## 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; see for example the introductions to these papers. 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 is funded by EPSRC from 2018-2020.

Parallels between injective von Neumann algebras and nuclear C*-algebras have been crucial to my older research on close operator algebras and the Kadison-Kastler perturbation problem, which asks whether sufficiently close algebras on the same Hilbert space are necessarily isomorphic. Some of my joint work on this problem for nuclear C*-algebras is described in this expository paper and some developments for non-amenable von Neumann algebras in this expository paper.

I also have interests in other topics across operator algebras and functional analysis: recent joint work has examined approximation properties for locally compact quantum groups.

## Editorial Work

## Postdoctoral researchers

- Joan Bosa, 2013-2015, 2016-17
- James Gabe, 2017-2019.
- Samuel Evington, 2018-2020.

## Research Students

- Liam Dickson, PhD, 2010-2014. thesis
- Tomasz Pierzchala, MRes, 2010-2013. thesis
- Jorge Castillejos-Lopez, PhD, 2012-2016. thesis
- Samuel Evington, PhD, 2013-2017. thesis

## Conferences and seminars organised

- Joint meeting of the Edinburgh Mathematical Society and Societat Catalana de Matematiques, ICMS, Edinburgh Sept 2017
- Intensive research programme: Operator Algebras, Dynamics and Interactions, CRM Barcelona, March - July 2017
- Workshop on C*-algebras: Geometry and Actions, Münster, July 2015.
- Focused Programme on C*-algebras, Münster, April - July 2015.
- Conference on operator algebras and applications in celebration of Erik Christensen and his work, Copenhagen, May 2015.
- Workshop on Structure and Classification of C*-algebras, Münster, April 2015.
- C Star: Classification, Structure, Amenability and Regularity, Glasgow, 2014.
- Glasgow-Aberdeen operator algebras meetings, 2013-present.
- The structure and classification of nuclear C*-algebras, ICMS, Edinburgh, 2013.
- The SaLT(eD) cellar
- Operators and Operator algebras in Edinburgh, Edinburgh 2009.

## Cogne, Italy

An interesting way down from an ice climb, 2011.