Research

My research focuses on operator algebras, this is a branch of functional analysis with connections to many other branches of pure mathematics. The central objects of study are C*-algebras and von Neumann algebras. These can be defined as *-subalgebras of the bounded operators on a Hilbert space which are closed in the norm and weak operator topology respectively, but also admit an abstract characterisation. Since every abelian C*-algebra is the algebra of continuous functions vanishing at infinity on a locally compact space, the study of C*-algebras should be thought of as non-commutative topology. Likewise, von Neumann algebras are the non-commutative analogue of measure spaces. I study both C*-algebras and von Neumann algebras and am particularly interested in the interplay and transfer of ideas between these different types of algebras.

There are strong parallels between recent developments in the fine structure of simple nuclear C*-algebras, and Connes ground breaking work on the structure of injective von Neumann factors in the 1970's. A key theme here is the development of coloured (i.e. higher dimensional) versions of von Neumann properties, in the topological setting of C*-algebras. My research in this direction was supported by EPSRC from Oct 2012 to June 2015, and was the main focus behind an Alexander von Humboldt foundation fellowship from 2015-18. The applicability of von Neumann methods in the setting of C*-algebras continues to grow; a major theme in my current research is the use of von Neumann ideas to give an abstract approach to classification and structure. This was funded by EPSRC from 2018-2020.

Postdocs

Students and thesis work

Editorial Work