Fundamental Questions in Theory of Algebraic and Kac-Moody Groups

研究将位于三个研究领域的接口：代数，几何和拓扑，数论。总体目标是研究关于约化代数群的基本问题，偶尔将结果和方法扩展到其他领域，如Kac-Moody群或2-群。第一个问题是从李代数Lie（G）到G的模的正特征积分。一些方法将被部署，希望能阐明一些旧的汉弗莱斯和维尔马的著作。另一个方向是研究这些群上的算术微分算子。这包括它们的几何形状、结构和表示理论。最后一个方向是研究形式为H\G/K的局部对称空间，其中G是局部紧群，K是它的紧子群，H是G中的格。一个特别的方向是根据几何性质H\B^2来研究G=PU（2，1）中的格，其中B^2=G/K是C^2中的单位球。以下问题令人感兴趣。产生光滑曲面H\B^2且c2=3（或6），c1^2=9（或18）的格的可积性类是什么？它们总是算术吗？我们能否对所有算术格进行分类，使得曲面H\B^2具有c2=6，c1^2=18？所有这些问题都有重要的潜在应用，尽管只是在基础科学领域。在短期或中期内，对社会和经济的好处是不可能的。

The research will lie on the interface of three research areas: Algebra, Geometry and Topology, Number Theory. The general aim is to study fundamental questions about reductive algebraic groups, occasionally expanding the results and the methods into other areas, such as Kac-Moody groups or 2-groups. The first question is integration of modules from the Lie algebra Lie(G) to G in positive characteristic. A number of methods will be deployed, hoping to shed light on some old conjectures by Humphreys and Verma. Another direction is the study of arithmetic differential operators on these groups. This include their geometry, their structure and their representation theory. The final direction is the study of locally symmetric 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. One particular direction is to study lattices in G=PU(2,1) in the light of geometric properties H\B^2 where B^2=G/K is the unit ball in C^2. The following questions are of interest. 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? All these questions have serious potential applications, although only in fundamental science. The benefits to society and economy are not likely in short or medium terms.