#### Availability

Due to existing commitments, I am unlikely to take any new PhD students in the 2020-21 cycle, although I am always willing to hear from applicants with a strong background in functional analysis. In any case, the list below of possible supervision areas is included for sake of background interest.

#### Current ideas for PhD projects

Here are three possible areas in which I am willing to supervise: each of these is not a specific PhD project, but a setting in which there are various possible research problems that a student could work on. If you would like to know more, then please feel free to get in touch.

- Algebras associated to group actions on L
^{p} spaces
- Cohomological invariants of Fourier algebras
- Properties of Banach-Kähler modules

#### Warning notes.

Themes 1 and 2 are likely to require some prior exposure to the basic theory of C*-algebras. However, some of the necessary background can be learned in the initial stages of the PhD, and depending on the individual strengths of an applicant there may be particular projects that require less functional-analytic machinery; some questions may be suitable for those with solid basic analysis and interests in the representation theory of finite groups.

It is possible that you are reading this and have had some exposure to various structural properties of Banach algebras known as "approximate amenability", "character amenability", or "module amenability". **I will not, for the foreseeable future, supervise on any of these three, nor on any hybrid of these.**

### The keyword version

Banach and operator algebras. Noncommutative harmonic analysis. Categorical and homological perspectives on functional analysis.

### The longer version

I am interested in a range of topics and problems on the interface between algebra and functional analysis. This tends to be driven by particular families of examples, but my preference is for finding general frameworks that unify common features of these examples, once one has investigated individual examples in depth. "Hacking out lots of estimates and then trying to exploit functorial behaviour" is not the most accurate description, nor the most catchy one, but it gives some idea of what I spend my research time doing.

In recent years I have worked intensively on various aspects of the Fourier algebras of locally compact groups: studying these objects involves a blend of (non-commutative) harmonic analysis and representation theory. Many of the developments over the last 20 years or so use the tools, or are guided by the philosophy, of the theory of *operator spaces* and *completely bounded maps* -- these may be thought of as enriched versions of Banach spaces and linear maps between them, where one allows matricial coefficients and not just scalar ones.

I am also interested in algebras of convolution operators arising from the canonical representation(s) of a group G on the Banach space L^{p}(G). When *p*=2 there is a rich theory available, from the world of C*-algebras and von Neumann algebras; for other values of *p* many basic structural questions remain unresolved, even for very explicit examples such as G=SL_{n}(R) or SL_{n}(Z).

Director of Studies for Study Abroad students

Mahmood Alaghmandan, 2013. (Supervised at the University of Saskatchewan, Canada, joint with E. Samei)

Bence Horvath, 2019. (Joint with N. J. Laustsen)

Christopher Menez, 2019.

Jan. 2014 - present: Lecturer at Lancaster University, United Kingdom

Aug. 2010 - Jan. 2014: Assistant Professor (tenure-track), University of Saskatchewan, Canada

Aug. 2008 - Jul. 2010: Postdoctoral researcher, Universite Laval, Canada

Feb. 2007 - Jul. 2008: Postdoctoral fellow, University of Manitoba, Canada

Sept. 2002 - Jun. 2006: PhD student, University of Newcastle upon Tyne, United Kingdom

Sept. 2001 - Jun, 2002: Editorial assistant, National Extension College, United Kingdom.

B.A. in Mathematics (2000, Cambridge).

CASM (2001, Cambridge).

PhD in Pure Mathematics (2006, Newcastle-upon-Tyne)

External examiner

S. Trotter (Leeds, PhD, 2016)

Y. Ji (Newcastle, PhD, 2018)

Internal examiner

E. Kastis (PhD, Lancaster, 2017)