LEAPS-MPS: The representation theory of combinatorial categories

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。在数学中，有限图指的是由线段连接的点的有限集合，这些点称为边。例如，一个三角形是一个有三个点和三条边的图，而字母“H”可以被看作是一个有6个点和5条边的图。尽管这一描述很简单，但图形已经证明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。本奖项的项目寻求在与上述两个应用相关的关键方面进一步发展图形研究的最新水平。特别是，一个项目试图了解图中大型独立(没有单一边连接)的点集合的大小，而另一个项目涉及大型网络随时间扩展的方式。最后一个项目考虑了一组机器人在图形上随机移动的场景，就像它是一条轨迹，同时不允许碰撞，并展示了由此可能产生的极其有趣的行为。除了这些研究关注的问题外，这笔补助金还将用于提高不同背景和技能水平的学生的教育标准。这包括在现有的学生研讨会上增加一个“在科学中成长”系列，为当地的AWM分会提供资金，使用资金将学生送到专门研究多样性的国家会议，并为夏季研究机会提供资金。这个项目建立在以前工作的基础上，利用组合范畴开发了一个研究高度对称图族的框架。这项工作适合于各种自然猜想，其中一个猜想暗示了这些族中图的独立数的某些正则行为。这些猜测构成了第一个拟议的项目。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性质(T)，似乎在这些家庭中表现稳定。我们相信，这个框架可以启发和扩展我们对具有性质(T)的群的理解，从而扩展我们对扩展图的理解。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。该模型有一个关联的协方差矩阵，该矩阵被猜测为唯一地识别树；这是配置空间的拓扑所缺乏的特征。此外，PI还为图形配置空间设计了一个随机模型，可以用来检测同源中是否存在奇异扭转。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.