Problems in Complex Analysis and CR Geometry

德安杰洛的建议涉及研究三个相关部分的复杂分析：积极性条件复杂的几何，CR映射，代数方面的次椭圆乘子理论。其中D '安杰洛的主要贡献，复杂的分析，近年来一直是一个系统的研究积极的条件厄米对称函数的复杂流形。这项工作已经合并了不同的问题，如复变模拟希尔伯特的第17个问题，适当的全纯映射之间的球，等距嵌入的全纯丛，和globalizable指标到一个连贯的主题。关于适当映射的工作导致了关于素性的令人惊讶的结果;不变CR映射提供了具有整数系数的大类多项式，这些多项式表现出与x + y的p次幂所满足的相同的显着同余性质。适当的映射也导致结果的复杂性理论之间的映射球。亚椭圆乘子理论的研究与科恩算法的复杂性和有效性有关，因此属于同一个领域。一元和多元复变理论是数学中引人注目的一部分，它提供了纯数学的一个美丽的例子，在整个工程和物理科学中都有应用。D '安杰洛的工作CR几何（几何的一部分，复杂的分析）澄清了一个最基本的问题，作用的厄米对称性和积极的条件。他对球之间的适当映射的研究导致了新的复杂性问题，甚至导致了新的素数测试。D 'Angelo已经出版了两本研究水平的书在几个复杂的变量，一个本科教科书（与D。West）以令人兴奋的习题集，在数学教育中发挥了积极的作用。他将在2005年夏天指导MSRI研究生的研究经验，这将使许多系的学生开始接触CR几何和本提案中描述的工作。这些问题的进展应影响复变理论，CR几何，并可能数论。

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.