Mathematical Sciences: Holomorphic Mappings

数学科学：全纯映射

• 批准号: 9622594
• 负责人: Sergey Pinchuk
• 金额: $ 7.2万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622594&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Holomorphic; Mappings

ABSTRACT Proposal: DMS-9622594 PI: Pinchuk The proposed research will be carried out in three directions. The first is the study of the conjecture that in complex n-space any proper holomorphic mapping between bounded domains with real analytic boundaries extends to a proper holomorphic mapping between neighborhoods of their closures. In dimension 1 this is a fundamental result known as the Schwarz Reflection Principle. For n1 the conjecture has been proved only in special cases. Pinchuk will use a geometric approach, which is based on the technique of Segre varieties, and is a powerful new tool to study various problems in domains with real analytic boundaries. The goal of the first part of the proposal is to develop the geometric reflection principle for higher dimensions and prove the conjecture above in general. The second direction is to apply the technique of Segre varieties together with other geometric methods to study rigidity phenomena of holomorphic mappings. Namely, Pinchuk plans to describe domains with noncompact automorphism groups. He will also attack the related problem that in rather general situations proper holomorphic self-mappings are biholomorphic. The third direction is the investigation of the so called Abhyankar Equations (AEs) with respect to the Jacobian Conjecture (JC). The JC (in the complex case, which is the principal one) claims that any locally inevitable (biholomorphic) polynomial mapping is globally inevitable. The JC is one of the most intriguing problems in mathematics. It has been intensively studied by many mathematicians (including very famous ones) since 1939. A number of partial results as well as faulty proofs has been published. Most of the experts believed and still believe that the JC is true. Therefore a recent counterexample to the real version of the JC, which was constructed by Pinchuk (using the Aes) came as a surprise. The JC has equivalent formulations in different areas of mathematics. Among others, its resolution ( positive or negative) will have implications for the finiteness of inversion formulas, for the existence of global solutions of certain differential equations, and for singularities of algebraic sets. The other problems of this proposal are natural, important, and known as hard problems in several complex variables. Their solution will became the essential part of the theory of holomorphic mappings.

摘要提案：DMS-9622594 PI：Pinchuk 拟议的研究将在三个方向进行。 第一个是研究复n-空间中有界域之间的任意真全纯映射都是真实的解析的 边界扩展到其闭包的邻域之间的适当全纯映射。在一维空间中，这是一个基本的结果，被称为施瓦茨反射原理。对于n1的猜想已被证明只有在特殊情况下。 Pinchuk将使用几何方法，这是基于技术的塞格雷品种，是一个强大的新工具，研究各种问题的域与真实的分析边界。第一部分的目标是发展更高维度的几何反射原理，并一般地证明上述猜想。第二个方向是 将塞格雷簇的技巧与其它几何方法结合起来研究全纯映射的刚性现象。也就是说，Pinchuk计划用非紧自同构群来描述域。 他还将攻击相关的问题，而在一般情况下适当的全纯自映射是双全纯的。第三个方向是关于雅可比猜想（JC）的所谓Abhyankar方程（AE）的研究。JC（在复杂的情况下，这是主要的）声称，任何局部不可避免的（双全纯）多项式映射是全球不可避免的。 JC是数学中最有趣的问题之一。 自1939年以来，许多数学家（包括非常著名的数学家）都对它进行了深入的研究。一些部分结果以及 发表了错误的校样。 大多数专家相信并且仍然相信JC是真实的。因此，最近平丘克（使用Aes）构造的JC的真实的版本的反例令人惊讶。 JC在不同的数学领域有等价的公式。其中，它的分辨率（正或负）将对反演公式的有限性，某些微分方程的整体解的存在性以及代数集的奇异性产生影响。 这个建议的其他问题是自然的，重要的，并被称为在几个复杂的变量的难题。它们的解将成为全纯映射理论的重要组成部分。