This project investigates combinatorial structures arising in algebraic geometry, particularly the Schubert calculus, whose origins are in the enumerative geometry of the late 19th century, but whose applications touch on modern mathematics such as quantum cohomology and quantum K-theory. The research will focus on the infinite-dimensional homogeneous spaces known as affine Grassmannians and affine flag varieties, which will be studied using combinatorial, algebraic, and geometric methods. In particular, explicit computational methods will be developed for various kinds of cohomology rings of these varieties. An example is the use of Groebner basis methods to obtain new formulae for the homology basis of the affine Grassmannian of a special linear group called k-Schur functions, which generalizes the classical expression of a Schur function as the weight generating function of Young tableaux. This project has applications to the theory of Macdonald polynomials and may provide new combinatorial formulae for Gromov-Witten invariants of flag manifolds.
The investigation is largely fueled by extensive computational experimentation. The robust implementation of algorithms derived from the project will lead to the development of new packages for computer algebra systems, which allow a user to build and manipulate computer implementations of mathematical objects. The dissemination of this software, through an open-source system, will not only advance the proposed research program, but will also have an outreach impact on the mathematics, physics, and computer science communities.
