CAREER: Algebraic structures in complex dynamics

最简单但非平凡的复杂动力系统是代数的：一个复变量中不可逆多项式的迭代，更一般地，复代数族的有理自映射。与这些全纯系统相关的某些解析量与代数结构有着惊人的联系。这个项目的例子说明了复杂动力学环境中丰富的代数结构：(1)通过其伴随树研究一维多项式，以及与模空间上正则函数场上的赋值的联系；(2)环簇上正则自映射的分析，这种映射的结果，以及与多势理论的联系；(3)正则自映射(如射影空间)的模空间的结构及其“动力”紧化(描述这种系统如何退化)的代数变体。这种动力系统最常见的例子之一是牛顿方法，这是一种迭代算法，最初在微积分课程中介绍。即使对于这个著名的、看似简单的例子，我们也只是最近才开始使用现代数学技术来理解它的结构和总体上的失败。以牛顿方法(其历史、应用和最新研究)为指导，我将为学生进行一系列教育项目：(1)本科生研究项目，重点是实例的计算机探索；(2)开发一门本科生动力学课程，介绍该方法的理论和最新应用；(3)为研究生举办两次研讨会；以及(4)定期举行学生研讨会。上述研究旨在探索混沌动力系统(如牛顿方法及其推广)与任何潜在的代数结构之间的相互作用。这样的结构为一个原本看起来难以处理的系统增加了额外的对称性或规律性元素。复杂动力系统这些方面的专家集中在法国、英国、日本和美国(在智利、西班牙、波兰和德国有较小的小组，仅举几例)；通过这个项目，首席研究员将把这些专家中的一些人聚集在一起，与学生一起工作，并澄清代数和动力学之间的这些联系。

The simplest yet non-trivial complex dynamical systems are algebraic: iteration of noninvertible polynomials in one complex variable and, more generally, rational self-maps of a complex algebraic variety. Certain analytic quantities associated to these holomorphic systems have surprising connections to algebraic constructions. Examples from this project that illustrate the rich algebraic structures in a complex dynamical setting include: (1) the study of one-dimensional polynomials by their associated trees and the connection with valuations on the field of regular functions on the moduli space; (2) an analysis of regular self-maps on toric varieties, the resultant of such mappings, and connections with pluri-potential theory; (3) the structure of the moduli spaces of regular self-maps (e.g. of projective spaces) and their "dynamical" compactifications (describing how such systems degenerate) as algebraic varieties.One of the most familiar examples of such a dynamical system is Newton's method, an iterative algorithm for finding roots of polynomials, first introduced in calculus courses. Even for this famous and seemingly simple example, we have only recently begun to understand its structure, and its failure in general, using modern mathematical techniques. With Newton's method (its history, its applications, and recent studies) as a guide, I will pursue a collection of educational projects for students: (1) undergraduate research projects, with an emphasis on computer exploration of examples; (2) the development of an undergraduate course in dynamics, to present both the theory and recent applications; (3) two workshops for graduate students; and (4) regular student seminars. The research described above is designed to explore the interplay between chaotic dynamical systems (such as Newton's method and its generalizations) and any underlying algebraic structures. Such structures add an extra element of symmetry or regularity to a system which might otherwise seem intractable. The experts in these aspects of complex dynamical systems are concentrated in France, England, Japan, and the US (with smaller groups in Chile, Spain, Poland, and Germany, to name a few); with this project, the Principal Investigator will bring together some of these experts to work with students and clarify these connections between algebra and dynamics.