Mapping Class Groups and Polynomials

这个项目的主要目标是研究曲面及其对称性。表面是一个二维空间，换句话说，是我们生活的世界的二维版本。表面有多种形状(例如，球的表面与甜甜圈的表面不同)，它们出现在许多不同的环境中，从物理到机器人，从数据分析到量子场论。曲面的对称性形成了一个美丽而丰富的理论，在过去的一个世纪里一直是密集研究的焦点。这些对称性与双曲几何之间的密切联系便于研究一些不是明显的几何问题，例如处理这些群的算法。曲面的映射类群是曲面的保定向同胚群的同伦类群。其中，映射类群编码了曲面基本群的外自同构群、曲面模空间的(Orbilold)基本群以及任意空间上曲面丛的同构类型。映射类群还与许多数学领域有联系，包括动力学、群论、数论、量子场论、表示论和代数几何，仅举几例。本研究的目标包括：(1)建立映射类群共轭问题的二次时间算法。在以前的NSF支持下所做的工作为Nielsen-瑟斯顿类型的映射类开发了一种二次时间算法，并且该理论的扩展有望为共轭问题提供类似的速度。(2)从映射类群的角度来理解多项式，包括曲面的哪些分支覆盖来自多项式的新方法。(3)刻画了映射类群的任意正规子群的结构。例如，提供了许多类型元素的正常闭包的描述，并发现了有限生成的直角Artin子群的新示例。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。

The main goal of this project is to study surfaces and their symmetries. A surface is a two-dimensional space, in other words, a two-dimensional version of the world we live in. Surfaces come in many shapes (for instance the surface of a ball is different from the surface of a doughnut) and they arise in many varied contexts, from physics to robotics to data analysis to quantum field theory. The symmetries of a surface form a beautiful and rich theory that has been the focus of intense study over the past century. The close connection between those symmetries and hyperbolic geometry facilitates the investigation of some questions that are not obviously geometric, such as algorithms for working with these groups.The mapping class group of a surface is the group of homotopy classes of orientation-preserving homeomorphisms of the surface. Among other things, the mapping class group encodes the outer automorphism group of the surface fundamental group, the (orbifold) fundamental group of the moduli space of the surface, and the isomorphism types of surface bundles over arbitrary spaces. The mapping class group also has connections to many, many areas of mathematics, including dynamics, group theory, number theory, quantum field theory, representation theory, and algebraic geometry, just to name a few. Goals of this research program include these: (1) Establish a quadratic-time algorithm for the conjugacy problem in the mapping class group. Work done under prior NSF support developed a quadratic-time algorithm for the Nielsen-Thurston type of a mapping class, and an extension of that theory is expected to give similar speed for the conjugacy problem. (2) Understand polynomials from the point of view of mapping class groups, including a new approach to the question of which branched covers of surfaces come from polynomials. (3) Describe the structure of an arbitrary normal subgroup of the mapping class group. For instance, descriptions are available of the normal closures of many types of elements, and new examples have been found of finitely generated right-angled Artin subgroups.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.