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.

动态的主要目标是了解模量空间。在这方面，最成功的努力是研究了二次多项式的模量空间，该空间包含曼德布罗特集（Mandelbrot Set），这是该主题中的基本对象。一个人希望在类似的程度上理解其他模量空间，例如给定程度的理性地图的模量空间。了解这些空间的分析和代数结构非常具有挑战性。原因之一是，很少有一维工具将其延续到更高的维度。该提案中概述的项目结合了拓扑，代数几何，复杂分析，Teichmueller理论以及曲线的非动力模量空间（以及各种紧凑型），以更好地理解复杂的动力学系统（在一个和几个变量中）及其相关的动力模量，从分析和Algebraic和Algebraic copt of Views of Vient of View。这些项目分为三个主要主题，每个主题都与瑟斯顿的拓扑特征理性地图有关，这是复杂动力学领域的中央定理。该提案中概述的研究计划将瑟斯顿的定理编织到这些研究主题中，以多种不同的方式。动态系统都在我们周围：行星的运动，天气，股票市场，我们所居住的生态系统。这些系统取决于多种参数，随着这些参数的变化，相应的系统受到影响。了解动态系统如何用不同的参数变化是一个非常复杂且精致的问题，在最简单的数学模型中甚至还没有完全理解。该提案中概述的研究概述了与某些动态系统相关的不同参数空间（或模量空间）之间的新连接，这些系统将被利用以进一步了解所讨论的空间。牛顿的方法是在不同科学领域中产生的一个动力系统，这是解决每个领域科学家使用的方程式的重要工具。围绕这个动态系统（在一个和几个变量中）尚未理解的动态系统（在一个和几个变量中）仍然存在许多基本问题。在某些情况下，该提案中概述的研究进展对这种动态系统具有影响。