The proposer will work on problems in representation theory of quantum affine algebras, infinite-dimensional Lie algebras and Macdonald polynomials. He will continue to develop the vertex operator approach to symmetric functions. In particular he will use this to study the realization of affine canonical basis in terms of Macdonald polynomials and will also investigate its application to crystal bases. In addition he will use the recent method of quantum wedge modules to study perfect crystal bases.
This project studies quantum affine algebras and vertex operator algebras, which are recent generalizations of Lie algebras and Lie groups. The study of Lie algebras and quantum groups is aimed at revealing more symmetry that existed in nature. This symmetry is crucial to the theory of quantum mechanics and quantum field theory, one of the most important scientific theories in this century.