Problems in Complex Analysis

首席研究员计划研究多个复变量和复杂动力学理论中的各种问题。与西伯尼教授一起，首席研究员将致力于全纯映射迭代理论的系统发展。这依赖于使用多势理论来构造不变度量，以及从几个复变量理论中构造广泛的函数论工具。动力系统理论中的一个主要问题是方程太难进行严格的研究。全纯映射具有足够的结构，因此许多结果都可以得到严格的证明，从而给出了一种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变量理论中的几个问题。这包括将Bochner-Hartogs延拓定理推广到更一般的流形和奇异空间中Cauchy-Riemann方程的解。这项建议的目标是利用复杂分析技术研究复杂分析和动力系统中的问题。复数被用来解决不能用实数来求解的代数表达式。随着时间的推移，复数被证明是解释基本物理现象所必需的，如电磁、机械系统中的振动等。复数分析研究依赖于复数的量的变化。动力系统是研究随时间变化的系统的学科。动力系统的例子很多，从天气到人口研究。一般来说，动力系统很难建模和理解。最简单的模型包括使用实代数式来表示系统。考虑复数上的代数表达式使人们能够在更高维的环境中看到和研究问题，并使用更强大的工具。在这个背景下，出现了一个非常错综复杂和引人入胜的场景。计算机生成的图片显示了这种现象的几何学：一种非常复杂的分形结构。正确理解这一几何学将使我们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.