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：丹Margalit这个项目的主要目标是研究表面和它们的对称性。 曲面是一个二维空间，换句话说，是我们生活的世界的二维版本。 表面有许多形状（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背景下，从物理学到机器人技术到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，在过去的世纪里一直是人们深入研究的焦点。 一个令人惊讶的现象是，有一种叫做曲线复合体的组合对象，它们本身看起来并不像一个曲面，但它们具有与曲面相同的对称性。 在过去的20年里，发现了许多这样的天体。 这个项目的第一个目标是（基本上）给出一个完整的列表，这样的曲线复合体。 这将是一个顶点在充分研究理论的对称性曲线复杂。 第二个项目是研究曲面对称性集合的某个重要子集--所谓的Torelli群。 这些对称性是重要的，因为它们与代数几何和表示论有很强的联系。 Torelli群的基本性质是未知的，尽管这个群已经被大量研究了50年。 这个项目旨在了解Torelli群的基本有限性-例如有限可呈现性。 这是曲面理论中的主要开放问题之一。 使用计算机辅助搜索，已经发现了这个问题的新立足点。 第三个项目是一个建议的算法快速计算的基本属性的一个单一的对称的表面。例如，该算法计算熵，这是在表面上实现的混合量。 其他此类算法也存在，但我们的算法要快得多。 例如，使用适当的大小概念（称为字长），现有的算法可以处理大小为30（或左右）的对称，我们的算法可以很容易地处理大小超过30，000的对称。 在这些项目的同时，首席研究员也将完成一本关于相关主题的本科生教科书，称为几何群论的办公时间，并将继续为研究生举办一个专业发展研讨会，拓扑学生研讨会。曲面的映射类群是曲面的保向同胚的同伦类群。 其中，映射类群编码曲面基本群的外自同构群、曲面模空间的（orbifold）基本群以及任意空间上曲面丛的同构类型。 映射类群也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。这个项目的目标有三个方面：（1）找到一个一般理论，当一个组合的、代数的或几何的对象与一个曲面相关联时，它的自同构群是扩展的映射类群;（2）确定映射类群的Torelli子群的有限性，特别是Torelli群是否是可表示的;（3）建立了计算伪Anosov映射类共轭不变量的多项式时间算法。 问题（1）是由伊万诺夫提出的;与布伦德尔一起，PI在这个问题上取得了实质性的进展。 问题（2）是映射类群理论中最重要的公开问题之一。 这是一个很难回答的问题，回到20世纪20年代德恩和尼尔森的工作。 Besthood、Lucarelli、Vogtmann和PI正在通过计算机辅助搜索取得重大进展。 已知问题（3）的各种算法，最显著的是Bestvina-Handel算法。 有了Yurttas，PI有了一个计算火车轨道的新算法，它可以在二次时间内工作;实际上它比Bestvina-Handel算法（我们推测它是双指数的）快得多。 这三个项目都解决了映射类群理论中的基本问题，在这三个案例中，PI和他的合作者已经取得了重大进展。除了这些研究目标，PI还建议继续开展两个对研究生和本科生有直接影响的主要项目。 第一个是拓扑学生研讨会，作为研究生拓扑研究会议以及专业发展研讨会的会议。 第二个是办公时间与几何群论，介绍文字几何群论的本科生。