Mathematical Sciences: Research in Complex Analysis and Complex Dynamics

数学科学：复分析和复动力学研究

• 批准号: 9401398
• 负责人: Eric Bedford
• 金额: $ 10.93万
• 依托单位: Indiana University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9401398&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Research; Complex; Analysis

9401398 Bedford This award supports mathematical research on problems arising in the theory of several complex variables. The work will be carried out in three directions. In the first, work will be done on applications of the methods of complex analysis to the dynamics of polynomial mappings for two complex variables. The mappings are assumed to be univalent. Central to this study will be the use of pluri-potential theory analogous to the impressive use of potential theory in the study of one-dimensional dynamics. In the second direction, work will continue on the classification of domains with noncompact automorphism groups. Since it is trivial to construct nonsmooth domains with such automorphism groups, attention will be restricted to domains with smooth boundaries. The first two domains addressed will be those which are convex and those with real analytic boundaries. In the third direction, the use of Levi flat surfaces will be studied in connection with the corona theorem for bounded analytic functions. A central goal of the work is the generalization of Wolff's d-bar approach to the corona problem on simply connected domains to multiply connected domains and Riemann surfaces. Several complex variables arose at the beginning of the century as a natural outgrowth of studies of functions of one complex variable. It became clear early on that the theory differed widely from it predecessor. The underlying geometry was far more difficult to grasp and the function theory had far more affinity with partial differential operators of first order. It thus grew as a hybrid subject combining deep characteristics of differential geometry and differential equations. Many of the fundamental structures were defined in the last three decades. Current studies still concentrate on understanding these basic mathematical forms. ***

9401398贝德福德 该奖项支持对多复变量理论中出现的问题进行数学研究。 这项工作将从三个方面展开。 在第一部分中，将研究复分析方法在两个复变量多项式映射动力学中的应用。 假设映射是单叶的。 这项研究的核心将是使用多势理论类似的令人印象深刻的使用势理论在一维动力学的研究。 在第二个方向，工作将继续对非紧自同构群域的分类。 由于构造具有这种自同构群的非光滑域是平凡的，因此将注意力限制在具有光滑边界的域上。 前两个域将是那些凸的和那些具有真实的解析边界。 在第三个方向，使用列维平面将研究与电晕定理有界解析函数。 一个中心目标的工作是推广沃尔夫的d-酒吧的方法电晕问题的简单连接域多连接域和黎曼曲面。 多复变出现在世纪初作为一个自然的产物研究的职能，一个复杂的变量。 很明显，这个理论与它的前身有很大的不同。 基本的几何是更难以把握和函数理论有更多的亲和力与偏微分算子的一阶。 因此，它成长为一个混合学科结合了微分几何和微分方程的深刻特点。 许多基本结构是在过去三十年中确定的。 目前的研究仍然集中在理解这些基本的数学形式。 ***