LEAPS-MPS: The representation theory of combinatorial categories

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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.