Mathematical Sciences: Representation Theory and AutomorphicForms

Speh教授将继续与J.Rohlf一起研究算术群自同构的Lefschetz数。她计划找到Lefschetz数的一个稳定化(在朗兰兹意义下)，然后利用这个稳定条件，利用Arthur-Selberg迹公式找到Lefschetz数的一个表达式。Barbasch教授将继续他关于实还原群的幂等表示、它们的特征标理论以及它们在酉对偶分类中的作用的工作。他将参与几个项目：(I)与J.Adams一起，他将研究幂等表示情况下的对偶还原对对应；(Ii)与A.Moy，他将研究非分支表示的酉对偶；(Iii)与S.Evens，他将研究轨道几何、类似的Shubert簇、和有限维表示中的重数。群论基本上是对称论。举一个简单的例子，当所讨论的系统在空间原点的位置改变时是不变的，自然就产生了翻译群。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”。Speh教授和Barbasch教授的项目将有助于我们对小组陈述的理解。

Professor Speh will continue her research with J. Rohlfs concerning Lefschetz numbers of automorphisms of arithmetic groups. She plans to find a stabilization (in the sense of Langlands) of the Lefschetz numbers and then use this to find an expression for the Lefschetz number using the Arthur-Selberg trace formulas. Professor Barbasch will continue his work on unipotent representations of real reductive groups, their character theory, and their role in the classification of the unitary dual. He will be involved in several projects: (i) with J. Adams he will study the dual reductive pair correspondence for the case of unipotent representations, (ii) with A. Moy he will study the unitary dual for unramified representations, and (iii) with S. Evens he will study relations between the geometry of orbits, similar Shubert varieties, and multiplicities in finite dimensional 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. The projects of Professors Speh and Barbasch will contribute to our understanding of group representations.