Problems in Complex Analysis

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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.