Exploring the Topology and Geometry of Dynamical Subvarieties

动力系统就在我们周围：它们支配着行星的运动、天气、股票市场和我们生活的生态系统。这些系统依赖于各种参数，并且当这些参数改变时，相应的系统受到影响。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，即使在最简单的数学模型中也无法完全理解。然而，在这方面有一个光辉的成功故事：那就是复二次多项式的参数空间或“模空间”。这个空间包含了著名的曼德尔布罗特集，它在过去的40年里得到了深入的研究。本提案中概述的研究探索了与特定动力系统相关的不同参数空间，以期在数学界理解Mandelbrot集所在空间的相同程度上理解它们。该提案还包含一个重要的外展部分，以支持U（M）的数学团，这是一个面向中学生和高中导师的免费数学夏令营。在这方面最成功的奋进一直是研究的模空间的二次多项式的曼德尔布罗特集的生活，一个基本对象的主题。最终，我们将努力理解任意阶有理映射的模空间，就像我们理解二次多项式的模空间一样。许多复杂分析的工具为一维环境中的关键突破铺平了道路，但这些工具并不能推广到更高的维度。因此，PI没有考虑整个模空间，而是遵循William Thurston的方法，研究模空间的子簇。要研究的最自然的子变种是那些来自动力学条件的子变种，比如对临界点的前向轨道施加组合约束。人们可以把瑟斯顿的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力学子簇的方法;也就是说，那些由后临界有限参数组成的子簇。继爱泼斯坦，PI将适应瑟斯顿的想法和建设，并应制定一个设置中，研究高维动力子品种的模空间。PI将通过研究二次有理映射模空间中的一维动态子簇和三次多项式模空间中的一维动态子簇来探索这一理论，该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的评估来支持。影响审查标准。

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.