Collaborative Research: Catalan Function and Schubert Calculus

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Combinatorics is an area of mathematics concerned with analyzing, organizing, and optimizing over discrete data. It is a fundamental tool in many scientific areas such as genomics, computer science, statistics, and physics. This project will develop combinatorial methods for attacking problems in algebra, geometry, and symmetric function theory, an area with applications to probability and statistical mechanics. A mutually beneficial component is the further development of the SAGE open-source mathematics software, a crucial tool for this investigation.This project spans combinatorial problems in representation theory, algebraic geometry, and physics. The inspiration comes from constructions such as tableaux and Bruhat posets which lie at the heart of classical mathematics such as Schubert calculus and the representation theory of the complex linear group. Fundamental objects in our investigations are the Schur functions and generalizations known as k-Schur functions, which are closely tied to (affine) Schubert calculus. This project will capitalize on PIs' recent work connecting k-Schur functions with certain Catalan symmetric functions to solve problems involving Macdonald polynomials, parabolic Hall-Littlewood polynomials, Gromov-Witten invariants, and the co(homology) and K-theory of affine Grassmannians.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.