Abstract: I will review the Sylvester Equation AX+XB  = C in finite and
infinite dimensions leading to recent results about solutions subject to
side conditions which are suggested by certain applications.

Abstract: A right Hilbert A-module, E, for some C*-algebra, A, is a generalization of a Hilbert space. Instead of the usual complex-valued inner product, we have an A-valued inner product. We also add a right action of A on E. While these slight modifications seem innocuous, many results of Hilbert space theory that one might expect to carry over, do not. Just as with Hilbert spaces, it is a certain class of operators which act on Hilbert A-modules which are of particular interest, as they form a C*-algebra. I will present an introduction to these modules and their corresponding operators.

Abstract (PDF file)

Abstract: Higher-rank graphs are a higher-dimensional analogue of
directed graphs introduced by Kumjian and Pask in 2000 as a
combinatorial model of Robertson and Steger's higher-dimensional
Cuntz-Krieger algebras associated to families of commuting matrices.
Formally, a higher-rank graph is a category satisfying some special
conditions. We will show some of the analogues of results for directed
graphs, and show a construction of higher-rank graphs in terms of
directed graphs that provides a different and potentially more intuitive
way of thinking them.

Abstract: Graph C*-algebras are a generalisation of Cuntz and Kriegers C*-algebras associated to topological Markov chains from their seminal work in 1980. We may associate a C*-algebra to a directed graph in such a way that many properties of the C*-algebra may be deduced from simpler combinatorial properties of the graph. There are several different methods of associating a C*-algebra to a directed graph. I will outline how we do this using groupoids, as a universal object, and as a couniversal object. I will present the two most frequently used results:

the gauge invariant uniqueness theorem, and the Cuntz-Krieger uniqueness theorem. I will then show compute some examples and present some interesting structure results.

Abstract: In the 1960's, Stephen Smale initiated an ambitious program to
understand the dynamics of a certain class maps on smooth manifolds, which he
called Axiom A. David Ruelle gave a definition of a Smale space to describe,
in purely topological terms, the dynamics of an Axiom A system on its
non-wandering set. I will discuss these definitions and give several concrete
examples. Later, Anthony Manning proved that the zeta function for such
systems was rational and Rufus Bowen conjectured that this was due to the
existence of an underlying homology theory. A partial solution to this was
given by Krieger and also Bowen and Franks when they constructed a very
beautiful invariant for shifts of finite type. I will discuss this invariant
and then show how it can be extended to all Smale spaces as a homology theory,
as predicted by Bowen.

Abstract: We describe a method of constructing finitely summable
Fredholm modules over the reduced C*-algebra of
a Gromov hyperbolic group. The method is to utilize the
action of the group on a certain geometrically defined `boundary'.

In the first talk we will introduce Gromov hyperbolic groups and
their boundaries, and the first major ingredient of the construction:
an analogue of the Poisson transform for hyperbolic groups.
In the second talk we will go into K-theory and K-homology and
derive the main result.

Abstract: We describe a method of constructing finitely summable
Fredholm modules over the reduced C*-algebra of
a Gromov hyperbolic group. The method is to utilize the
action of the group on a certain geometrically defined `boundary'.

In the first talk we will introduce Gromov hyperbolic groups and
their boundaries, and the first major ingredient of the construction:
an analogue of the Poisson transform for hyperbolic groups.
In the second talk we will go into K-theory and K-homology and
derive the main result.

Abstract: The aim of the talk is to give an overview of recent
developments in semigroup C*-algebras. The emphasis will lie on two
aspects: Amenability and K-theory. We will also discuss several concrete
examples.

Abstract: A fatithful action of a group G on the space X* of words on
a finite set  X of 'symbols', is self-similar if for every
(g,x) in G x X there exists (h,y) \in G x X such that g. xw = y h.w.
These actions have been studied by Nekrashevych, who introduced a class
of C*-algebras associated to various completions of a natural bimodule
that arises naturally from the self-similarity.
I will start by reviewing these C*-algebras and their associated Toeplitz extensions;
both have natural periodic dynamics and on the second part I will discuss the structure
of the equilibrium state space for these C*-dynamical systems.
This is (current)  joint work with Iain Raeburn, Jacqui Ramagge, and Michael Whittaker.