Complex Analysis and Geometry

首席研究员提议研究涉及无限维对象的复杂分析和几何问题。该项目有三个主题。第一个是研究某些无限维流形及其回路空间的层上同调群。有限维中的大量经验和无限维中的小得多但不断增长的经验表明，上同调问题将是无限维中分析和几何的核心。首要的问题是：（1）无限维理论与有限维理论有何不同，两者又有何相似之处？(2)从一个流形的循环空间的上同调群中，可以收集到关于它的什么样的信息？要考虑的一个具体问题是赋予无限维流形的上同调群一个合适的拓扑，并询问是否可以证明像勒雷同构定理或塞尔对偶定理的相对版本的定理。该项目的第二个主题涉及埃尔米特全纯向量丛（在有限维流形上）在不一定适当的映射下的直接图像。人们所能期望的最好结果是，直接像将是希尔伯特空间的光滑场，希尔伯特空间是向量丛的最近推广。主要研究者将探索以下主题：几何条件，暗示直接图像确实是希尔伯特空间的光滑场;直接图像的曲率;以及希尔伯特空间全纯场中的柯西-黎曼方程理论。第三个主题是关于给定Kahler流形上的Kahler度量空间。这是一个无限维的黎曼流形，该项目将研究它的测地线。数学的作用之一是提供描述我们周围世界的术语。随着我们发现越来越复杂的自然和社会现象，描述仍然保持简单至关重要。数学通过引入新的概念来实现这一点。这里有一个与这个项目有关的例子。在真实的世界中的数字是用真实的数字来衡量的，自从解析几何的发明以来，我们知道真实的世界的数字（例如，曲线、曲面）可以由真实的变量的函数来描述。然而，振荡现象中的真实的世界量，例如交流电中的电压，可以用复数来描述得简单得多。引入复数需要投入一些时间和精力，但是一旦这样做了，对现象的描述就变得完全透明了。到目前为止，我们已经很好地理解了复数在大量问题中是不可或缺的。类似地，人们常常被迫从真实的变量的函数和它们所描述的图形过渡到复变量的函数和相关的几何图形。真实的世界现象通常涉及大量的参数来控制，并且它们的处理需要处理大量变量的函数。事实再次证明，引入一个新的心理结构，无限多变量的函数，往往简化了对这些现象的理解。这个项目，继续早期的研究principal调查员，是针对发现的基本性质的函数的无穷多个复杂的变量和相关的数字，所谓的无限维复杂的流形。该项目的一部分将调查什么轴承无限维理论有限维，更接近家庭的问题。该项目中的一些具体问题直接受到量子理论的启发，并有可能影响理论物理。通过让研究生参与，该项目还将有助于向年轻人介绍数学研究。

The principal investigator proposes to work on problems of complex analysis and geometry that involve infinite dimensional objects. The project has three main themes. The first is to study the sheaf cohomology groups of certain infinite-dimensional manifolds and their loop spaces. The huge amount of experience from finite dimensions and the much smaller, but growing, amount of experience in infinite dimensions suggest that cohomological questions will be central to analysis and geometry in infinite dimensions. The overarching questions are: (1) How is the infinite-dimensional theory different from the finite-dimensional theory, and how are the two similar? (2) What kind of information concerning a manifold can be gleaned from the cohomology groups of its loop spaces? A concrete problem to be considered is to endow the cohomology groups of infinite-dimensional manifolds with a suitable topology and to ask whether theorems like Leray's isomorphism theorem or a relative version of Serre duality theorem can be proved. The second theme of the project concerns direct images of Hermitian holomorphic vector bundles (over finite-dimensional manifolds) under not-necessarily-proper maps. The best one can expect is that direct images will be smooth fields of Hilbert spaces, a recent generalization of vector bundles. The principal investigator will explore topics such as the following ones: geometric conditions that imply the direct image is indeed a smooth field of Hilbert spaces; the curvature of the direct image; and the theory of the Cauchy-Riemann equations in holomorphic fields of Hilbert spaces. The third theme concerns the space of Kahler metrics on a given Kahler manifold. This is an infinite-dimensional Riemannian manifold, and the project will investigate its geodesics.One of the roles of mathematics is to provide the terms in which to describe the world around us. As we are discovering more and more complicated phenomena, both natural and societal, it is critical that the descriptions nevertheless stay simple. Mathematics achieves this by introducing new notions. Here is an example that is pertinent to this project. Quantities in the real world are measured by real numbers, and since the invention of analytic geometry we know that real world figures (e.g., curves, surfaces) can be described by functions of real variables. Yet real world quantities in oscillatory phenomena, for example voltage in alternating current, can be described much more simply in terms of complex numbers. It takes some investment of time and energy to introduce complex numbers, but once this is done, the descriptions of the phenomena become fully transparent. By now we understand well that complex numbers are indispensable in a huge number of problems. Similarly, one is often forced to pass from functions of real variables and the figures they describe to functions of complex variables and the associated geometric figures. Real world phenomena often involve a vast number of parameters to control, and their treatment requires working with functions of a vast number of variables. It again turns out that introducing a new mental construct, functions of infinitely many variables, oftentimes simplifies the understanding of those phenomena. This project, continuing earlier research of the prinicipal investigator, is directed towards uncovering fundamental properties of functions of infinitely many complex variables and the associated figures, so-called infinite-dimensional complex manifolds. Part of the project will investigate what bearing the infinite-dimensional theory has on finite-dimensional, closer-to-home problems. Some of the concrete problems in the project are directly motivated by quantum theory and have the potential to impact theoretical physics. By involving graduate students, the project will also serve to introduce young people to mathematical research.