Fundamental Questions in Theory of Algebraic and Kac-Moody Groups

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.