Combinatorics of Macdonald Polynomials and Schubert Calculus

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Combinatorics is an active and central branch of pure and applied mathematics. Because the field is concerned with the development of tools for analyzing, organizing, and arranging discrete data, combinatorial methods are essential in many scientific areas such as genomics, computer science, statistics, and physics. The methods can often be traced back to research inspired by problems in algebra and geometry. For example, the RSA public-key encryption algorithm is based on a combinatorial result in modular arithmetic. This research project is devoted to developing combinatorial techniques for attacking problems that connect to algebraic and geometric areas such as symmetric function theory, a subject with applications to probability and statistical mechanics. The investigation will make use of a computational symbolic algebra system and will further develop the SAGE open-source mathematics software suite. This research project spans combinatorial problems in representation theory, algebraic geometry, and physics. The inspiration comes from the central importance of constructions such as tableaux and Bruhat posets in the classical studies of representations of the complex linear group and Schubert calculus. The precision of combinatorics is carried to more abstract problems by a distinguished Schur basis for the algebra of symmetric functions. The project concerns variations on the classical theories that address subtle questions about geometric Gromov-Witten invariants, symmetric functions over a field of parameters, bi-graded representations, and quantum integrable systems. A priority is to develop the necessary combinatorial framework for the contemporary set of problems. The venture goes hand in hand with an investigation of refined physical, geometric, and algebraic theories attached to the combinatorics. The resulting combinatorial constructions will lend computational facility to related application areas.