Abstract: The main goal of this lecture is to illustrate in one extended example some basic topological techniques in the C*-algebraic approach to index theory. The quantization commutes with reduction phenomenon (which I shall explain from scratch in the talk) was first explored by Guillemin and Sternberg within the context of Kahler geometry. A great deal has been achieved since then, but I want to return to the original complex-geometric context and examine there a remarkable Dirac operator approach developed by Tian and Zhang. This was originally framed within the context of symplectic geometry, but it simplifies considerably in the Kahler case, especially from the C*-algebra point of view.

Abstract: Torus manifold is a compact oriented $2n$-dimensional $T^n$-manifold with fixed points. We can define a labelled graph from the given torus manifold as follows: vertices are fixed points; edges are invariant $2$-dim sphere; edges are labelled by tangential representations around fixed points. This labelled graph is called a torus graph (this may be regarded as the generalization of special class of GKM graph). It is known that the equivariant cohomology of torus manifold can be computed by using combinatorial data of torus graphs. In this talk, we study when torus actions of torus manifolds can be induced from non-abelian compact connected Lie group (i.e., when torus actions can be extended to non-abelian group actions). To do this, we introduce root systems of torus graphs. By using this root system, we characterize what kind of compact connected non-abelian Lie group (whose maximal torus is $T^n$) acts on torus manifold. This is a joint work with Mikiya Masuda.