Complex Manifolds and Meromorphic Mappings

摘要：Shiffman将继续他在复杂变异和亚纯映射方面的研究。他将使用他以前在电流和内万林纳理论方面的工作中的技术来研究几个复杂变量的有理映射的动力学。他的哲学包括考虑映射的迭代序列，并观察迭代次数增加而不是半径增加的值的分布，就像Nevanlinna理论中的情况一样。Shiffman还将应用值分布理论和代数几何的一些新技术来研究单复变量的唯一性定理，以期更好地理解代数流形的双曲性概念。他将继续研究他的发现的含义，即Polya的唯一性定理和一些新的在一个复杂变量中的唯一性结果可以用某些拟射影曲面的相对双曲性来重新表述。动力学是一门研究过程时间演化的学科，它可以像一个阻尼摆一样简单，也可以像天气一样复杂，受可以用数学公式表述的物理定律支配。该项目主要涉及非线性系统；这些都是表现出混沌行为的系统，理解混沌系统的行为在生物和物理科学的许多领域都是至关重要的，比如对流体湍流、化学反应和生物节律的研究。该项目的另一个组成部分涉及理解复杂代数流形的几何，它在量子场论中起着重要作用，并为各种物理现象提供模型。

Proposal: DMS-9800479 Principal Investigator: Bernard Shiffman Abstract: Shiffman will continue his research on complex varieties and meromorphic mappings. He will use techniques from his previous work on currents and Nevanlinna theory to study the dynamics of rational maps in several complex variables. His philosophy involves considering the sequence of iterates of a map and looking at the distribution of values for increasing numbers of iterations instead of for increasing radius, as would be the case in Nevanlinna theory. Shiffman will also apply some new techniques of value distribution theory and algebraic geometry to investigate uniqueness theorems in one complex variable, with a goal towards obtaining a better understanding of the concept of hyperbolicity for algebraic manifolds. He will continue to study the implications of his discovery that Polya's uniqueness theorem and some newer uniqueness results in one complex variable can be reformulated in terms of the relative hyperbolicity of certain quasi-projective surfaces. Dynamics is the study of the time evolution of processes, which can be as simple as a damped pendulum or as complicated as the weather, governed by physical laws that can be mathematically formulated. The project is concerned primarily with non-linear systems; these are the systems that exhibit chaotic behavior, and understanding how chaotic systems behave is of fundamental importance in many areas of the biological and physical sciences, such as the study of turbulence of fluids, chemical reactions, and biological rhythms. Another component of the project involves understanding the geometry of complex algebraic manifolds, which play an important role in quantum field theory and provide models for diverse physical phenomena.