Complex analysis and geometry

数学的作用之一是提供术语来描述我们周围的世界。当我们发现和处理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——曲线、曲面等——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。引入复数需要一些投入，但一旦完成，描述就会变得完全透明。到目前为止，我们很清楚，在科学和工程中出现的大量问题中，复数是必不可少的。同样，从实变量函数及其所描述的图形过渡到复变量函数及其相关的几何图形往往是有利的。本研究将处理复变量函数和复几何图形（称为复流形）的基本性质。一个部分是由微观世界的量子描述中出现的问题驱动的，并试图理解这种描述在多大程度上独立于数学描述构建时被迫做出的有些武断的选择。事实证明，这个问题和其他在流形研究中出现的问题有一个共同的推广，答案似乎取决于曲率的概念。PI将研究各种情况下的曲率；正是这个概念统一了本研究的三个组成部分。——如前所述，项目中的一些具体问题是由量子理论直接激发的，并且有可能影响理论物理。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。用更专业的术语来说，这个项目的一个组成部分是研究希尔伯特空间的场，厄米向量束的推广。它们作为全纯矢量束的直接像出现，PI将寻求将直接像的曲率与矢量束的曲率联系起来。第二个分量是由常数标量曲率的卡勒度规问题引起的。这些度量的存在唯一性与所有Kahler度量空间的几何形状有关，即无限维黎曼流形，特别是与该空间中的测地线有关。PI将研究这些测地线和控制它们的偏微分方程。第三部分是关于全纯矢量束中的一般厄密度量及其曲率。度规可以是奇异的，也可以是无限秩的束。这里的问题是如何将在线束情况下很容易理解的现象推广到这种情况。

One role of mathematics is to provide the terms in which to describe the world around us. As we are discovering and dealing with more and more complicated phenomena, both natural and societal, it is critical that the description nevertheless stay simple. Mathematics achieves this by introducing new notions. Here is an example, pertinent for 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---curves, surfaces, etc.---can be described by functions of real variables. Yet real world quantities in oscillatory phenomena, for example voltage in an alternating current, can be described much more simply in terms of complex numbers. It takes some investment to introduce complex numbers, but once done, the description becomes fully transparent. By now we understand well that complex numbers are indispensable in a vast number of problems that arise in science and engineering. Similarly, it is often advantageous to pass from functions of real variables and the figures they describe to functions of complex variables and the associated geometric figures. This research will deal with fundamental properties of functions of complex variables and of complex geometric figures, known as complex manifolds. One component is motivated by problems that arise in the quantum description of the micro-world, and seeks to understand to what extent this description is independent of the somewhat arbitrary choices one is forced to make as the mathematical description is constructed. It turns out that this problem and various others that arise in the study of manifolds have a common generalization, and the answer promises to depend on the notion of curvature. The PI will study curvature in various situations; it is the concept that unifies the three components of this research.---As said, some of the concrete problems in the project are directly motivated by quantum theory, and have a potential to impact theoretical physics. By involving graduate students, the project will also serve to introduce young people to mathematical research.In more technical terms, one component of this project studies fields of Hilbert spaces, generalizations of hermitian vector bundles. They arise as direct images of holomorphic vector bundles, and the PI will seek to connect the curvature of the direct image with the curvature of the vector bundle. A second component is motivated by the problem of Kahler metrics of constant scalar curvature. The existence and uniqueness of these metrics is related to the geometry of the space of all Kahler metrics, an infinite dimensional Riemannian manifold, in particular to the geodesics in this space. The PI will study these geodesics and the partial differential equation that governs them. The third component is about general hermitian metrics in holomorphic vector bundles and their curvature. The metric can be singular and the bundle of infinite rank. The questions here concern how to generalize to this setting the phenomena well understood in the case of line bundles.