The proposed research studies affine Kac-Moody algebras using Kashiwara's theory of crystals. A crystal here is a combinatorial object (a set along with some operations) associated to each highest weight representation of a symmetrizable Kac-Moody algebra. Kashiwara's construction makes heavy use of the associated quantized universal enveloping algebra, but the crystals themselves can often be realized by other means. Some such realizations are purely combinatorial, and others use non-trivial geometry. In finite type, one realization which has generated a lot of interest uses the Mirkovic-Vilonen (MV) polytopes developed by Anderson and by Kamnitzer. Along with his collaborators, the P.I. is currently developing a version of this combinatorics for symmetric affine types, using quiver varieties. Affine MV polytopes have been sought since the finite type polytopes first appeared, so their construction is itself an important development. The P.I. will investigate the connection between these new polytopes and various algebraic and geometric structures related to affine algebras. This work may also lead to a notion of MV polytope in all (not necessarily symmetric) affine types. The proposed research also considers ways of extracting other combinatorial realizations from the geometry of quiver varieties, and develops applications of crystal theory to Macdonald polynomials and Demazure characters.
This proposal addresses important questions in the theory of affine algebras, and will be of interest to a number of people in that field. Affine algebras are important in mathematical physics, so there is potential for cross-disciplinary impact. The proposal will also fund undergraduate research projects, and more generally contribute to the training and developing of young mathematicians. There are several questions related to this research that can be studied in terms of explicit realizations of crystals. These are ideal for undergraduate research projects, as only a limited amount of background is required in order to approach the questions, yet they can provide an entry point into a rich representation-theoretic story. The proposal will also support student seminars which are designed both to train students in advanced subjects and to develop their skills as presenters. Finally, the P.I. will continue to work with programs such as math circles aimed at high school students.