Geometric Representation Theory: Interface of Algebra, Geometry and Topology.

一般目的是研究与简单代数群有关的紧空间的几何和拓扑。该项目将重新审视Atiyah、Borel、Bott、Hirzebruch等人的一些经典论文，试图用现代的方法来提高我们对空间的理解。一个特别的问题是研究与sl_2嵌入到简单李代数有关的自同构李代数。该项目将研究其表征理论和分类。第二个方向是研究H\G/K形式的空间，其中G是一个局部紧群，K是它的紧子群，H是G中的一个格，设G=PU（2,1）。H\B^2的几何性质是什么其中B^2=G/K是C^2中的单位球。产生光滑表面H\B^2且c2=3（或6），c1^2=9（或18）的格的可通约性类别是什么？他们总是算术吗？我们能否对所有算术格进行分类使得曲面H\B^2有c2=6 c1^2=18？第三组空间是K\G/H，其中所有群都是紧的。我们能否找到有效的方法来处理这些空间的拓扑不变量？所有这些问题在基础科学中都有重要的潜在应用。对社会和经济的好处不太可能在短期或中期实现。

The general aim is to study geometry and topology of compact spaces related to simple algebraic groups. The project will revisit some of the classical papers by Atiyah, Borel, Bott, Hirzebruch, etc. trying to use the modern methods to improve our understanding of the spaces. One particular question is to investigate the automorphic Lie algebras related to the sl_2 embeddings into a simple Lie algebra. The project will look into its representation theory and categorification.The second direction is to study spaces of the form H\G/K where G is a locally compact group, K is its compact subgroup, H is a lattice in G. Say G=PU(2,1). What are the geometric properties of H\B^2 where B^2=G/K is the unit ball in C^2. What are commensurability classes of lattices that produce a smooth surface H\B^2 with c2=3 (or 6), c1^2=9 (or 18)? Are they always arithmetic? Can we classify all arithmetic lattices such that the surface H\B^2 has c2=6, c1^2=18?The third set of spaces are K\G/H where all the groups are compact. Can we develop efficient methods of working with topological invariants of these spaces?All these questions have serious potential applications in fundamental science. The benefits to society and economy are not likely in short or medium term.