Complex Manifolds and Meromorphic Mappings

建议：DMS-9800479首席研究员：Bernard Shiffman摘要：Shiffman将继续他对复数簇和亚纯映射的研究。他将使用他之前关于洋流和内瓦林纳理论的工作中的技术来研究几个复变量中有理映射的动力学。他的哲学包括考虑地图的迭代序列，并查看迭代次数增加而不是半径增加时的值分布，就像内万林纳理论中的情况一样。Shiffman还将应用值分布理论和代数几何的一些新技术来研究一个复变量的唯一性定理，目的是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较新的唯一性导致一个复变量可以根据某些拟射影曲面的相对双曲性重新表述。动力学是一门研究过程的时间演化的学科，它可以简单到有阻尼力的钟摆，也可以复杂到天气，受可以用数学表示的物理定律的支配。该项目主要涉及非线性系统；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具有基本重要性，例如研究流体的湍流、化学反应和生物节律。该项目的另一个组成部分涉及了解复杂代数流形的几何，这在量子场论中发挥着重要作用，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.