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
Daniel Drimbe (2023-)
Julian Kranz (2023-)
Sam Evington (2018-2020)
James Gabe (2017-2019)
Joan Bosa (2013-2015, 2016-2017)
Students and thesis work
Jakub Curda (2023-)
Brian Chan (2023-)
Shanshan Hua (2021-)
K-stability of Z-stable C*-algebras. preprint.
Robert-Mihai Neagu (2020-). Co-supervised by Jamie Gabe.
The admissible KMS bundles on classifiable C*-algebras, preprint.
Inclusions of real rank zero (with Gabe), preprint
An Elliott intertwining approach to classifying actions of C*-tensor categories (with Girón Pacheco), preprint.
A note on when quasidiagonal traces are amenable, J. Op. Theory, to appear.
On topologically zero-dimensional morphisms (with Castillejos Lopez), J. Funct. Anal., 296(9), 110368, 2024.
Sergio Girón Pacheco (2018-2023). DPhil. Co-supervised by Sam Evington. Thesis
An Elliott intertwining approach to classifying actions of C*-tensor categories (with Neagu), preprint.
A classification of anomolous actions through model action absorption, preprint.
Anomolous symmeteries of classifiable C*-algebras (with Evington), Studia Math, 270, 73-101, 2023.
The Cuntz-Topelitz algebras have nuclear dimension 1 (with Easo et. al), J. Funct. Anal., 279(7), 108690, 2020.
Sam Evington (2013 - 2017). Phd. Thesis
Locally trivial W*bundles (with Pennig), Int. J. Math., 27 (11), 2016.
Jorge Castillejos Lopez (2012 - 2016). PhD. Thesis
Decomposable approximations and approximately finite dimensional C*-algebras, Math. Proc. Cam. Phil. Soc., 162(1), 1-12, 2017.
Tomasz Pierzchala (2010 -2013). MRes. Thesis
Liam Dickson (2010 - 2014). PhD. Thesis.
A Kadison-Kastler row metric and intermediate subalgebras, Int. J. Math., 25(8), 1450082, 2014.
Editorial Work
2023 - Present. Journal of the London Mathematical Society
2023 - Present. London Mathematical Society Student Text Series.
2023. Special issue of the Münster Journal of Mathematics dedicated to the memory of Eberhard Kirchberg
2021 - Present. Quarterly Journal of Mathematics
2013 - 2022. Proceedings A of the Royal Society of Edinburgh.