LEAPS-MPS: The representation theory of combinatorial categories

LEAPS-MPS：组合类别的表示理论

基本信息

批准号：
2137628
负责人：
Eric Ramos
金额：
$ 10.54万
依托单位：
Bowdoin College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-09-01 至 2023-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2137628&HistoricalAwards=false
关键词：
LEAPS MPS representation theory combinatorial

项目摘要

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). In mathematics, a finite graph refers to a finite collection of points with line segments, called edges, connecting them. For example, a triangle is a graph with three points and three edges, while the letter "H" could be viewed as a graph with 6 points and 5 edges. Despite the simplicity of this description, graphs have proven themselves to be one of the most important tools in the modern mathematical toolkit, being critical in applications to, for instance, large networks and robotics. The projects of the current award seek to further the state of the art in the study of graphs in key ways related with the two aforementioned applications. In particular, one project seeks to understand the sizes of large independent (without a single edge connecting them) collections of points within the graph, whereas another relates to ways in which large networks expand over time. The final project considers scenarios of a collection of robots randomly moving on the graph as if it were a track, while not being allowed to collide, and showing the extremely interesting behavior that can result from this. In addition to these research concerns, this grant will be used in furthering educational standards for students of various backgrounds and skill levels. This includes attaching a "Growing Up in Science" series to existing student seminars, supplying funding for the local AWM chapter, using funds to send students to national conferences which specialize in diversity in research, and funding for summer research opportunities.This project builds on previous work, which developed a framework for studying families of highly symmetric graphs using combinatorial categories. This work lends itself to a variety of natural conjectures, including one that would imply certain regular behaviors in the independence numbers of graphs in these families. These conjectures comprise the first proposed project. The second project applies a similar categorical framework to families of discrete groups, including automorphism groups of free groups and integral special linear groups. It has been observed that various group theoretic properties, such as Kazhdan's property (T), seem to behave stably in these families. It is our belief that this framework can illuminate and expand upon our understanding of groups with property (T) and thereby our understanding of expander graphs. Finally, recent work has presented a model for random braiding in the configuration space of a tree. This model has an associated covariance matrix, which has been conjectured to uniquely identify the tree; a feature which the topology of the configuration space lacks. Furthermore, the PI has also devised a random model for graph configuration spaces that may be used to detect the presence of exotic torsions in homology.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.

该奖项是根据2021年《美国救援计划法》（公法117-2）全部或部分资助的。在数学中，有限的图是指具有线段的有限点集合，称为边缘，连接它们。例如，三角形是一个具有三个点和三个边缘的图形，而字母“ H”可以看作是具有6分和5个边缘的图。尽管这种描述很简单，但图表已证明自己是现代数学工具包中最重要的工具之一，在例如大型网络和机器人技术的应用中至关重要。当前奖项的项目旨在以与上述两个应用程序相关的关键方式研究图表的状态。特别是，一个项目试图理解图表内的大型独立（无单一的边缘连接）集合的大小，而另一个项目与大型网络随着时间的推移而扩展的方式相关。最终项目考虑了一系列机器人在图表上随机移动的场景，好像是一条轨道一样，同时不允许碰撞，并显示出可能由此产生的非常有趣的行为。除了这些研究问题外，该赠款还将用于为各种背景和技能水平的学生提供教育标准。这包括将“在科学中成长”系列附加到现有的学生研讨会，为当地AWM章节提供资金，使用资金将学生送往专门从事研究多样性的国家会议，并为夏季研究机会提供资金。先前的工作开发了一个框架，用于使用组合类别研究高度对称图的家庭。这项工作将自己带入了各种自然猜想，其中包括暗示这些家庭独立数量中某些规则行为的猜想。这些猜想是第一个提议的项目。第二个项目将类似的分类框架应用于离散群体的家族，包括自由组的自动形态群体和整体特殊线性群体。已经观察到，在这些家庭中，各种群体理论特性（例如Kazhdan的财产（T））似乎表现稳定。我们的信念是，该框架可以照亮和扩展我们对具有财产（t）群体的理解，从而对扩张器图的理解。最后，最近的工作提出了一个在树的配置空间中随机编织的模型。该模型具有关联的协方差矩阵，该矩阵已被猜想以唯一识别树。配置空间拓扑缺乏的功能。此外，PI还为图形配置空间设计了一个随机模型，该模型可用于检测同源性中异国情调的扭转。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子和广泛的评估来进行评估。影响审查标准。