Professor Casian will attempt to extend his recent proof of the Kazhdan-Lusztig conjecture for Kac-Moody algebras in the symmetrizable case in various directions. One direction will involve Schubert varieties of finite dimension instead of those of finite codimension or infinite dimension. Another will entail trying to remove the symmetrizability assumption. In addition Professor Casian's research will focus on weight filtrations on Harish-Chandra modules. This research involves the theory of group representations. Group theory is basically the theory of symmetry. To take a simple example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group.
