The importance of symmetric functions and the representation theory of Hecke algebras and the symmetric groups derives in part from their applicability in a wide range of scientific and mathematical disciplines. Within the theory of symmetric functions, this programme will focus on a particular topic, the Macdonald polynomials, which have especially wide-ranging mathematical interconnections. The goal of the program will be to unify the diverse approaches to the study of these polynomials.
n the 1980's, I.G.Macdonald formulated a series of conjectures which predicted the constant terms of expressions that involve an important new class of symmetric functions called the Macdonald polynomials. Since their introduction, these conjectures and polynomials have been a central topic of study in Algebraic Combinatorics. Of particular note has been the variety of approaches used in efforts to solve the conjectures or to find an algebraic or geometric interpretation for the Macdonald polynomials themselves. Different approaches involve double affine Hecke algebras, homology of nilpotent Lie algebras, generalized traces of Lie algebra representations and diagonal actions of the symmetric group on polynomial rings in two sets of variables. In this programme we will attempt to unify these different approaches to the Macdonald Conjectures in a way that allows for a significant interpretation of the Macdonald polynomials and settles some of the outstanding conjectures that have resulted from this work.
Links with other areas such as algebraic geometery, Lie algebras, non-commutative algebra, mathematical physics and mathematical statistics will be emphasised. Workshops will be arranged in order to foster existing and potential applications in these and other subjects.