Geometric Function Theory in Several Complex Variables

这个项目有两个主要的研究方向。第一个是在解析泛函或超函数意义下关于一般边值理论的工作的继续。在目前的状态下，这一理论澄清了函数边值研究中的几个问题，特别是定义的内在性质。他们在全球背景下澄清的观点需要在当地版本中进一步研究。在非真实分析背景下的理论仍有待完成。第二个方向是多势理论和复杂几何的交汇。一个仍然遥远的目标是将Poletsky的圆盘理论可能扩展到几乎复杂的流形的情况，它在数学中的作用正在迅速增长。该方法的主要研究领域是几个复变量。这一领域因其与数学的许多领域的联系而丰富。特别是在几个复变量和部分微分方程理论之间有着显著的相互作用，该理论本身就是研究物理和工程中许多过程的最基本的工具。上面讨论的第一个方向正是在那个背景下。在应用上，定量研究往往被定性研究所取代，要么是因为定量研究太难，要么是因为定性研究实际上更有启发性。这是拓扑学必不可少的领域。在几个复杂变量和拓扑学之间存在着几个重要的联系。上面讨论的第二个方向符合这个背景。变形理论是一个重要的主题，对几乎复杂结构的研究已经发挥了至关重要的作用。

There are two main research directions in this project.The first one is the continuation of the work on a general theory of boundary values, in the sense of analytic functionals or hyperfunctions. In its present state this theory has clarified several points in thestudy of boundary values of functions, in particularviz the intrinsic character of the definitions. Pointswhich they have clarified in the global setting needfurther study in the local version. A theory in the non real analytic setting is still to be done. The second direction is at the confluent of pluri-potentialtheory and complex geometry. A still remote goal isthe possible extension of Poletsky's theory of disksto the case of almost complex manifolds, whose rolein Mathematics is rapidly growing. It also leadsto the study of approximate solutions to the d-baroperator.The primary field of this proposal is SeveralComplex Variables. This field is rich due to itsconnections to many fields in Mathematics.There is in particular a remarkable interplay betweenSeveral Complex Variables and the theory ofPartial Differential Equations, which itself isthe most basic tool in the study of many processesin Physics and Engineering. The first direction discussedabove fits in that setting. In applications quantitative studies have often to be replaced by qualitative studies, either because thequantitative study is too difficult or because thequalitative study is in fact more illuminating. Thisis where the field of Topology is essential.Several important connections exist between SeveralComplex Variables and Topology. The second directiondiscussed above fits in that setting. Deformation theoryis a major theme, in which the study of almost complexstructures has already played a crucial role.