This proposal spans combinatorial problems in representation theory, topology, algebraic geometry, and physics. A common thread is drawn by the connection of each to the weak and strong (Bruhat) orders on the type-A affine Weyl group. The underlying idea is that there are general methods for attacking the wide variety of problems that arise in this affine setting. Inspiration comes from the role that tableaux, Bruhat posets, and crystal bases play in theories such as symmetric functions, Schubert calculus and quantum algebras. The aim is to establish a combinatorial backbone that supports new theories in the same way. An emphasis is on applications to Gromov-Witten invariants, conformal field theories, Schubert calculus, and Macdonald polynomials. A guiding light will be the connection of each of these applications to the k-Schur functions.
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 rudimentary 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 an elementary combinatorial result in modular arithmetic. This project is devoted to developing combinatorial techniques for attacking problems that connect to algebraic and geometric areas such as 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, as the use of a computer algebra system is crucial for this investigation.
