CAREER: Polynomials, Geometry, and Dynamics

动力系统在我们周围无处不在：它们控制着行星的运动、天气、股市和我们生活的生态系统。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。了解动力系统如何随不同的参数变化是一个非常复杂和微妙的问题，即使在最简单的数学模型中也不能完全理解。这项建议中概述的研究探索了与特定动力系统相关的参数空间的不同视角之间的联系。例如，观察这些参数空间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，而多项式的根是一个数学领域，先验地与动力学没有明显的联系。调查这些神秘的、有些令人惊讶的联系是这项提议中概述的主要研究目标之一。该提案还包含一个重要的推广部分，通过到密歇根大学数学系的数学实地考察与高中生分享数学，并通过一系列研讨会让研究生和博士后有机会相互学习，随后将在密歇根大学举行同伴会议。复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研究二次多项式的模空间，Mandelbrot集就是其中的一个；Mandelbrot集是复杂动力学中的普遍对象。Mandelbrot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(即具有整数系数的某一多项式集合的根)中存在可区分的代数整数，并且这些点在边界上是稠密的。在这个建议的第一部分，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)。我们利用这些根的拓扑闭包，(在其他人的工作之后)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。大部分讨论都是以与二次多项式类似的方式展开的，除了有一些惊人的差异和许多有趣的问题需要探索几何、动力学、代数和伽罗瓦理论的联系。这一建议的第二部分是围绕代数数据和瑟斯顿关于有理映射的拓扑刻画的，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映射有关的是三种不同类型的线性算子。这些运算符自然地出现在瑟斯顿定理的背景下，每个运算符作用于有限维矢量空间。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。这项建议探讨了这些算子(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)与原始有理映射的几何和动力学之间的联系。

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.