Complex analysis and Geometry in Infinite Dimensions

无限维中的复杂分析和几何

• 批准号: 0700281
• 负责人: Laszlo Lempert
• 金额: $ 46.99万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700281&HistoricalAwards=false
• 关键词: Complex; analysis; Geometry; Infinite; Dimensions

The project's focus is on complex analysis and geometry in infinite-dimensional manifolds. The latter topic especially represents largely uncharted territory. The principal investigator will explore it by considering fundamental results of the finite-dimensional theory and asking how they generalize to the infinite-dimensional setting. In some cases one already has a sense of what the generalization should be, and the challenge is actually to prove it; in other cases even the terms in which to formulate a generalization must be discovered. Two notions are central to the project. One is the concept of a complex loop space. Such a space is obtained by starting with a (finite-dimensional) complex manifold. The collection of all loops in this manifold is an infinite-dimensional complex manifold known as a complex loop space. The other central idea is that of a cohesive sheaf, an infinite-dimensional generalization of the all-important coherent sheaves of the finite-dimensional theory that the principal investigator and a collaborator have recently introduced. A large part of the project is the study of these two notions, especially their confluence: namely, cohesive sheaves over loop spaces, and their cohomology groups. The project will also seek applications in other areas of mathematics for the results that the project expects to obtain.The location of a point or point-mass in three-dimensional space can be described by its coordinates, that is, by three numbers. Accordingly, since Descartes we have known that curves and surfaces in space can be described by functions of three variables. Now to specify the position of a more complicated object, say a Frisbee, the three coordinates of its center of mass do not suffice. The orientation of its axis also has to be given, which will involve two more numbers. In the end, one needs five numbers to specify the position of the Frisbee, and one says that all possible positions of the Frisbee constitute a five-dimensional space (or manifold). The flight of the Frisbee through the air then corresponds to a curve in this manifold. Studying the positions of even more complicated (for example, hinged) objects, one is led to even higher dimensional manifolds, and, if the object lacks any rigidity -- think of a rubber band -- to infinite-dimensional ones. The project is concerned with fundamental properties of such infinite-dimensional manifolds and of the attendant functions of infinitely many variables. A main goal is to understand how local information on these manifolds and functions can be assembled into global information. The mathematical construct that tells us to what extent this is possible is called a sheaf cohomology group, and such groups will be the main objects of the research. Various components of this work have first arisen in other parts of mathematics and in theoretical physics (quantum field theory and string theory), so the project, if successful, should have some relevance to those disciplines. However, this is going to be fundamental research, and the principal investigator does not expect immediate applications outside mathematics. On the other hand, graduate students will be involved in the research. The project will thus contribute to the training of future researchers and educators.

该项目的重点是无限维流形中的复杂分析和几何。后一个主题尤其代表了很大程度上未知的领域。首席研究员将通过考虑有限维理论的基本结果并询问它们如何推广到无限维设置来探索它。在某些情况下，人们已经知道概括应该是什么，而挑战实际上是证明它；在其他情况下，甚至必须发现用来表述概括的术语。该项目的核心有两个概念。一是复循环空间的概念。这样的空间是通过从（有限维）复流形开始获得的。该流形中所有循环的集合是一个无限维复流形，称为复循环空间。 另一个中心思想是凝聚层，这是首席研究员和合作者最近引入的有限维理论中最重要的凝聚层的无限维推广。该项目的很大一部分是研究这两个概念，特别是它们的融合：即循环空间上的内聚滑轮及其上同调群。该项目还将寻求该项目期望获得的结果在其他数学领域的应用。三维空间中的点或点质量的位置可以通过其坐标来描述，即通过三个数字来描述。因此，自笛卡尔以来我们就知道空间中的曲线和曲面可以用三变量的函数来描述。现在要指定一个更复杂的物体（例如飞盘）的位置，其质心的三个坐标是不够的。还必须给出其轴的方向，这将涉及另外两个数字。最后，需要五个数字来指定飞盘的位置，就表示飞盘所有可能的位置构成了一个五维空间（或流形）。飞盘在空气中的飞行则对应于该流形中的一条曲线。研究更复杂（例如，铰接的）物体的位置，我们会得到更高维的流形，并且，如果物体缺乏任何刚性（想象一根橡皮筋），就会得到无限维的流形。该项目涉及这种无限维流形的基本性质以及无限多变量的伴随函数。主要目标是了解如何将这些流形和函数的局部信息组装成全局信息。告诉我们这在多大程度上是可能的数学构造称为束上同调群，此类群将成为研究的主要对象。这项工作的各个组成部分首先出现在数学的其他部分和理论物理（量子场论和弦理论）中，因此该项目如果成功，应该与这些学科有一定的相关性。然而，这将是基础研究，首席研究员并不期望立即应用于数学之外。另一方面，研究生将参与研究。因此，该项目将有助于培训未来的研究人员和教育工作者。