This project concerns the development of a vast extension of Schubert calculus to affine Grassmannians and affine flag varieties, called ``affine Schubert calculus". Classical Schubert calculus, a branch of enumerative algebraic geometry concerned with counting subspaces satisfying certain intersection conditions, is the outcome of the solution to Hilbert's Fifteenth problem. In the modern formulation, Schubert calculus is usually interpreted in cohomology theories of homogeneous spaces, most notably flag varieties. The full development of affine Schubert calculus will solve long-standing open problems in Macdonald theory and have an impact on physical questions, such as generalizations of Wess-Zumino-Witten conformal field theory models and extensions of Calogero-Sutherland quantum mechanical models whose eigenfunctions are k-Schur functions. The new approach to affine Schubert calculus is made possible by the recent discovery of certain explicitly defined symmetric functions called k-Schur functions. The k-Schur functions, which arose in the study of the seemingly unrelated Macdonald theory, were recently shown to be connected to the geometry and topology of the affine Grassmanian. The novel combinatorics of k-Schur functions will be exploited to deduce formulae for various multiplicities, including intersection multiplicities in the affine Grassmannian and the affine flag manifold. Some of these multiplicities are known to occur in Macdonald theory and as Verlinde fusion coefficients for the WZW model.
This many-faceted project involves and ties together various problems from combinatorics, geometry, representation theory, physics, and computation. The main questions that will be addressed can be viewed from several points of view: a geometric perspective (questions such as "how many lines are there satisfying a number of generic intersection conditions?"), a combinatorial perspective ("how many elements are in given sets and what properties do these sets have?"), a physics perspective ("how do fields correlate?"), and computational aspects ("are there efficient algorithms for calculating these numbers or objects?"). The project is an international cooperative research venture, with core group members located in Canada, the United States, Chile, and France, and interdisciplinary, involving mathematicians, physicists, and computer scientists. Graduate students will receive professional training by direct involvement in the research and will benefit from interaction with the research team. A summer school at the Fields institute for graduate students is also planned at the conclusion of the project. 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. The dissemination of this new software through an open-source computational package, will not only advance the proposed research program but will also have an outreach impact on the mathematics, physics, and computer science communities.
