Sims, Aidan D. Professor

Cartan Subalgebras
Shortly after the invention of graph C*-algebras in the late 1990s, the development of Leavitt path algebras began a fascinating interplay between functional analysis and abstract algebra that is still the subject of intensive study. Recently, Cartan subalgebras have emerged as a fundamental articulation point between the two, linking both to the study of groupoids. This project will explore questions of existence and structure of Cartan subalgebras in operator algebras, and continue to develop the theory of Cartan subalgebras in abstract algebra.

Twisted operator algebras associated to higher-rank graphs
Higher-rank graphs are elementary combinatorial objects akin to directed graphs in which directed paths have a k-dimensional shape instead of a 1-dimensional length. We use them as models for operator algebras, which are a formalism for infinite-dimensional linear algebra. Recent developments regarding the structure of k-graphs have led to a new construction of twisted operator algebras from higher-rank graphs. This project will investigate the structure of these twisted operator algebras, focussing on the key differences between the twisted and untwisted situations.

Equilibrium states of operator-algebraic dynamical systems
When operator algebras are used as models for quantum statistical mechanical systems, the time evolution of the system is described by an action of the real numbers by automorphisms of the operator algebra. Equilibria of the physical system, are described by a special class of states on the operator algebra which are called KMS states. The definition of a KMS state makes sense for operator-algebraic dynamics that do not arise from physical systems, and the KMS states seem to contain useful information about the operator algebra. This project will investigate the presence of KMS states on operator algebras constructed from discrete structures, and will examine what information about the discrete structure the associated KMS states encode.

Groupoids and operator algebras
A groupoid is like a group except that it has many identity elements, called units, no two of which can sensibly be multiplied. Groupoids turn out to be excellent models for operator algebras because many operator-algebraic properties have a neat characterisation given a suitable groupoid model. This project will focus on the representation theory of operator algebras associated to the étale groupoids that are most closely related to discrete groups.

Operator algebras and topological graphs
A topological graph is like a directed graph except that the spaces of vertices and of edges each comprise a topological space. Topological graphs are very flexible in the sense that it is easy to construct lots of examples by drawing pictures. They are particularly good for constructing continuous fields of operator algebras: take an ordinary directed graph, make a copy of it for each point in a topological space X, and then glue them all together using the topology of X. This project will investigate how continuous fields of this sort can be twisted using topological data from X to construct interesting fields of operator algebras, and to what extent the twisting data can be recovered from the operator algebra.

Room 39C.195

University of Wollongong, Northfields Ave

Wollongong

NSW

2522

AUSTRALIA

My general area of research is in mathematics, specifically in the areas of functional analysis, operator algebras, and noncommutative geometry. More specifically, I study C*-algebras. These infinite-dimensional analogues of matrix algebras first arose out of efforts in the 1930's and 1940's to provide a mathematical framework for quantum mechanics. These days C*-algebra theory is a very active area of mathematical research in its own right, and enjoys deep connections with other areas of mathematics such as symbolic dynamics, ergodic theory, group theory and even number theory. It also remains relevant to mathematical physics through noncommutative geometry and through applications of K-theory to the study of topological phases of matter.

I am particularly interested in studying C*-algebras associated to combinatorial objects like directed graphs, to algebraic objects like groupoids, and to analytic objects like Hilbert bimodules. Generally speaking I'm interested in understanding how to represent a given class of algebraic or combinatorial objects C*-algebraically, and investigating how the structural features of the resulting C*-algebras relate to the properties of the generating objects. Some of my key interests are in computing the primitive-ideal structure and regularity properties (for example, quasidiagonality, pure infiniteness and nuclear dimension) of these sorts of C*-algebras.

I am also interested in understanding C*-algebraic representations of dynamical systems; in particular in characterising the KMS equilibrium states of C*-dynamical systems.

My research interests also include purely algebraic analogues of groupoid algebras, and in the extent to which groupoid models can explain the striking parallels between the algebraic and analytic theories.

A full list of my articles and preprints is available on the arXiv at https://arxiv.org/a/0000-0002-1965-6451.html

Cartan Subalgebras
Shortly after the invention of graph C*-algebras in the late 1990s, the development of Leavitt path algebras began a fascinating interplay between functional analysis and abstract algebra that is still the subject of intensive study. Recently, Cartan subalgebras have emerged as a fundamental articulation point between the two, linking both to the study of groupoids. This project will explore questions of existence and structure of Cartan subalgebras in operator algebras, and continue to develop the theory of Cartan subalgebras in abstract algebra.

Twisted operator algebras associated to higher-rank graphs
Higher-rank graphs are elementary combinatorial objects akin to directed graphs in which directed paths have a k-dimensional shape instead of a 1-dimensional length. We use them as models for operator algebras, which are a formalism for infinite-dimensional linear algebra. Recent developments regarding the structure of k-graphs have led to a new construction of twisted operator algebras from higher-rank graphs. This project will investigate the structure of these twisted operator algebras, focussing on the key differences between the twisted and untwisted situations.

Equilibrium states of operator-algebraic dynamical systems
When operator algebras are used as models for quantum statistical mechanical systems, the time evolution of the system is described by an action of the real numbers by automorphisms of the operator algebra. Equilibria of the physical system, are described by a special class of states on the operator algebra which are called KMS states. The definition of a KMS state makes sense for operator-algebraic dynamics that do not arise from physical systems, and the KMS states seem to contain useful information about the operator algebra. This project will investigate the presence of KMS states on operator algebras constructed from discrete structures, and will examine what information about the discrete structure the associated KMS states encode.

Groupoids and operator algebras
A groupoid is like a group except that it has many identity elements, called units, no two of which can sensibly be multiplied. Groupoids turn out to be excellent models for operator algebras because many operator-algebraic properties have a neat characterisation given a suitable groupoid model. This project will focus on the representation theory of operator algebras associated to the étale groupoids that are most closely related to discrete groups.

Operator algebras and topological graphs
A topological graph is like a directed graph except that the spaces of vertices and of edges each comprise a topological space. Topological graphs are very flexible in the sense that it is easy to construct lots of examples by drawing pictures. They are particularly good for constructing continuous fields of operator algebras: take an ordinary directed graph, make a copy of it for each point in a topological space X, and then glue them all together using the topology of X. This project will investigate how continuous fields of this sort can be twisted using topological data from X to construct interesting fields of operator algebras, and to what extent the twisting data can be recovered from the operator algebra.

Room 39C.195

University of Wollongong, Northfields Ave

Wollongong

NSW

2522

AUSTRALIA