## Teaching

I'm taught 3H analysis of differentiation and integration, and 2 sections of 2B Linear algebra in 2016-17. I'm also the stream leader for the SMSTC pure analysis stream.

## Projects

I'm prepared to supervise a range of projects at level 5 related to functional analysis and operator algebras and other projects on the interface between algebra and analysis. Some possible projects are described below; I'm also happy to discuss other possible projects with students.

*Classification of AF-algebras via K-theory*. In a way that can be made precise, C*-algebras can be viewed as non-commutative analogues of locally compact topological spaces and so invariants and ideas from algebraic topology play a key role in the study of C*-algebras. One of the most successful examples of this is the use of K-theory to classify certain C*-algebras. The origins of this programme lie in George Elliott's classification of the approximately finite dimensional (AF) C*-algebras in the 70's (these algebras are those than can be suitably approximated by sums of matrix algebras), and the classification programme remains a major research theme today. The aim of this project would be to understand the basics of C*-algebras and their K-theory and give Elliott's classification of AF-algebras. A good starting point to get a feel for what this is about is the book 'An introduction to K-theory for C*-algebras' by Rordam, Larsen and Lausten, though we'd also need to look at some other texts in order to get a feeling for C*-algebras.

*von Neumann algebras*. A von Neumann algebra is a weakly closed, *-subalgebra of the bounded operators on a Hilbert space. These algebras may be viewed as non-commutative analogues of measure spaces. This project aims to understand the basics of von Neumann algebra theory, including Murray and von Neumann's classification of factors (the building blocks of von Neumann algebras) into types by means of comparison of projections. This leads to the important continuous dimension of the projections in a II_{1} factor (in contrast to B(H), where the dimension of a projection is a natural number or infinity, a projection in a II_{1} factor has dimension in [0,1]). Possible books include Kadison and Ringrose's `Fundamentals of the Theory of Operator Algebras', the first volume of Takeaski's `The theory of operator algebras'.

*Amenable groups*. A group is amenable if it admits an invariant mean. Such a mean can be used to `average' over the group even when the group is not finite. Examples of amenable groups include all finite groups, all abelian groups and all compact groups. There are many characterisations of amenability, which can be explored in the project. Some topics which might be worth considering include Tarski's characterisation of amenability for discrete groups as those that are not paradoxical, and the unitarisability of uniformly bounded group representations. There's a number of possible sources, such as Patterson's book 'Amenability' and Pisiers book on simiarity problems (Pisier also has a recent expository paper giving a detailed account of very recent work of Monod and Ozawa on unitarisability).

*Rigidity properties for groups*. This project is concerned with a ridigity property for locally compact groups, known as property T, introduced by Kazhdan in 1967 and exploited widely since then. The aim will be to learn about the topology on the equivalence classes of unitary representations of groups, use this to define property T and give some examples (including SL(3,Z)) and applications. A good source for this project is the book 'Kazhdan's Property T' by Bekka, de la Harpe and Valette, which has a lengthy appendix with the required background on unitary group representations.

*Free probability*. Free probability theory was invented by Voiculescu in the 90's in the context of von Neumann algebras, and has rapidly grown since. In classical probability theory, independence of random variables corresponds to the algebraic notion of a tensor product; free independence arises when this tensor product is replaced by a free product. The classical theorems of probability, such as the central limit theorem, have analogues in free probability with the semicircular distribution replacing the normal distribution. Nica and Speicher's book `Lectures on the Combinatorics of Free Probability' provides a nice account of the theory from a combinatorial view point.

*Entropy of dynamical systems*. Wikipedia. The aim of this project is to look at measure preserving dynamical systems in the abstract setting (so the project won't have much to do with the dynamical systems arising in the dynamical systems course). In particular I'd like to focus on Bernouli shifts, which are a certain type of dynamical system for which Kolmogorov first introduced the entropy in order to show that all Bernouli shifts are isomorphic. One might then go on to consider how entropy can be used to classify Bernouli shifts and then look at topological versions of this notion.

The level 4 projects I offer change from year to year, but are normally either analytic or combinatorial in nature.

### Old courses

- 2015-16: 4H Functional Analysis, 5M advanced functional analysis
- 2014-15: 3H Analysis of differentiation and integration
- 2013-14: 2E Introduction to real analysis.
- 2012-13: 2E Introduction to real analysis.
- 2011-12: 4H Galois theory, 3H Analysis of differentiation and integration, 2E Introduction to real analysis
- 2010-11: 4H Galois Theory, 3H Analysis 1
- 2009-11: 4H Galois Theory, 3H Analysis 1
- 2008-09: 3H Introductory Analysis, 3P Real and complex analysis
- 2007-08: 5M Complex Analysis 2, 4H Algebraic Topology, 2U Basic Analysis

## Biking, Fort William

The last lap of the 24 hour bike relay. A suprisingly sunny event for October. 2010.