Complex Manifolds and Meromorphic Mappings

提案：DMS-9800479主要研究者：Bernard Shiffman 翻译后摘要：Shiffman将继续他的研究复品种和亚纯映射。 他将使用技术从他以前的工作电流和Nevanlinna理论来研究动态的理性地图在几个复杂的变量。他的哲学涉及考虑序列的迭代地图和寻找分布的价值观越来越多的迭代，而不是增加半径，因为这将是在Nevanlinna理论的情况。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，目标是更好地理解代数流形的双曲性概念。他将继续研究的影响，他的发现，波利亚的唯一性定理和一些较新的唯一性结果在一个复杂的变量可以重新制定的相对双曲性的某些准射影曲面。 动力学是对过程的时间演化的研究，这些过程可以像阻尼摆一样简单，也可以像天气一样复杂，由可以用数学公式表示的物理定律支配。该项目主要关注非线性系统;这些系统表现出混沌行为，了解混沌系统的行为在生物和物理科学的许多领域都具有根本的重要性，例如流体湍流，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代数流形的几何，复代数流形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.