Mapping Class Semigroups and the Classification of Conformal Dynamical Systems

动力系统是描述系统随时间演化的数学对象。它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建模。 在某些情况下，我们有最简单类型的分类，并知道它们是如何排列的。 例如，我们现在对真实的二次情形有了很好的理解。这一成就是来自不同领域的许多研究人员工作的结果。它仍然是一个中心的挑战，给一个分类的例子以外的真实的二次设置。 这个项目利用来自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，以及称为保形的更一般的类型。 在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。 虽然我们对这些更一般的系统的理解是穷人，我们有许多新的工具，从代数到分析，用它来研究它们。 该项目应用技术，从完善的理论映射类组的分类问题。 映射类群是二维物体的对称性，数学家和物理学家都在研究它。这个项目概括了映射类组的概念，包括这些二维复杂的动力系统。 它还应用了最近发展的自相似群理论的新工具。这一研究领域有着丰富的可及性问题。 谁从事这些主题的学生会来欣赏数学的本质统一和研究的兴奋。这将有助于培养一批受过广泛训练的数学人才。 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。 复杂动力系统的组合基础是由A. Douady，J. Hubbard，and W.瑟斯顿由于缺乏一个足够自然的代数框架，基本问题的进展被推迟，直到L。Bartholdi和V. Nekrashevych在2006年。这些自相似分组技术现在是标准的。这一新领域的最新发展与映射类群理论的发展是并行的。自然对象--球面的分支自覆盖，其分支点的前向轨道形成有限集--可以被有效地看作是可数半群中分支映射类的表示。这个半群又是纯映射类群上的一个偏集，这使得它的组合结构非常丰富。就像映射类群一样，在半群方面，与泰希穆勒理论、动力学、几何和算法问题有着同样非常深刻的联系。对这些联系的更充分的探索是本项目的重点。基本的动力学有限性的结果，和发展一个适当的概念，相对双曲，目前缺乏。有趣的是，相对双曲性的概念似乎自然地导致与粗糙几何和算术动力学在Berkovich spaces.This奖项反映了NSF的法定使命的连接，并已被认为是值得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的支持。

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.