Complex analysis and Geometry in Infinite Dimensions

该项目的重点是在无限维流形的复杂分析和几何。后一个话题在很大程度上代表了未知领域。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，人们已经对概括应该是什么有了一种感觉，而挑战实际上是证明它；在其他情况下，甚至必须发现用来形成概括的术语。有两个概念是这个项目的核心。一个是复循环空间的概念。这样的空间是通过从一个（有限维）复流形开始得到的。这个流形中所有循环的集合是一个称为复循环空间的无限维复流形。另一个中心思想是内聚束，这是对有限维理论中所有重要的内聚束的无限维概括，这是首席研究员和一位合作者最近介绍的。该项目的很大一部分是对这两个概念的研究，特别是它们的融合：即环空间上的内聚束和它们的上同调群。该项目还将为该项目期望获得的结果寻求在其他数学领域的应用。一个点或点质量在三维空间中的位置可以用它的坐标来描述，即用三个数字来描述。因此，从笛卡尔开始，我们就知道空间中的曲线和曲面可以用三个变量的函数来描述。现在要指定一个更复杂的物体的位置，比如飞盘，它的质心的三个坐标是不够的。它的轴的方向也必须给定，这将涉及到另外两个数字。最后，一个人需要五个数字来指定飞盘的位置，一个人说飞盘的所有可能位置构成一个五维空间（或流形）。飞盘在空中的飞行对应于这个流形中的曲线。研究更加复杂的位置(例如,铰链)对象,一个是导致甚至高维流形,如果对象没有任何刚性——把橡皮筋——无限维度的。这个项目是关于这种无限维流形和伴随的无限多变量函数的基本性质。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结构被称为束上同调群，这样的群将是研究的主要对象。这项工作的不同组成部分首先出现在数学和理论物理的其他部分（量子场论和弦理论），所以这个项目，如果成功，应该与这些学科有一些关联。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以外的领域。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。

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.