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

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.