LEAPS-MPS: The representation theory of combinatorial categories

该奖项全部或部分根据2021年美国救援计划法案（公法117-2）资助。在数学中，有限图指的是一个有限的点的集合，线段称为边，连接它们。例如，三角形是一个有三个点和三条边的图形，而字母“H”可以被视为一个有6个点和5条边的图形。尽管这种描述很简单，但图已经被证明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中至关重要。当前奖项的项目旨在以与上述两个应用相关的关键方式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。最后一个项目考虑了一组机器人在图上随机移动的场景，就好像它是一个轨道，同时不允许碰撞，并显示了由此产生的非常有趣的行为。除了这些研究问题，这笔赠款将用于促进各种背景和技能水平的学生的教育标准。这包括附加一个“在科学中成长”系列现有的学生研讨会，提供资金的地方AWM章，使用资金送学生参加全国会议，专门在研究的多样性，并资助夏季的研究机会。这个项目建立在以前的工作，开发了一个框架，研究家庭的高度对称图使用组合类别。这项工作本身适合于各种各样的自然结构，包括一个将意味着在这些家庭中的图的独立数的某些规则的行为。这些设备构成第一个拟议项目。第二个项目将类似的范畴框架应用于离散群的族，包括自由群和积分特殊线性群的自同构群。已经观察到，各种群论性质，如Kazhdan性质（T），似乎在这些家族中表现稳定。我们相信，这个框架可以阐明和扩展我们对具有性质（T）的群的理解，从而我们对扩展图的理解。最后，最近的工作提出了一个模型的随机编织的配置空间的树。该模型有一个相关的协方差矩阵，它已被证明是唯一识别的树，配置空间的拓扑结构缺乏的功能。此外，PI还设计了一个图配置空间的随机模型，可用于检测同源性中奇异扭转的存在。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.