Problems in Complex Analysis

首席研究员计划研究多复变量和复杂动力学理论中的各种问题。 与教授Sibony，首席研究员将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建不变测度，以及从多复变量理论中获得的广泛的函数理论工具。动力系统理论中的一个主要问题是方程太难，难以进行严格的研究。全纯映射有足够的结构，使许多结果可以严格证明，从而使一个想法的现象，也可以发生在更复杂的系统。首席研究员还计划研究多复变量理论中的几个问题。这包括推广的Bochner-Hartogs延伸定理更一般的流形和解决方案的柯西-黎曼方程在奇异空间。 这个建议的目标是研究问题，在复杂的分析和动力系统使用技术的复杂分析。复数被引入来解决用真实的数无法解决的代数表达式。随着时间的推移，复数已被证明是必要的，以解释基本的物理现象，如电磁学，机械系统中的振动等复分析研究的变化量，取决于复数。动态系统是研究随时间变化的系统。动力系统的例子从天气到人口的研究。动力学系统通常很难建模和理解。最简单的模型涉及使用真实的代数表达式来表示系统。考虑复数上的代数表达式允许人们在更高维的环境中使用更强大的工具来看待和研究问题。在这种情况下，一个非常复杂和迷人的场景出现了。计算机生成的图片显示了这种现象的几何形状：一种非常复杂的分形结构。正确理解这种几何学将导致更好地理解真实的动力系统。

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 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 and 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.