LEAPS-MPS: The representation theory of combinatorial categories

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。在数学中，有限图指的是有线段（称为边）连接点的有限集合。例如，三角形是一个有3个点和3条边的图，而字母“H”可以看作是一个有6个点和5条边的图。尽管这种描述很简单，但图已经证明了自己是现代数学工具包中最重要的工具之一，在大型网络和机器人等应用程序中至关重要。当前奖项的项目旨在通过与上述两个应用程序相关的关键方式进一步推动图形研究的艺术状态。特别是，一个项目试图理解图中大型独立（没有一条边连接它们）点集合的大小，而另一个项目则涉及大型网络随时间扩展的方式。最后的项目考虑了这样的场景：一组机器人在图形上随机移动，就像它是一条轨道一样，同时不允许碰撞，并展示了由此产生的极其有趣的行为。除了这些研究问题外，这笔拨款将用于提高不同背景和技能水平的学生的教育水平。这包括在现有的学生研讨会上附加一个“在科学中成长”系列，为当地的AWM分会提供资金，用资金派学生参加专门研究多样性的国家会议，以及为夏季研究机会提供资金。该项目建立在先前的工作基础上，该工作开发了一个使用组合范畴研究高度对称图族的框架。这项工作有助于各种自然的猜想，包括在这些族的图的独立数中暗示某些规则的行为。这些猜想构成了第一个提出的项目。第二个项目将类似的范畴框架应用于离散群族，包括自由群的自同构群和积分特殊线性群。已经观察到，各种群论性质，如Kazhdan的性质(T)，似乎在这些族中表现稳定。我们相信，这个框架可以阐明并扩展我们对具有属性(T)的群的理解，从而扩展我们对展开图的理解。最后，最近的工作提出了一个树形空间中的随机编织模型。该模型具有关联的协方差矩阵，该协方差矩阵已被推测为唯一识别树；配置空间的拓扑结构所缺乏的特性。此外，PI还设计了一个图组态空间的随机模型，该模型可用于检测同调中奇异扭转的存在。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.