Complex Analysis and CR Geometry

首席研究员将研究几个复变量中的十个相关问题。这些问题大致可分为复分析中的正性条件和CR几何两个方面。积极性条件的想法已经产生了影响的主要研究者的工作复变量类似物希尔伯特的第十七个问题，他的工作与卡特林等距嵌入，他的工作与瓦罗林稳定标准的埃尔米特度量。这里的一个主要问题是一个非线性柯西-施瓦茨不等式，无论是度量的全纯线丛和一般设置。第二个领域主要涉及球面和超二次曲面之间的CR映射。两个主要的问题出现：群不变CR映射和光滑CR映射的复杂性在不同的维度之间的领域。群不变映射与数论和组合数学有着惊人的联系。首席研究员打算探索与数论的联系。例如，他将调查某些三次和高阶丢番图方程时出现的考虑CR映射从透镜空间超二次。这些方程涉及的注入性的结果，是小学在最简单的情况下，二项式系数出现，但相当微妙的一般。他将继续发展CR映射的复杂性概念，研究真全纯映射，并寻求与亚Riemannian几何的联系。至少一个世纪以来，复维映射定理在数学、物理和工程中发挥了至关重要的作用。在更高维度的情况更加微妙，新的现象出现。最终，高维复分析的应用将渗透到所有科学领域，就像现在的一维复分析一样。本研究的关键出发点是从单位圆过渡到单位球面。近年来，不同维度的球之间的CR映射的作用变得更加突出。CR映射是复解析映射的边界模拟。他们的研究导致了分析、几何和代数的不寻常的结合。进展导致复变量类似希尔伯特的第十七个问题，等距嵌入定理，一种新的复杂性理论，螺旋CR结构。 最近的调查导致数论问题，形成拟议的工作的基础。这项工作将继续影响复杂性分析，特别是通过关于正性条件的部分，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。提议者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方案，2006年在AIM为研究人员举办了一个讲习班。他将在PCMI为研究生开设相关材料的课程。最后，他介绍了他的博士后研究员Jiri Lebl和他目前的研究生Dusty Grundmeier对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.