Mathematical Sciences: Geometry and Representations of Lie Groups and Algebras

Casian教授将尝试在各个方向上扩展他最近对Kac-Moody代数的Kazhdan-Lusztig猜想的证明。一个方向将涉及有限维的舒伯特变换，而不是有限余维或无限维的舒伯特变换。另一种方法是试图消除对称性假设。此外，Casian教授的研究将集中在Harish-Chandra模块的权重过滤。本研究涉及群体表征理论。群论基本上就是对称理论。举一个简单的例子，当所讨论的系统在空间原点位置的变化下是不变的，平移组自然产生。虽然群是抽象的对象，但特殊情况需要对称群的具体实现或“表示”。

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.