Complex analysis and geometry

数学的一个作用是提供描述我们周围世界的术语。随着我们发现和处理越来越复杂的自然和社会现象，描述保持简单是至关重要的。数学通过引入新概念实现了这一点。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衡量的，自从解析几何的发明以来，我们知道现实世界的图形-曲线、曲面等--可以用实变量的函数来描述。然而，振荡现象中的真实世界的量，例如交流电中的电压，可以更简单地用复数来描述。引入复数需要一些投资，但一旦完成，描述就会变得完全透明。到目前为止，我们已经很好地了解到，在科学和工程中出现的大量问题中，复数是不可或缺的。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通常是有利的。这项研究将处理复变量函数和复数几何图形的基本性质，即复流形。一个组成部分的动机是在微观世界的量子描述中出现的问题，并试图理解这种描述在多大程度上独立于一个人在构建数学描述时被迫做出的有点武断的选择。事实证明，这个问题和流形研究中出现的其他各种问题都有一个共同的推广，而答案有望取决于曲率的概念。PI将研究各种情况下的曲率；它是将这项研究的三个组成部分统一起来的概念。-正如所说，该项目中的一些具体问题直接受到量子理论的推动，并有可能影响理论物理。通过让研究生参与，该项目还将向年轻人介绍数学研究。用更专业的术语来说，该项目的一个组成部分研究希尔伯特空间的领域，即厄米特向量丛的推广。它们作为全纯向量丛的直接映象出现，PI将寻求将直接映象的曲率与向量丛的曲率联系起来。第二个部分是由恒定标量曲率的Kahler度量问题引起的。这些度量的存在和唯一性与所有Kahler度量空间的几何有关，这是一个无限维黎曼流形，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式。第三部分是关于全纯向量丛中的一般Hermite度量及其曲率。度量可以是奇异的，也可以是无限秩丛。这里的问题涉及如何将在线束的情况下很好地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.