Complex Analysis and CR Geometry

首席研究员将研究几个复杂变量中的十个相关问题。问题大致分为两个领域，复分析中的正性条件和CR几何。关于正性条件的想法已经影响了首席研究员对希尔伯特第十七问题的复变量类比的研究，他与卡特林关于等距嵌入的研究，以及他与瓦罗林关于厄米度规稳定性准则的研究。这里主要关注的是一个非线性Cauchy-Schwarz不等式，它既适用于全纯线束上的度量，也适用于一般情况下的度量。第二个领域主要涉及球面和超二次曲面之间的CR映射。出现了两个主要问题：群不变的CR映射和不同维球之间光滑CR映射的复杂性。群不变映射与数论和组合学有着惊人的联系。首席研究员打算探索与数论的联系。例如，他将研究在考虑从Lens空间到超二次曲面的CR映射时出现的某些三次和高阶丢芬图方程。这些方程包含一个注入性结果，在最简单的情况下是初等的，在二项式系数出现的情况下，但通常是相当微妙的。他将继续发展CR映射的复杂性概念，研究适当的全纯映射，并寻求与亚黎曼几何的联系。在一个复杂维度上的映射定理在数学、物理和工程中发挥了至关重要的作用，至少有一个世纪了。在更高维度的情况是更加微妙和新的现象出现。最终，高维复分析的应用将渗透到所有的科学领域，就像一维复分析现在所做的那样。本研究的关键是从单位圆过渡到单位球。近年来，不同维度球间CR映射的作用越来越突出。CR映射是复解析映射的边界类似物。他们的研究导致了分析、几何和代数的不同寻常的结合。进展导致了希尔伯特第十七问题的复杂变量类比，等距嵌入定理，一种新的复杂性理论，以及螺旋CR结构。最近的调查导致了数论问题，这些问题构成了拟议工作的基础。这项工作将继续影响复杂分析，特别是通过有关正条件的部分，但它也将扩大CR几何的范围，包括复杂性理论和数论的应用。这位提案人围绕这些主题组织了几次会议，2005年在MSRI为研究生举办了一个项目，2006年在AIM为研究人员举办了一个研讨会。他将在PCMI为研究生提供相关材料的课程。最后，他向他的博士后研究员Jiri Lebl和他现在的研究生Dusty Grundmeier介绍了CR几何可能的新应用。

The principal investigator will study ten related problems in several complex variables. The problems divide roughly into two areas, positivity conditions in complex analysis and CR geometry. The ideas on positivity conditions have already had an impact on the principal investigator's work on complex variables analogues of Hilbert's seventeenth problem, his work with Catlin on isometric imbedding, and his work with Varolin on stability criteria for Hermitian metrics. One of the main concerns here is a nonlinear Cauchy-Schwarz inequality, both for metrics on holomorphic line bundles and in a general setting. The second area primarily concerns CR mappings between spheres and hyperquadrics. Two main issues arise: group-invariant CR mappings and the complexity of smooth CR mappings between spheres in different dimensions. The group-invariant mappings exhibit surprising connections with number theory and combinatorics. The principal investigator intends to explore the connection with number theory. For instance, he will investigate certain cubic and higher order Diophantine equations that arise when considering CR mappings from Lens spaces to hyperquadrics. These equations involve an injectivity result that is elementary in the simplest case, where binomial coefficients arise, but quite subtle in general. He will continue to develop notions of complexity for CR mappings, to study proper holomorphic mappings, and to seek connections with sub-Riemannian geometry.Mapping theorems in one complex dimension have played a crucial role in mathematics, physics, and engineering for at least a century. The situation in higher dimensions is much more subtle and new phenomena arise. Eventually the applications of higher dimensional complex analysis will permeate all of science, as one-dimensional complex analysis does now. The crucial point of departure in this research is to pass from the unit circle to the unit sphere. The role of CR mappings between spheres in different dimensions has become more prominent in recent years. CR mappings are boundary analogues of complex analytic mappings. Their study leads to an unusual combination of analysis, geometry, and algebra. Progress has led to complex variables analogues of Hilbert's seventeenth problem, to isometric imbedding theorems, to a new kind of complexity theory, and to helical CR structures. Recent investigations have led to number-theoretic questions that form the foundation for the proposed work. This work will continue to impact complex analysis, especially via the portion concerning positivity conditions, but it will also broaden the scope of CR Geometry to include applications to complexity theory and to number theory. The proposer has organized several meetings around these topics, a program in 2005 at MSRI for graduate students and a workshop in 2006 at AIM for researchers. He will be giving a course at PCMI to graduate students on related material. Finally, he has introduced both his postdoctoral fellow Jiri Lebl and his current graduate student Dusty Grundmeier to the possible new applications of CR geometry.