- 10:00--10:50 Yoshikata Kida (Tokyo): On a treeing arising from the Baumslag-Solitar group
Abstract: To a probability-measure-preserving action of a countable group, the transformation-groupoid is associated. This talk is focused on the Baumslag-Solitar group. After reviewing invariants and structure of the associated groupoids, I will report that some of them admit a quotient with a treeing, which is an analogue of a free-generating-set of a group, and its application.
- 11:10--12:00 Jacek Brodzki (Southampton): A differential complex for CAT(0) spaces and K-amenability
Abstract: In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to $C^*$-algebra $K$-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of $p$-adic groups.
This talk will present an extension of the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces which relies on the beautiful geometry of nonpositively curved spaces. The utility of this construction is demonstrated through a new proof of $K$-amenability for groups that act properly on bounded-geometry CAT(0)-cubical spaces. This talk is based on joint work with Nigel Higson and Erik Guentner.
- 14:00--14:50 Serge Richard (Nagoya): Levinson's theorem: topological, complex, and infinite
Abstract: During this seminar we shall present the most recent findings about Levinson's theorem. This theorem, first obtained by Levinson in 1949, turns out to be an index theorem in scattering theory. Recently, its topological version has been extended for systems having either a finite number of complex eigenvalues or an infinite number of bound states. Part of these developments are based on a recent work about Schroedinger operators with inverse square potential on the half-line.
- 15:10--16:00 Bai-Ling Wang (ANU): Dirac operators on Orientifolds and orientifold K-theory
Abstract: An orientifold is a proper Lie groupoid $(X_1, X_0, \epsilon)$ with a groupoid homomorphism $\epsilon$ from the space $X_1$ of arrows to $Z_2 = {\pm 1}$. It is a generalization of the notion of orbifold and manifold with involution. An arrow is called even or odd if its value under $\epsilon$ is $+1$ or $-1$. Orientifold vector bundle is a complex vector bundle over $X_0$ on which the even arrows in $X_1$ act complex linearly and the odd arrows act complex anti-linearly. In this talk, I will focus on one particular example (an action orientifold) coming from an action of a Lie group $G$ on a manifold $X$ with a group homomorphism $\epsilon: G \to Z_2$. I will discuss a notion of Dirac operator on a compact Spin^c orientifold whose index provides an element in the unitary-antiunitary representation ring of $(G, \epsilon)$.