Geometric Function Theory in Several Complex Variables

摘要：Rosay教授的首要目标是与E.L. Stout合作，继续研究边界值（超功能意义上的）理论。事实上，两人已经开始发展一种全新的边界值理论。这是函数理论的研究，与几个复杂变量和偏微分方程有很强的联系。Rosay的项目还需要继续他对复杂n空间生物全纯态动力学的各种问题的研究（与P. Ahern和F. Forstneric合作）。其他几个主题（例如，柯西问题的唯一性，非线性d杆方程）也将被探讨。罗赛的主要研究领域是复数变量函数理论。该领域是数学中的核心领域，与近似理论和偏微分方程等领域有直接联系，这些领域与直接应用非常接近（例如，图像处理，建模，科学计算）。应该强调的是，数学结果乍一看可能相当抽象，实际上可以通过创造新的计算工具的可能性而导致非常实际的应用。（即使像基础科学中通常的那样，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变为更实用或至少适用的东西是可能的，人们只需要看看当前项目的例子。斯托特和罗赛最近给出了一个全新的证明，证明了一个非常经典和基本的定理，即佩利-维纳定理。这个新的证明不太可能在经典的环境中找到，因为这种方法只有在广义和更抽象的“解析泛函”环境中才变得自然。然而，尽管它是在广义的背景下产生的，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联系得更紧密。从某种角度来看，这种发展并不令人惊讶。几个世纪以来，人们都知道，许多基本数学现象（如幂级数展开的收敛性）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，许多计算最好是用复数变量完成的，即使人们对用实数表示的最终答案感兴趣。当然，正如在任何其他科学中一样，没有人能声称在复杂分析领域的所有研究都将导致应用，而发现的应用往往是完全出乎意料的。但复杂分析显然为纯粹数学家和应用数学家提供了一个必不可少的工具。

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.