Group Theoretical, Combinatorial, and Dynamical Aspects of Mapping Class Groups

映射类组的群理论、组合和动力学方面

• 批准号: 1510556
• 负责人: Dan Margalit
• 金额: $ 22.81万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1510556&HistoricalAwards=false
• 关键词: Group; Theoretical; Combinatorial; Dynamical; Aspects

AbstractAward: DMS 1510556, Principal Investigator: Dan MargalitThe 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. One surprising phenomenon is that there are combinatorial objects called curve complexes - looking nothing themselves like a surface - that have the same symmetries as a surface. Many such objects have been discovered in the past twenty years. The first goal of this project is to give (essentially) a complete list of such curve complexes. This will be a capstone in the well-studied theory of symmetries of curve complexes. The second project is to study a certain centrally important subset of the set of symmetries of a surface - the so-called Torelli group. These symmetries are significant because of their strong connections to algebraic geometry and representation theory. Basic properties of the Torelli group are unknown, despite the fact that this group has been studied heavily for fifty years. This project aims to understand the basic finiteness properties of the Torelli group - for instance finite presentability. This is one of the main open problems in the theory of surfaces. Using a computer-aided search, new footholds have been found into this problem. The third project is a proposed algorithm for quickly computing the basic properties of a single symmetry of a surface. For instance, this algorithm computes the entropy, which is the amount of mixing being achieved on the surface. Other such algorithms exist, but ours is much faster. For instance, using an appropriate notion of size (called word length), the existing algorithms can handle symmetries of size 30 (or so) and our algorithm can very easily handle symmetries of size upwards of 30,000. In conjunction with these projects, the principal investigator will also be completing a textbook for undergraduates on a related subject, called Office Hours with a Geometric Group Theorist, and also will continue to run a professional development workshop for graduate students, the Topology Students Workshop.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. The goals laid out in this project are threefold: (1) find a general theory for when a combinatorial, algebraic, or geometric object associated to a surface has the extended mapping class group as its group of automorphisms; (2) determine the finiteness properties of the Torelli subgroup of the mapping class group, specifically whether or not the Torelli group is finitely presented; and (3) establish a polynomial-time algorithm to compute the conjugacy invariants for a pseudo-Anosov mapping class. Problem (1) was conceived by Ivanov; with Brendle, the PI has made substantial progress on this question. Problem (2) is one of the most important open problems in the theory of mapping class groups. It is a very hard question going back to the work of Dehn and Nielsen in the 1920s. Bestvina, Lucarelli, Vogtmann, and the PI are making significant progress by performing a computer-aided search. Various algorithms for Problem (3) are known, most notably the Bestvina-Handel algorithm. With Yurttas the PI has a new algorithm for computing train tracks that works in quadratic time; in practice it is much quicker than the Bestvina-Handel algorithm (which we conjecture to be doubly exponential). All three projects address fundamental questions in the theory of mapping class groups and in all three cases the PI and his collaborators have already made significant headway. In addition to these research goals, the PI also proposes to continue work on two major projects that have direct impact on graduate and undergraduate students. The first is the Topology Students Workshop, a conference that serves both as a research conference in topology for graduate students as well as a professional development workshop. The second is Office Hours with a Geometric Group Theorist, an introductory text on Geometric Group Theory for undergraduates.

摘要奖：DMS 1510556，首席研究员：Dan Margalit 该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），并且它们产生在许多不同的背景下，从物理学到机器人技术，从数据分析到量子场论。 表面的对称性形成了一个美丽而丰富的理论，它一直是过去一个世纪深入研究的焦点。 一个令人惊讶的现象是，存在一种称为复合曲线的组合对象，它们本身看起来一点也不像曲面，但它们具有与曲面相同的对称性。 在过去的二十年里，已经发现了许多这样的物体。 该项目的第一个目标是（本质上）给出此类曲线复合体的完整列表。 这将成为经过深入研究的复合曲线对称理论的顶峰。 第二个项目是研究曲面对称性集的某个核心重要子集 - 所谓的托雷利群。 这些对称性非常重要，因为它们与代数几何和表示论密切相关。 托雷利群的基本性质尚不清楚，尽管该群已被深入研究了五十年。 该项目旨在了解 Torelli 群的基本有限性属性 - 例如有限可表现性。 这是曲面理论中主要的开放问题之一。 使用计算机辅助搜索，已经找到了解决这个问题的新立足点。 第三个项目是提出的算法，用于快速计算表面单一对称性的基本属性。例如，该算法计算熵，即表面上实现的混合量。 还存在其他类似的算法，但我们的速度要快得多。 例如，使用适当的大小概念（称为字长），现有算法可以处理大小为 30（左右）的对称性，而我们的算法可以非常轻松地处理大小超过 30,000 的对称性。 结合这些项目，首席研究员还将完成一本相关主题的本科生教科书，名为“与几何群理论家的办公时间”，并将继续为研究生举办一个专业发展研讨会，即拓扑学生研讨会。表面的映射类群是表面的保向同胚的同伦类群。 除此之外，映射类群对表面基本群的外自同构群、表面模空间的（轨道折叠）基本群以及任意空间上的表面丛的同构类型进行编码。 映射类组还与许多许多数学领域有联系，包括动力学、群论、数论、量子场论、表示论和代数几何等等。该项目的目标有三个：（1）找到与表面相关的组合、代数或几何对象何时将扩展映射类组作为其自同构组的一般理论； (2)确定映射类群的Torelli子群的有限性，具体是Torelli群是否有限呈现； (3)建立多项式时间算法来计算伪阿诺索夫映射类的共轭不变量。 问题（1）是伊万诺夫提出的；与 Brendle 合作，PI 在这个问题上取得了实质性进展。 问题(2)是映射类群理论中最重要的开放问题之一。 回顾 Dehn 和 Nielsen 在 20 年代的工作，这是一个非常困难的问题。 Bestvina、Lucarelli、Vogtmann 和 PI 通过执行计算机辅助搜索取得了重大进展。 问题(3)的各种算法是已知的，最著名的是Bestvina-Handel算法。 借助 Yurttas，PI 拥有了一种计算二次时间火车轨道的新算法；实际上，它比 Bestvina-Handel 算法（我们推测该算法是双指数算法）快得多。 所有三个项目都解决了映射阶级群体理论中的基本问题，并且在所有三个案例中，PI 和他的合作者都已经取得了重大进展。除了这些研究目标外，PI 还建议继续开展对研究生和本科生有直接影响的两个主要项目。 第一个是拓扑学学生研讨会，该会议既是研究生拓扑学研究会议，也是专业发展研讨会。 第二本是《几何群理论家的办公时间》，这是一本针对本科生的几何群理论入门课本。