My research
I am interested in Lie groups, equivariant symplectic geometry and related algebraic topology. More specifically, I am interested in geometric quantization of q-Hamiltonian spaces.
Freed-Hopkins-Teleman Theorem (FHT) asserts that the equivariant twisted K-homology of a compact connected Lie group G with torsion-free fundamental group(with ring structure being Pontryagin product) is isomorphic to Verlinde algebra of G, which is the ring of positive energy representations of the loop group LG, with ring structure being the fusion product. Motivated by this result, Meinrenken realized the quantization of q-Hamiltonian G-manifolds as equivariant K-homology pushforward induced by the G-valued moment maps. This quantization scheme has several advantages in that it does not involve quantizing the corresponding Hamiltonian LG-space, which is an infinite dimensional Banach manifold, and does not mention any twisted Dirac operator at all.
The main theme of my thesis is about generalizing FHT and Meinrenken's result by replacing K-homology with KR-homology, a refined version of K-homology which takes into account the diffeomorphic involution on the manifold M and the automorphic involution of G which is compatible with both the G-action and the involution on M. I would like to see how different automorphic involutions on G yield different(and expectedly interesting) ring structures of the equivariant twisted K-homology of G.