Geometric Function Theory in Several Complex Variables

这个项目有两个主要的研究方向：第一个是继续关于解析泛函或超函数意义上的一般边值理论的工作。在其目前的状态，这一理论已经澄清了几个点的研究边界值的功能，特别是viz的内在性质的定义。他们在全球背景下阐明的观点需要在本地版本中进一步研究。在非真实的分析环境中的理论仍有待完成。第二个方向是复势理论与复几何的融合。一个仍然遥远的目标是可能的扩展Poletsky的理论disks的情况下几乎复杂的流形，其作用在数学正在迅速增长。这一建议的主要领域是多复变函数。这一领域是丰富的，因为它与数学的许多领域都有联系，特别是在多复变量和偏微分方程理论之间有着显著的相互作用，偏微分方程理论本身就是研究物理和工程中许多过程的最基本工具。上面讨论的第一个方向适合这种情况。在实际应用中，定量研究往往被定性研究所取代，这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。这就是拓扑学的本质所在。在多复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适合这种设置。变形理论是一个重要的研究课题，其中对几乎复杂结构的研究已经起到了至关重要的作用。

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.