in the sense of hyperfunctions). Indeed, the two have started developing an entirely new theory of boundary values. This is research in function theory, with strong connections to several complex variables and partial differential equations. Rosay's project also entails the continuation of his study of various questions on the dynamics of biholomorphisms of complex n-space (collaborations with P. Ahern and F. Forstneric). Several other topics (e.g., uniqueness in the Cauchy problem, the nonlinear d-bar equation) will also be explored.Rosay's main area of activity is the theory of functions of several complex variables. This field is a central one in mathematics, with direct ties to such areas as approximation theory and partial differential equations that are very close to immediate applications (e.g., image processing, modeling, scientific computation). It should be stressed that mathematical results that may at first glance look quite abstract can in fact lead to very practical applications, by creating the possibility of new computational tools. (Even if, as usual in fundamental science, benefits may take many years to surface.) In order to illustrate how a shift from what seems to be purely theoretical to something more applied or at least applicable is possible, one need look no further than the present project for an example. Stout and Rosay recently gave a totally new proof of a very classical and fundamental theorem known as the Paley-Wiener theorem. This new proof was unlikely to be found in the classical setting, because the approach only became natural in the generalized and more abstract setting of "analytic functionals." However, despite the fact that it arose in a generalized setting, it turns out that the new proof is constructive in nature, and therefore much more closely linked to the possibility of practical computations. From one perspective, this development is not so surprising. It has been known for centuries that the full explanation of many basic mathematical phenomena (as fundamental as the convergence of power series expansion) is to be found only in the realm of complex numbers. It has likewise been known for centuries that many computations are best done with complex variables, even if one is interested in a final answer expressed in terms of real numbers. Of course, no more so than in any other science can one claim that all the research in the field of complex analysis will lead to applications, and often the applications that are found will come as complete surprises. But complex analysis clearly furnishes an essential tool to both pure and applied mathematicians.

提案：DMS-9877194 首席研究员：Jean-Pierre Rosay 摘要：Rosay 教授的首要目标是与 E.L. 合作继续他的研究。斯托特，边界值理论（在超函数的意义上）。事实上，两人已经开始发展一种全新的边界值理论。这是函数论的研究，与多个复变量和偏微分方程有密切的联系。 Rosay 的项目还需要继续研究有关复 n 空间双全纯动力学的各种问题（与 P. Ahern 和 F. Forstneric 合作）。还将探讨其他几个主题（例如，柯西问题的唯一性、非线性 d 杆方程）。Rosay 的主要活动领域是多个复变量的函数理论。该领域是数学的核心领域，与逼近论和偏微分方程等领域有直接联系，这些领域非常接近直接应用（例如图像处理、建模、科学计算）。应该强调的是，乍一看可能相当抽象的数学结果实际上可以通过创造新的计算工具的可能性而带来非常实际的应用。 （即使像基础科学中通常那样，好处可能需要很多年才能显现出来。）为了说明如何从看似纯粹的理论转向更实用或至少适用的东西是可能的，我们不需要再进一步看一下以当前项目为例。斯托特和罗赛最近对一个非常经典和基本的定理（称为佩利-维纳定理）给出了全新的证明。这种新的证明不太可能在经典环境中找到，因为这种方法只有在“分析泛函”的广义和更抽象的环境中才变得自然。然而，尽管它是在广义背景下出现的，但事实证明，新的证明本质上是建设性的，因此与实际计算的可能性更加紧密地联系在一起。从某个角度来看，这一发展并不令人意外。几个世纪以来，人们都知道，许多基本数学现象（如幂级数展开式的收敛性一样基本）的完整解释只能在复数领域中找到。同样，几个世纪以来人们都知道，许多计算最好使用复数变量来完成，即使人们对以实数表示的最终答案感兴趣。当然，与任何其他科学一样，我们也不能声称复杂分析领域的所有研究都会带来应用，而且通常发现的应用会完全令人惊讶。但复分析显然为纯粹数学家和应用数学家提供了重要的工具。