Mapping Class Semigroups and the Classification of Conformal Dynamical Systems

动力系统是描述系统随时间演变的数学对象。他们模拟了许多类型的行为，从动物神经元到流行病再到天气模拟。在某些情况下，我们对最简单的类型进行了分类，并知道它们是如何排列的。例如，我们现在对实二次型的情况有了很好的理解。这一成就是来自不同领域的许多研究人员工作的结果。这仍然是一个中心挑战，给出一个例子的分类超出了真正的二次方背景。这个项目使用了几个不同数学领域的工具来应对这一挑战。它专注于涉及复数的系统，以及更一般的称为共形的类型。这个项目中研究的共形动力系统包括新发现的更奇特的例子。尽管我们对那些更一般的系统的理解较差，但我们有许多新的工具，从代数到分析，用来研究它们。该项目将成熟的映射班组理论中的技术应用到分类问题上。映射类群是二维对象的对称性，数学家和物理学家都在研究它。这个项目以一种包括这些二维复杂动力系统的方式推广了映射类群的概念。它还应用了最近发展起来的自相似群理论中的新工具。这一研究领域有丰富的可获得性问题。参与这些主题的学生将逐渐体会到数学和研究的激动人心的本质统一。这将有助于培养一批受过广泛培训的数学人才。为这项研究设计的特殊软件允许丰富的实验和技术技能的发展。A·杜阿迪、J·哈伯德和W·瑟斯顿奠定了复杂动力系统的组合基础。直到2006年L.Bartholdi和V.Nekrashevych引入了新的技术，缺乏足够自然的代数框架才推迟了基本问题的进展。这些自相似的分组技术现在是标准技术。这一新领域的最新发展与映射类群理论的发展是平行的。自然对象-其分支点的前向轨道形成有限集的球面的分枝自覆盖-可以被有效地视为表示可数半群中的分枝映射类。这个半群是纯映射类群上的二重集，这使得组合结构变得非常丰富。就像映射类群一样，在半群方面，也有与Teichmueller理论、动力学、几何和算法问题类似的非常深的联系。更充分地探索这些联系是本项目的重点。目前还缺乏基本的动力学有限性结果，以及相对双曲性的适当概念的发展。有趣的是，相对双曲性的概念似乎自然而然地导致了与伯克维奇空间的粗略几何和算术动力学的联系。这一奖项反映了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 mathematical objects that describe the evolution of systems over time. They model many types of behavior, from animal neurons to epidemics to weather modeling. In some cases, we have a classification of the simplest types, and know how they are arranged. For example, we have now a good understanding in the real quadratic case. This achievement was the result of work by many researchers from different fields. It remains a central challenge to give a classification of examples beyond the real quadratic setting. This project takes up this challenge using tools from several different areas of mathematics. It focuses on systems that involve complex numbers, and on more general types called conformal. The conformal dynamical systems studied in this project include newly discovered, more exotic examples. Though our understanding of those more general systems is poorer, we have many new tools, from algebra to analysis, with which to study them. The project applies techniques from the well-developed theory of mapping class groups to the classification problem. Mapping class groups are symmetries of two-dimensional objects and are studied by both mathematicians and physicists. This project generalizes the notion of a mapping class group in a way that includes these two-dimensional complex dynamical systems. It also applies new tools from the recently developed theory of self-similar groups. This research area has an abundance of accessible problems. Students who engage with these topics will come to appreciate the essential unity of mathematics and the excitement of research. This will contribute to the development of a pool of mathematical talent that is broadly trained. Special software designed for this study allows for rich experimentation and the development of technical skills. The combinatorial foundations of complex dynamical systems were laid by A. Douady, J. Hubbard, and W. Thurston. The lack of a sufficiently natural algebraic framework delayed progress on fundamental problems until new techniques were introduced by L. Bartholdi and V. Nekrashevych in 2006. These selfsimilar group techniques are now standard. The recent developments in this new field are paralleling those in the theory of mapping class groups. The natural objects–branched self-covers of the sphere whose forward orbits of branch points form a finite set—may be fruitfully regarded as representing branched mapping classes in a countable semigroup. That this semigroup is in addition a biset over the pure mapping class group makes the combinatorial structure immensely rich. Just as with mapping class groups, on the semigroup side, there are similarly very deep connections to Teichmueller theory, dynamics, geometry, and algorithmic questions. The fuller exploration of these connections is the focus of this project. Basic dynamical finiteness results, and the development of an appropriate notion of relative hyperbolicity, are currently lacking. Intriguingly, the notions of relative hyperbolicity seem to lead naturally to connections with both coarse geometry and arithmetic dynamics on Berkovich spaces.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.