The theory of Macdonald symmetric functions is one of the most active areas in enumerative combinatorics. At the heart of this theory is the interpretation, due to Haiman, of Macdonald polynomials in terms of the algebraic geometry of families of points in the complex plane, that is, of the Hilbert scheme. The PI, joint with Grojnowski, proposes to generalize this to higher dimensions. In order to avoid the algebraic geometric difficulties involved, they study an analog of a small piece of the usual 2-dimensional theory, the theory of the two variable Catalan polynomial. The project will continue Fishel and Grojnowski's program for defining the combinatorial d-dimensional analogs of the Catalan polynomial. It will explain detailed computations of the higher dimensional algebraic geometry, and motivated by this, detailed computations with a family of poset structures on the 1-skeleton of the associahedron, a simplicial complex whose vertices are Dyck paths. The PI expects that further study of these structures will lead to a combinatorial description of the geometric quantities, and to a generalization of much of the theory of Macdonald symmetric functions.
The theory of Macdonald symmetric functions is already a vast body of work, with exciting applications and interactions with representation theory, algebraic geometry, statistics, and even physics (for example, in studying ``the topological vertex'' of string theory). The PI expects that, just as with the Macdonald polynomials, these higher dimensional symmetric functions will arise in diverse mathematical fields.
