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.

动力系统就在我们周围：它们控制着行星的运动、天气、股市、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，查看这些参数空间的最有趣的方法之一是用代数方法；也就是说，这些参数空间与多项式的根密切相关，这是一个与动力学没有明显联系的数学领域。调查这些神秘且有些令人惊讶的联系是该提案中概述的主要研究目标之一。该提案还包含一个重要的外展部分，即通过对密歇根大学数学系的数学实地考察与高中生分享数学知识，并为研究生和博士后提供通过一系列研讨会相互学习的机会，这些研讨会将在随后在密歇根大学举行了同伴会议。复杂动力学领域的一个主要目标是理解动力学模空间。在这方面最成功的努力是对曼德尔布罗特所设定的二次多项式模空间的研究。曼德尔布罗特集是复杂动力学中的通用对象。曼德尔布罗特集的大部分结构实际上可以通过代数数据来揭示；也就是说，Mandelbrot集合中存在可区分的代数整数（即具有整数系数的某个多项式集合的根），并且这些点在边界上是密集的。在该提案的第一部分中，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（在其他人的一些工作之后）开发迭代函数系统的动力学理论，以更好地理解该集合的结构。大部分讨论的展开方式与二次多项式的讨论方式相似，只是存在一些令人惊讶的差异，并且仍然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。该提案的第二部分以代数数据和瑟斯顿的有理图拓扑表征为中心，这是复杂动力学中最重要的定理之一。与黎曼球面上的后临界有限有理图相关的是三种不同类型的线性算子。这些算子自然地出现在瑟斯顿定理的设置中，每个算子都作用于有限维向量空间。这些算子的特征多项式具有有理系数，因此相应的特征值是代数的。该提案探讨了这些算子（及其特征值）彼此之间可能存在的联系，以及这些算子（及其特征值）与原始有理图的几何和动力学之间的联系。