Problems in Complex Analysis

复杂分析中的问题

• 批准号: 9803286
• 负责人: John Fornaess
• 金额: $ 14.56万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803286&HistoricalAwards=false
• 关键词: Problems; Complex; Analysis

The principal investigator plans to work on various problems in the theory of several complex variables and complex dynamics. With Professor Sibony, the principal investigator will work on a systematic development of the theory of iterations of holomorphic maps. This depends on the use of pluripotential theory to construct invariant measures as well as a broad range of function theoretic tools from the theory of several complex variables. One of the main problems in the theory of dynamical systems is that the equations are too difficult for rigorous study. Holomorphic maps have enough structure so that many results can be proved rigorously thereby giving an idea of phenomena that can occur also in more complicated systems. The principal investigator also plans to work on several problems in the theory of several complex variables. This includes generalizations of the Bochner-Hartogs extension Theorem to more general manifolds and to solutions of the Cauchy-Riemann equations in singular spaces. The principal investigator plans to continue work with Professor Gavosto on investigation of iterations of holomorphic maps near homoclinic tangencies and other bifurcations relying heavily on use of the computer for graphics and rigorous proofs. The Principal Investigator plans also to continue joint works with Brendan Weickert, Greg Buzzard and Araceli Bonifant on various questions in the theory of complex dynamics. The goals of this proposal are to study problems in complex analysis and in dynamical systems using techniques of complex analysis. Complex numbers were introduced to solve algebraic expressions that could not be solved with real numbers. With time, complex numbers have proved to be necessary to explain fundamental physical phenomena like electromagnetisms, vibrations in mechanical systems, etc. Complex analysis studies the changes of quantities that depend on complex numbers. Dynamical systems is the study of systems that change over time. Examples of dynamical systems range from the weather to the study of populations. Dynamical systems are in general very hard to model and to understand. The simplest models involve using real algebraic expressions to represent the system. Considering the algebraic expressions over the complex numbers allows one to see and study the problems in a higher dimensional setting using more powerful tools. In this setting, a very intricate and fascinating scenario appears. Computer generated pictures show the geometry of this phenomenon: a very complicated fractal structure. Proper understanding of this geometry will lead to a better understanding of real dynamical systems.

首席研究员计划研究多复变量和复动力学理论中的各种问题。首席研究员将与 Sibony 教授一起致力于全纯映射迭代理论的系统发展。这取决于使用多能理论来构造不变测量以及来自多个复杂变量理论的广泛函数理论工具。动力系统理论的主要问题之一是方程太难进行严格的研究。 全纯映射具有足够的结构，因此可以严格证明许多结果，从而给出在更复杂的系统中也可能发生的现象的想法。首席研究员还计划研究几个复杂变量理论中的几个问题。 这包括将博赫纳-哈托格斯扩张定理推广到更一般的流形以及奇异空间中柯西-黎曼方程的解。首席研究员计划继续与 Gavosto 教授合作，研究同宿切线和其他分岔附近的全纯图的迭代，这在很大程度上依赖于使用计算机进行图形和严格的证明。首席研究员还计划继续与 Brendan Weickert、Greg Buzzard 和 Araceli Bonifant 合作研究复杂动力学理论中的各种问题。该提案的目标是使用复杂分析技术来研究复杂分析和动力系统中的问题。引入复数是为了解决用实数无法解决的代数表达式。随着时间的推移，复数已被证明是解释电磁学、机械系统振动等基本物理现象所必需的。复分析研究取决于复数的量的变化。动力系统是对随时间变化的系统的研究。动力系统的例子包括从天气到人口研究。动力系统通常很难建模和理解。最简单的模型涉及使用实代数表达式来表示系统。 考虑复数的代数表达式可以让人们使用更强大的工具在更高维度的环境中查看和研究问题。在这种背景下，出现了一个非常复杂且引人入胜的场景。计算机生成的图片显示了这种现象的几何形状：非常复杂的分形结构。 正确理解这种几何形状将有助于更好地理解真实的动力系统。