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.

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.

We study the classification of Hilbert space contractions belonging to the class $C_{0}$ via their relation with their Jordan models.This classification is carried out in two different settings: up to similarity and up to unitary equivalence, both of which are stronger than the usual quasisimilarity relation that is known to always hold between a $C_{0}$ contraction and its model.

We obtain positive results under a variety of assumptions, ranging from function theoretic (the Vasyunin approach using the Carleson condition for sequences of inner functions) to operator algebraic (the Arveson approach using boundary representations of operator algebras).

The general aim of Alain Connes' noncommutative geometry is to develop selected geometric ideas using techniques from Hilbert space theory, with the goal of then transporting these concepts to new and unfamiliar contexts. But in this lecture I'll stay mostly with the familiar. I'll start with Hermann Weyl's theorem on eigenvalue asymptotics, examine it from the noncommutative-geometric point of view, and explain how a tentative idea of what a noncommutative geometric space might actually be starts to emerge in the process.

This talk is about the decomposition of the regular representation of a group like SL(2,R) into its irreducible constituents. The decomposition has both continuous and discrete parts, and more specifically my talk is about the continuous part, which arises through so-called parabolic induction. I'll describe a Hilbert bimodule construction, due to Pierre Clare, that places parabolic induction in a noncommutative-geometric, or C*-algebraic, context. It is also possible to construct an opposite "parabolic restriction" bimodule, although this is not at all trivial. The question of whether the induction and restriction bimodules are adjoint to one another has some interesting geometric aspects. This will be the main focus of the talk.

In cooperatively breeding species, individuals help to raise offspring that are not their own. We use two population-genetic models to study the advantage of this kind of helpful behaviour in social groups with high reproductive skew. Our first model does not allow for competition among relatives to occur, but our second model does. Specifically, our second model assumes a competitive hierarchy among nestmates, with non-breeding helpers ranked higher than their newborn siblings. For each model we obtain an expression for the change in inclusive fitness experienced by a helpful individual in a selfish population. The prediction suggested by each expression is confirmed with computer simulation. When model predictions are compared to one another, we find that helping emerges under a broader range of conditions in the second model. Although competition among kin occurs in our second model, we conclude that the life-history features associated with this competition also act to promote the evolutionary transition from solitary to cooperative breeding.