# Notes

These are mostly notes I have produced for various talks. Occasionally I may present something which isn't research, but which I thought I'd like to present in my own way.

- arXiv:1308.1073 [math.FA]: "The approximation property implies that convolvers are pseudo-measures", a writeup of a result of Cowling's.
- The group algebra derivation problem [PDF]: A self-contained proof of the group derivation problem, using the new proof by Bader et al.
- "Functorial properties of weakly almost periodic functionals on the measure algebra". This has been superceeded by a later paper, as the referee noticed a much, much better way of proving the main result. However, perhaps the techniques used in this paper might be of interest: in particular, is there a non-commutative generalisation? This paper is available on the ArXiV, and will not otherwise be published (be aware that material in the first and final section is dealt with better in arXiv:0904.0436 (current version)).
- "Weakly almost periodic functionals, representations, and operator spaces": This paper shows that a completely contractive dual Banach algebra is isometric to a weak*-closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result has been shown independently by F. Uygul (math.FA/0702200 on the arXiv), using rather similar arguments to my presentation, which will hence remain only on the arXiv.
- Bases in Banach spaces [PDF]: Introduction to Schauder bases, basic sequences, unconditional bases, and the Gowers dichotomy theorem.
- Interpolation spaces [PDF]: Some notes on Beauzamy's book.

# Thesis

Perhaps half, on page count, of my thesis has been published, so people might be interested in the remained (which is often technical, and may or may not be published in expanded form in the future).

# Older notes

These are various, mostly half-finished, projects from the past, when I was an undergrad student at Cambridge. I'm including LaTeX source code, in case anyone wishes to continue my work.

### Galois Theory

These are, verbatim, the notes presented by Dr. P.M.H. Wilson in Michaelmas term, 1999. I LaTeXed them, added a few notes of explaination (to account for my lack of understanding of the course Groups, Rings and Fields). Briefly, the notes cover field extensions, separability, algebraic closures, normal extensions and Galois extensions, Galois groups of finite fields, cyclotomic extensions, kummar theory, solving by radicals and the insolubility of the general quintic by radicals.

- Local version in Acrobat (395Kb) format.
- Latex source available upon request.

### Probabilistic Combinatorics

These notes are based upon the course given by Dr. Thomason in Lent 2001. I have expanded somewhat upon his explainations though, mainly to help myself understand where he gets various estimates and values from. Very incomplete.

- Local version in Acrobat (243Kb) format.
- All the source code in a zip file.

### Algebraic Topology

These are based on the lecture course given by Dr. N.I. Shepherd-Barron in Michaelmas term, 1999. They are not complete, and currently contain revision of basic point set topology, homotopy theory, quotient spaces, covering spaces and sketches of simplical complex theory.

- Local version in Postscript (409Kb) format.
- All the source code in a zip file.

### DeRham Cohomology

These were going to be some brief notes on smooth manifolds, with detailed notes on the DeRham Cohomology and applications thereof. Based mainly upon the lecture course given by Dr. Barden in Lent term 2000, but not a direct transcript of those lectures, as Dr. Barden has already given such a set of notes out. Very incomplete.

- Local version in Acrobat (146Kb) format.
- All the source code in a zip file.

### Groups, Rings and Fields

These are based (well, are a direct copy) of the lecture course given by Dr. M. Hyland in Lent 1999. Incomplete.

- Local version in Acrobat (318Kb) format.
- All the source code in a zip file.