THEMATIC PROGRAMS

Gabor Szabo, University of Muenster
The Rokhlin dimension of topological $\mathbb{Z}^m$-actions

We study the topological variant of Rokhlin dimension for topological dynamical systems $(X,\alpha,\mathbb{Z}^m)$ in the case where $X$ is assumed to have finite covering dimension. Finite Rokhlin dimension in this sense is a property that implies finite Rokhlin dimension of the induced action on C*-algebraic level, as was discussed in a recent paper by Ilan Hirshberg, Wilhelm Winter and Joachim Zacharias. In particular, it implies (in this context) that the transformation group C*-algebra has finite nuclear dimension. For a single aperiodic homeomorphism, finite Rokhlin dimension follows easily from a recent result by Yonathan Gutman, which in turn uses an important property shown by Elon Lindenstrauss. We show that their methods can be pushed to work in the realm of actions of countably infinite groups. In the particular case of free $\mathbb{Z}^m$-actions on a fixed space $X$, application of said methods yields a uniform bound of Rokhlin dimension that depends only on $m$ and $\operatorname{dim}(X)$.

Mike Whittaker, University of Wollongong
C*-algebras associated to self-similar actions and Zappa-Szép product semigroups

A self-similar action (G,X) consists of a group G and finite alphabet X along with an action of the group on the free monoid X* consisting of all finite words in the letters of X. Mark Lawson showed that a self-similar action could be realised as a Zappa-Szép product semigroup X* x G. A universal Toeplitz algebra can be constructed for a self-similar action with a Cuntz-Pimsner algebra quotient. We show that
a generalised self-similar Toeplitz algebra can be constructed for a Zappa-Szép product with a boundary quotient generalising the Cuntz-Pimsner algebra of a self-similar action. We use this to describe new presentations of C*-algebras associated to semigroups constructed by Xin Li. Some examples include the Baumslag-Solitar groups and the affine semigroup over the natural numbers.
This is joint work with Nathan Brownlowe, Jacqui Ramagge, and Dave Robertson.

Andrew Dean, Lakehead University
Classifying actions of the reals on ${C^*}$-algebras

We shall consider some classes of actions of the reals on ${C^*}$-algebras and develop invariants to classify them up to equivariant isomorphism.

Bingbing Liang, SUNY at Buffalo
Mean Dimension and von Neumann-L${\ddot{u}}$ck Rank

Mean dimension is a numerical invariant in topological dynamics and related to entropy. The von Neumann-Luck rank is a L^2-invariant and related to L^2 Betti number.
I will establish an equality between the von Neumann-Luck rank of a module of the integral group ring of a amenable group and the mean dimension of the associated algebraic action. Also I will give some applications of this equality. This is a joint work with Hanfeng Li.

Ian Putnam, University of Victoria
Spectral triples for subshifts

Building on ideas of Christensen and Ivan, we give a construction for spectral triples for the crossed products of the group of integers on a Cantor set which arise from subshifts. We examine their basic properties.
(Joint work with A. Julien)