Combinatorial Representation Theory from Knot Theory and Algebraic Geometry

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The core of this project is the problem of incomplete data: situations where we only have partial information and need either to estimate the missing data or to find ways to solve the problem based only on our fragmentary knowledge. In this project, we use tools from geometry as well as from combinatorics, namely the sophisticated counting techniques that allow us to analyze patterns in diverse applications from DNA sequencing to optimization. The work contributes significantly to developing a more diverse workforce in mathematics, both increasing the pipeline for women and underrepresented minorities and strengthening retention further downstream. This includes incorporating student researchers at all stages of their careers into the PI's lab.The specific research addressed in this proposal is: 1) combinatorially analyzing the transition matrices between two important bases of irreducible symmetric-group representations, the web basis and the tableau basis; 2) describing the geometry and topology of components of Springer fibers and, more generally, Hessenberg varieties; 3) computing the (equivariant) cohomology rings of Hessenberg varieties; and 4) describing generalized splines for different edge-labeled graphs. The PI is an expert in Hessenberg varieties and the GKM approach to equivariant cohomology; a sequence of the PI's papers on these subjects, together with new work by Harada, Rhoades, and others, leave the field poised on the brink of new discoveries.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.