CAREER: Polynomials, Geometry, and Dynamics

职业：多项式、几何和动力学

• 批准号: 1452392
• 负责人: Sarah Koch
• 金额: $ 44.69万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2021-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1452392&HistoricalAwards=false
• 关键词: CAREER; Polynomials; Geometry; Dynamics

Dynamical systems are all around us: they govern 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 explores connections between different perspectives on parameter spaces associated to particular dynamical systems. For example, one of the most intriguing ways to view these parameter spaces is algebraically; that is, these parameters spaces are intimately related of roots of polynomials, an area of mathematics that, a priori, has no obvious connection to dynamics. Investigating these mysterious and somewhat surprising connections is one of the main research goals outlined in this proposal. This proposal also contains a substantial outreach component to share mathematics with high school students through mathematical fieldtrips to the Mathematics Department at the University of Michigan, and to give graduate students and postdocs opportunities to learn from each other through a series of workshops, which will be followed by companion conferences at the University of Michigan. A major goal in the field of complex dynamics is to understand dynamical moduli spaces. The most successful endeavor in this regard has been the study of the moduli space of quadratic polynomials where the Mandelbrot set lives; the Mandelbrot set is a universal object in complex dynamics. Much of the structure of the Mandelbrot set can actually be revealed through algebraic data; that is, there are distinguished algebraic integers in the Mandelbrot set (ie, roots of a certain collection of polynomials with integer coefficients), and these points are dense in the boundary. In the first part of this proposal, we consider collections of algebraic integers (or roots of collections of polynomials) arising naturally in the parameter space of an iterated function system. We take the topological closure of these roots and (following some work of others) develop a dynamical theory of the iterated function system to better understand the structure of this set. Much of the discussion unfolds in a way parallel to that for quadratic polynomials except there are some surprising differences and still many interesting questions to explore which link geometry, dynamics, algebra, and Galois theory. The second part of this proposal is centered around algebraic data and Thurston's Topological Characterization of Rational Maps, one of the most important theorems in complex dynamics. Associated to a postcritically finite rational map on the Riemann sphere, are three different kinds of linear operators. These operators naturally arise in the setting of Thurston's theorem, each acting on a finite-dimensional vector space. The characteristic polynomials of these operators have rational coefficients, so the corresponding eigenvalues are algebraic. This proposal explores possible connections that these operators (and their eigenvalues) have with each other, and connections that these operators (and their eigenvalues) have with the geometry and dynamics of the original rational map.

动力系统无处不在：它们控制着行星、天气、股票市场以及我们赖以生存的生态系统的运行。这些系统依赖于各种参数，随着这些参数的变化，相应的系统也会受到影响。理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不能完全理解。本提案中概述的研究探讨了与特定动力系统相关的参数空间的不同观点之间的联系。例如，观察这些参数空间的最有趣的方法之一是代数；也就是说，这些参数空间与多项式的根密切相关，这是一个数学领域，先验地说，与动力学没有明显的联系。调查这些神秘而又有些令人惊讶的联系是本提案中概述的主要研究目标之一。该提案还包含一个实质性的扩展部分，通过到密歇根大学数学系进行数学实地考察，与高中生分享数学，并通过一系列研讨会为研究生和博士后提供相互学习的机会，随后将在密歇根大学举行相关会议。复杂动力学领域的一个主要目标是理解动力模空间。在这方面最成功的努力是对二次多项式的模空间的研究，其中曼德尔布罗集合存在；Mandelbrot集合是复杂动力学中的一个通用对象。曼德布洛特集合的大部分结构实际上可以通过代数数据揭示出来；即在Mandelbrot集合中存在可区分的代数整数（即某一系数为整数的多项式集合的根），这些点在边界上是密集的。在本建议的第一部分，我们考虑在迭代函数系统的参数空间中自然产生的代数整数集合（或多项式集合的根）。我们采用这些根的拓扑闭包，并（继其他人的一些工作之后）发展了迭代函数系统的动力学理论，以更好地理解这个集合的结构。除了有一些令人惊讶的差异，还有许多有趣的问题需要探索，这些问题将几何、动力学、代数和伽罗瓦理论联系起来。这个建议的第二部分是围绕代数数据和Thurston的Rational映射的拓扑表征，这是复杂动力学中最重要的定理之一。与黎曼球上的后临界有限有理映射相关的是三种不同的线性算子。这些算子自然出现在瑟斯顿定理的集合中，每一个都作用于有限维的向量空间。这些算子的特征多项式具有有理系数，因此对应的特征值是代数的。本文探讨了这些算子（及其特征值）彼此之间的可能联系，以及这些算子（及其特征值）与原始有理映射的几何和动力学之间的可能联系。