Complex Variables Analogues of Hilbert's Seventeenth Problem

提案：DMS-9970024首席研究员：约翰P. D '安杰洛摘要：D'安杰洛的项目将继续他的调查复变量版本的希尔伯特的第十七个问题（由于卡特林和D '安杰洛），并给予它的应用复杂的几何，分析和代数。他提出导出全纯丛的Catlin-D 'Angelo等距嵌入定理的推论，并将结果推广到某些退化度量。这一工作直接关系到不同维球之间的真全纯映射的分类问题。希尔伯特第十七问题的复变量版本已经证明有以下后果。假设给定复欧几里得空间上的一个多项式，该多项式在闭单位球上不为零。这个多项式必须是球之间的有理适当映射（减少到最低项）的分母，但目标维度需要选择足够大。因此，限制目标尺寸约束了球之间的适当映射的复杂性。Faran的工作进一步证明了这一点。德安杰洛试图证明一个一般定理提供了一个尖锐的界限的程度适当的合理映射之间的单位球在不同的层面。这将为一个关于不同维CR流形之间CR映射的更一般猜想提供证据。大卫希尔伯特在1900年国际数学家大会上的著名演讲中，提出了一系列研究问题，极大地影响了本世纪数学的发展。阿廷解决了希尔伯特原来的第十七问题在1925年左右证明，多项式在几个真实的变量的价值是非负的必须是一个总和平方的合理职能。自然模拟这一结果的多项式的几个复变量不成立。鉴于复变理论在物理、工程和纯数学中的重要性，很自然地会考虑复多项式的希尔伯特问题的其他翻版。近年来，D 'Angelo发现这种类似物在研究不同复杂维度的球之间的“适当映射”中起着至关重要的作用。D 'Angelo和Catlin后来建立了一个版本的阿廷定理在复杂的设置。它的应用程序适当的映射，使人们能够理解的作用，单位领域在更高的层面，从而grushes一个多维的工具，研究静电，扩散和流体流动。最近，Catlin和D 'Angelo证明了一个主要结果，通过将它们投射到一个抽象的框架中，即所谓的全纯向量丛的框架中，推广了他们早期的定理。证明给出了一个意想不到的应用伯格曼核函数（分析的一部分）的问题，代数，以及几何重新解释希尔伯特的问题。这个结果可能对现代数学物理学（如弦论和超对称）中全纯向量丛起重要作用的部分有相当大的意义。D 'Angelo是1999年伯格曼奖的赢家，这项工作在引文中提到。D 'Angelo目前的建议提出了这些想法的各种扩展和应用。

Proposal: DMS-9970024Principal Investigator: John P. D'AngeloAbstract: D'Angelo's project will continue his investigation of a complex variables version of Hilbert's seventeenth problem (due to Catlin and D'Angelo) and give applications of it to complex geometry, analysis, and algebra. He proposes to derive corollaries of the Catlin-D'Angelo isometric imbedding theorem for holomorphic bundles, and to extend the result to certain degenerate metrics. This work bears directly on the problem of classifying proper holomorphic mappings between balls in different dimensions. The complex variables version of Hilbert's seventeenth problem already proved has the following consequence. Suppose one is given a polynomial on complex Euclidean space that does not vanish on the closed unit ball. This polynomial must then be the denominator of a rational proper mapping (reduced to lowest terms) between balls, but the target dimension needs to be chosen sufficiently large. Thus restricting the target dimension constrains the complexity of a proper map between balls. Work of Faran gives further evidence of this. D'Angelo seeks to prove a general theorem providing a sharp bound for the degree of a proper rational mapping between unit balls in different dimensions. This will provide evidence for a more general conjecture about CR mappings between CR manifolds of different dimensions.In his famous address to the International Congress of Mathematicians in 1900, David Hilbert posed a series of research problems that have greatly influenced the development of mathematics in this century. Artin solved Hilbert's original Seventeenth Problem around 1925 by proving that a polynomial in several real variables whose values are nonnegative must be a sum of squares of rational functions. The natural analogue of this result for polynomials of several complex variables does not hold. Given the importance of complex variable theory in physics, engineering, and pure mathematics, it is natural to consider other facsimiles of Hilbert's problem for complex polynomials. In recent years, D'Angelo discovered that such analogues play a crucial role in the study of objects known as "proper mappings" between balls in different complex dimensions. D'Angelo and Catlin later established a version of Artin's theorem in the complex setting. Its application to proper mappings enables one to understand the role of the unit sphere in higher dimensions and thus furnishes a multidimensional tool for the study of electrostatics, diffusion, and fluid flow. More recently, Catlin and D'Angelo have proved a major result that generalizes their earlier theorems by casting them in an abstract framework, the framework of so-called holomorphic vector bundles. The proof gives an unexpected application of the Bergman kernel function (a part of analysis) to a problem in algebra, as well as a geometric reinterpretation of Hilbert's problem. The result may have considerable significance for the parts of modern mathematical physics (such as string theory and supersymmetry) where holomorphic vector bundles play a large role. D'Angelo is the 1999 winner of the Bergman Prize, and this work is mentioned in the citation. D'Angelo's current proposal suggests various extensions of and applications for these ideas.