Complex Dynamics and Moduli Spaces

复杂动力学和模空间

• 批准号: 1300315
• 负责人: Sarah Koch
• 金额: $ 13.82万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2014-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300315&HistoricalAwards=false
• 关键词: Complex; Dynamics; Moduli; Spaces

A major goal in dynamics is to understand moduli spaces. The most successful endeavor in this regard has been the study of the moduli space of quadratic polynomials, which contains the Mandelbrot Set, a fundamental object in the subject. One hopes to understand other moduli spaces to a similar extent, like the moduli space of rational maps of a given degree. Understanding the analytic and algebraic structure of these spaces is quite challenging; one reason is that few of the one-dimensional tools carry over to higher dimensions. The projects outlined in this proposal incorporate topology, algebraic geometry, complex analysis, Teichmueller theory, and the nondynamical moduli spaces of curves (and various compactifications thereof) to better understand complex dynamical systems (in one and several variables) and their associated dynamical moduli spaces from both analytic and algebraic points of view. The projects are organized into three main topics, and each topic is related in some way to Thurston's Topological Characterization of Rational Maps, a central theorem in the field of complex dynamics. The research program outlined in this proposal weaves Thurston's theorem into these research topics in a variety of different ways. Dynamical systems are all around us: the motion of the planets, the weather, the stock market, the ecosystems in which we live. These systems depend on a variety of parameters, and as these parameters change, the corresponding system is affected. Understanding how dynamical systems change with different parameters is a very complicated and delicate question which is not even completely understood in the simplest of mathematical models. The research outlined in this proposal forges new connections between different parameter spaces (or moduli spaces) associated to certain dynamical systems, which will be exploited to further understand the spaces in question. One dynamical system that arises across different scientific fields is Newton's Method, an essential tool for solving equations that is employed by scientists in every field. There are still many fundamental questions surrounding this dynamical system (in one and several variables) that have yet to be understood. Progress on the research outlined in the proposal has implications for this dynamical system in certain cases.

动力学的一个主要目标是理解模空间。在这方面最成功的努力是对二次多项式模空间的研究，其中包含曼德尔布罗特集，这是该学科的基本对象。人们希望以类似的程度理解其他模空间，例如给定次数的有理映射的模空间。理解这些空间的解析和代数结构非常具有挑战性；原因之一是很少有一维工具可以扩展到更高维度。本提案中概述的项目结合了拓扑、代数几何、复分析、Teichmueller 理论和曲线的非动力模量空间（及其各种紧化），以更好地理解复杂动力系统（在一个和多个变量中）及其相关的动力模量空间从分析和代数的角度来看。这些项目分为三个主要主题，每个主题都以某种方式与瑟斯顿的有理图拓扑表征（复杂动力学领域的中心定理）相关。本提案中概述的研究计划以各种不同的方式将瑟斯顿定理融入到这些研究主题中。动力系统就在我们周围：行星的运动、天气、股票市场、我们生活的生态系统。这些系统取决于多种参数，随着这些参数的变化，相应的系统就会受到影响。了解动力系统如何随不同参数变化是一个非常复杂和微妙的问题，甚至在最简单的数学模型中也无法完全理解。该提案中概述的研究在与某些动力系统相关的不同参数空间（或模空间）之间建立了新的联系，这将被用来进一步理解所讨论的空间。牛顿法是跨不同科学领域出现的一种动力系统，它是各个领域的科学家所使用的求解方程的重要工具。围绕这一动力系统（在一个或多个变量中）仍有许多基本问题有待理解。在某些情况下，提案中概述的研究进展会对这个动力系统产生影响。