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.

Abstractaward：DMS 1510556，主要研究人员：Dan Margalit该项目的主要目标是研究表面及其对称性。 表面是一个二维空间，换句话说，我们生活在世界上的二维版本。表面有多种形状（例如，球的表面与甜甜圈的表面不同），它们在许多不同的环境中，从物理学到机器人到机器人到数据分析到量子场理论。 表面的对称性形成了一种美丽而丰富的理论，在过去的一个世纪中一直是激烈研究的重点。 一个令人惊讶的现象是，有一个称为曲线复合物的组合物体 - 看起来像表面像表面一样 - 具有与表面相同的对称性。 在过去的二十年中，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）（本质上）完整的曲线复合体列表。 这将是曲线复合物对称性的精心对称性理论中的一个顶点。 第二个项目是研究表面 - 所谓的Torelli组的一组对称对称性的某种集中重要子集。 这些对称性很重要，因为它们与代数几何学和表示理论有着牢固的联系。 尽管对该组进行了五十年的研究，但Torelli组的基本特性尚不清楚。 该项目旨在了解Torelli组的基本有限属性 - 例如有限的存在性。 这是表面理论中的主要开放问题之一。 使用计算机辅助搜索，已经在此问题中发现了新的立足点。 第三个项目是提出的算法，用于快速计算表面单个对称性的基本特性。例如，该算法计算熵，这是在表面上实现的混合量。 存在其他这样的算法，但是我们的算法更快。 例如，使用适当的大小概念（称为单词长度），现有算法可以处理30号尺寸（左右）的对称性，并且我们的算法很容易处理大小的对称性，大小超过30,000。 结合这些项目，首席研究人员还将完成一本在相关主题上的本科生的教科书，该学科与几何学理论家一起称为办公时间，并将继续为研究生工作室，拓扑学生工作室进行专业发展研讨会。 除其他事项外，映射类组编码表面基本组的外部自动形态组，表面模量空间的（Orbifold）基本组（Orbifold）基本组以及表面束的同构类型在任意空间上。 映射类小组还与许多数学领域有联系，包括动态，群体理论，数字理论，量子场理论，表示理论和代数几何形状，仅举几例。该项目中提出的目标是三重：（1）找到与表面相关的组合，代数或几何对象的一般理论，将扩展的映射类组作为其一组自动形态； （2）确定映射类组的Torelli子群的有限性能，特别是托雷利组是否有限呈现； （3）建立一个多项式时间算法来计算伪-Anosov映射类的共轭不变性。 伊万诺夫构想了问题（1）；在布伦德（Brendle）的情况下，PI在这个问题上取得了重大进展。 问题（2）是映射课程组理论中最重要的开放问题之一。 这是一个非常棘手的问题，可以追溯到1920年代Dehn和Nielsen的工作。 Bestvina，Lucarelli，Vogtmann和Pi通过执行计算机辅助搜索而取得了重大进展。 问题（3）的各种算法是已知的，最著名的是BestVina Handel算法。 使用Yurttas，PI具有用于计算二次时间工作的计算火车轨道的新算法。在实践中，它比BestVina Handel算法快得多（我们认为这是双重指数的）。 这三个项目都涉及映射课程组理论中的基本问题，在所有三种情况下，PI及其合作者都已经取得了重大进展。除了这些研究目标外，PI还建议继续在两个直接影响研究生和本科生的主要项目上工作。 第一个是拓扑学生研讨会，该会议既是研究生拓扑的研究会议，又是专业发展研讨会。 第二个是与几何群体理论家的办公时间，这是本科生几何群体理论的介绍性文本。