Exploring the Topology and Geometry of Dynamical Subvarieties

探索动力学子类型的拓扑和几何

• 批准号: 2104649
• 负责人: Sarah Koch
• 金额: $ 40.75万
• 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: 2021
• 资助国家: 美国
• 起止时间: 2021-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2104649&HistoricalAwards=false
• 关键词: Exploring; Topology; Geometry; Dynamical; Subvarieties

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 that is not even completely understood in the simplest of mathematical models. However, there is one shining success story in this regard: that is, the parameter space, or "moduli space", of complex quadratic polynomials. This space contains the famous Mandelbrot Set, which has been thoroughly studied over the last 40 years. The research outlined in this proposal explores different parameter spaces associated to particular dynamical systems, with a view toward understanding them to the same extent that the mathematical community understands the space where the Mandelbrot set lives. This proposal also contains a significant outreach component to support the Math Corps at U(M), a free math Summer Camp for middle school students and high school mentors.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, a fundamental object in the subject. Ultimately, we strive to understand the moduli space of rational maps of arbitrary degree to the same extent that we understand the moduli space of quadratic polynomials. Many tools from complex analysis that pave the way for key breakthroughs in the one-dimensional setting do not carry over to higher dimensions. So instead of considering the whole moduli space, PI follows an approach initiated by William Thurston and investigate sub-varieties of moduli space. The most natural sub-varieties to study are those that come from dynamical conditions, like imposing combinatorial constraints on the forward orbits of critical points. One may view Thurston’s Topological Characterization of Rational Maps as a first step. It provides a way to understand zero-dimensional dynamical sub-varieties; that is, those that consist of postcritically finite parameters. Following Epstein, the PI will adapt Thurston’s ideas and constructions and shall develop a setting in which to study higher-dimensional dynamical sub-varieties of moduli spaces. PI will explore this theory by working with one-dimensional dynamical sub-varieties in the moduli space of quadratic rational maps, and one-dimensional dynamical sub-varieties in the moduli space of cubic polynomials, where there are already very challenging and fundamental problems concerning their topology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动力系统就在我们周围：它们控制着行星的运动、天气、股票市场、我们生活的生态系统，这些系统取决于各种参数，随着这些参数的变化，相应的系统就会受到影响。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无法完全理解。然而，在这方面有一个闪亮的成功故事：那就是参数空间或“模”。复二次的“空间”该空间包含著名的曼德尔布罗特集合，该集合在过去 40 年中得到了深入研究，探索了与特定动力系统相关的不同参数空间，以期像数学一样理解它们。该提案还包含支持 U(M) 数学队的重要外展内容，这是一个针对中学生和高中导师的免费数学夏令营。复杂的动力学是理解动态模空间在这方面最成功的努力是研究曼德尔布罗特集所在的二次多项式的模空间，这是该学科的基本对象，最终，我们努力理解有理映射的模空间。任意程度与我们理解二次多项式模空间的程度相同，许多为一维设置中的关键突破铺平道路的复杂分析工具并不能延续到更高维度。在模空间中，PI 遵循 William Thurston 发起的一种方法，研究模空间的子种类，最自然的研究子种类是那些来自动态条件的子种类，例如对临界点的前向轨道施加组合约束。将瑟斯顿的有理图拓扑表征视为第一步，它提供了一种理解零维动态子类型的方法；即由后临界有限参数组成的子类型。 Epstein 表示，PI 将采用 Thurston 的思想和结构，并开发一种研究模空间的高维动力学子簇的环境，PI 将通过使用模空间中的一维动力学子簇来探索该理论。二次有理图和三次多项式模空间中的一维动力学子品种，其中已经存在关于其拓扑的非常具有挑战性和基本的问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。