Exploring the Topology and Geometry of Dynamical Subvarieties

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。然而，在这方面有一个闪亮的成功故事：那就是复二次多项式的参数空间，或称“模空间”。这个空间包含著名的曼德尔布洛特集，在过去的40年里已经被彻底研究过了。这项提案中概述的研究探索与特定动力系统相关的不同参数空间，以期像数学界理解曼德布罗特集所在的空间一样理解它们。这项提案还包含了一个重要的外展部分，以支持密歇根大学数学团，这是一个为中学生和高中导师举办的免费数学夏令营。复杂动力学领域的一个主要目标是了解动态模空间。在这方面最成功的努力是研究曼德布罗特集所在的二次多项式的模空间，二次多项式的模空间是本课题的基本对象。最后，我们努力理解任意次有理映射的模空间，就像我们理解二次多项式的模空间一样。许多来自复杂分析的工具，为一维环境中的关键突破铺平了道路，但并没有延续到更高的维度。因此，PI没有考虑整个模空间，而是遵循了威廉·瑟斯顿的方法，研究了模空间的子簇。要研究的最自然的子品种是那些来自动力条件的子品种，比如对临界点的前向轨道施加组合约束。人们可以把瑟斯顿对有理映射的拓扑刻画看作是第一步。它提供了一种理解零维动态子簇的方法；也就是，那些由后临界有限参数组成的子簇。在爱泼斯坦之后，PI将采用瑟斯顿的思想和结构，并将开发一个研究模空间的高维动态子种类的环境。PI将探索这一理论，在二次有理映射的模空间中使用一维动态子变体，在三次多项式的模空间中使用一维动态子变体，其中关于它们的拓扑已经存在非常具有挑战性的基本问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.