Problems in Complex Analysis and CR Geometry

D’angelo的建议涉及复分析的三个相关部分的研究：复几何中的正性条件、CR映射和次椭圆乘子理论的代数方面。D’angelo近年来对复流形分析的主要贡献之一是系统地研究了复流形上厄米对称函数的正条件。这项工作融合了各种各样的问题，如希尔伯特第17问题的复变量模拟，球之间的适当全纯映射，全纯束的等距嵌入，以及可全局度量，成为一个连贯的主题。对固有映射的研究得出了一个关于原性的惊人结果；不变的CR映射提供了大量具有整数系数的多项式类，它们具有与x + y的p次幂所满足的相同的显著同余性质。适当的映射还导致了球之间映射的复杂性理论的结果。亚椭圆乘法器理论的研究工作与Kohn算法的复杂性和有效性有关，因此属于同一一般领域。单变量和多变量复变理论是数学中一个引人注目的部分；它为纯数学提供了一个美丽的例子，它在整个工程和物理科学中都有应用。D’angelo在CR几何（复分析的几何部分）方面的工作阐明了一个最基本的问题，即厄米对称和正条件的作用。他关于球之间的适当映射的工作导致了新的复杂性问题，甚至是一个新的素数测试。D'Angelo出版了两本关于几个复杂变量的研究水平的书籍，一本带有令人兴奋的问题集的本科教科书（与D. West合作），并在数学教育中发挥了积极的作用。他将于2005年夏季在MSRI指导研究生的研究经验，该研究将引导来自许多部门的学生进入CR几何和本提案中描述的工作。这些问题的进展将影响复变理论、CR几何以及可能的数论。

D'Angelo's proposal concerns research in three related parts of complex analysis: positivity conditions in complex geometry, CR mappings, and algebraic aspects of subelliptic multiplier theory. One of D'Angelo's primary contributions to complex analysis in recent years has been a systematic study of positivity conditions for Hermitian symmetric functions on complex manifolds. This work has merged diverse issues such as a complex variables analogue of Hilbert's 17th problem, proper holomorphic mappings between balls, isometric imbedding for holomorphic bundles, and globalizable metrics into a coherent subject. The work on proper mappings led to a surprising result about primality; invariant CR mappings provide large classes of polynomials with integer coefficients exhibiting the same remarkable congruence properties satisfied by the p-th power of x plus y. Proper mappings also led to results in the complexity theory of mappings between balls. The work on subelliptic multiplier theory is related to the complexity and effectiveness of Kohn's algorithm, and thus fits into the same general area.Complex variable theory in one and several variables is a striking part of mathematics; it provides a beautiful example of pure mathematics, which has provided applications throughout engineering and the physical sciences. D'Angelo's work in CR geometry (a geometric part of complex analysis) has clarified one of the most basic issues, the roles of Hermitian symmetry and positivity conditions. His work on proper mappings between balls has led to new sorts of complexity questions, and even to a new primality test. D'Angelo has published two research level books in several complex variables, one undergraduate textbook (with D. West) with exciting problem sets, and has played an active role in mathematics education. He will be guiding a research experience for graduate students at MSRI in summer 2005 that will initiate students from many departments into CR geometry and the workdescribed in this proposal. Progress on these problems should impact complex variable theory, CR geometry, and possibly number theory.