RUI: Combinatorial Models in Representation Theory, Geometry, and Analysis

Graphs are essential tools used to model and analyze networks of all kinds: digital, electrical, social, epidemiological, etc. In many applications, the edges of a graph are labeled with additional data that indicates the capacity of the edge. The PI seeks to optimize some aspect of the graph depending on the application.

For instance, if the edge labels of a subway map indicate the capacity of each line at different times of day, one might want to maximize the total number of passengers that can be transported between two stations during rush hour; while if the edge labels of a road map indicate cost to plow the road, one might want to minimize the cost to plow paths between all essential services. This project analyzes two kinds of edge-labeled graphs: webs, which arise in knot theory and representation theory as well as combinatorics; and algebraic splines, which arise in analysis and applied math, especially in data compression or data interpolation. Undergraduate students and postdocs will be involved in the project.

Previous constructions of webs rely on local information about edge-labelings and seek to identify global properties (including bases of the associated representation, or coefficients in a knot-theoretic state sum). With Russell, the PI proposes a new model of webs called ''stranding'' that identifies many of these global properties as paths through the graph.

This unlocks exciting and promising new attacks on open questions in the field. In splines, the PI will develop and expand on recent progress with Nazir and Schilling towards the longstanding lower bound conjecture for the dimension of splines. Simultaneously, the PI continues projects to analyze a special family of splines that model equivariant cohomology, with applications towards the Stanley-Stembridge conjecture in algebraic combinatorics.

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.