Complex Analysis and Geometry

首席研究员建议研究涉及无限维物体的复杂分析和几何问题。该项目有三个主题。首先研究了一类无限维流形的轴上同调群及其环空间。有限维的大量经验和无限维的少得多但不断增长的经验表明，上同调问题将是无限维分析和几何的核心。首要问题是：(1)无限维理论与有限维理论有何不同，两者有何相似之处？(2)流形的哪些信息可以从它的环空间的上同调群中得到？要考虑的一个具体问题是赋予无限维流形的上同构群一个合适的拓扑，并问是否可以证明像Leray同构定理或Serre对偶定理的一个相对版本这样的定理。该项目的第二个主题涉及在非必要固有映射下的厄密全纯向量束的直接像（在有限维流形上）。我们能期望的最好结果是直接图像将是希尔伯特空间的光滑场，希尔伯特空间是向量束的最新推广。首席研究员将探索以下主题：暗示直接像确实是希尔伯特空间的光滑场的几何条件；直像的曲率；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的卡勒流形上的卡勒度量的空间。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术语来描述我们周围的世界。当我们发现越来越复杂的自然和社会现象时，保持简单的描述是至关重要的。数学通过引入新概念来实现这一点。这里有一个与这个项目相关的例子。现实世界中的量是用实数来测量的，自从解析几何发明以来，我们知道现实世界中的图形（如曲线、曲面）可以用实数变量的函数来描述。然而，振荡现象中的实际量，例如交流电中的电压，可以用复数来更简单地描述。引入复数需要投入一些时间和精力，但一旦这样做了，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，人们经常被迫从实变量函数和它们所描述的图形过渡到复变量函数和相关的几何图形。现实世界的现象通常涉及大量需要控制的参数，对它们的处理需要使用大量变量的函数。再一次证明引入一个新的思维结构，无限多变量的函数，通常会简化对这些现象的理解。该项目延续了首席研究员早期的研究，旨在揭示无限多个复杂变量和相关图形的函数的基本性质，即所谓的无限维复杂流形。项目的一部分将研究无限维理论对有限维问题的影响。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学。通过研究生的参与，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.