CAREER: Algebraic structures in complex dynamics

职业：复杂动力学中的代数结构

• 批准号: 0747936
• 负责人: Laura DeMarco
• 金额: $ 55.97万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0747936&HistoricalAwards=false
• 关键词: CAREER; Algebraic; structures; complex; dynamics

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.

最简单但不平凡的复杂动力系统是代数的：一个复杂变量中不可逆多项式的迭代，更一般地说，是复杂代数簇的有理自映射。 与这些全纯系统相关的某些解析量与代数构造有着令人惊讶的联系。 该项目的例子说明了复杂动态环境中丰富的代数结构，包括：（1）通过相关树研究一维多项式以及与模空间正则函数域的评估之间的联系； (2) 对环面变体的规则自映射、此类映射的结果以及与多势理论的联系进行分析； （3）正则自映射（例如射影空间）的模空间的结构及其作为代数簇的“动态”紧化（描述此类系统如何退化）。这种动态系统最熟悉的例子之一是牛顿的系统方法，一种求多项式根的迭代算法，首次在微积分课程中引入。 即使对于这个著名且看似简单的例子，我们最近才开始使用现代数学技术来理解它的结构以及它的一般失败。 以牛顿方法（其历史、应用和最近的研究）为指导，我将为学生开展一系列教育项目：（1）本科生研究项目，重点是计算机对实例的探索； (2) 开发动力学本科课程，介绍理论和最新应用； (3) 两个研究生工作坊； (4) 定期的学生研讨会。 上述研究旨在探索混沌动力系统（例如牛顿方法及其推广）与任何潜在代数结构之间的相互作用。 这种结构为系统添加了额外的对称性或规律性元素，否则系统可能看起来很棘手。 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（智利、西班牙、波兰和德国等也有较小的群体）；通过这个项目，首席研究员将召集其中一些专家与学生一起工作，并阐明代数和动力学之间的联系。