The Taught Course Centres for PhD students in the Mathematical Sciences were set up as the result of a recommendation of the last-but-one International Review of Mathematics. The review panel said that the highly specialised nature of British PhDs meant that graduates lacked the background knowledge to compete internationally, and recommended that more coursework be introduced into PhD studies.
As a result, EPSRC, the research council responsible for mathematics, called for proposals to run Taught Course Centres. The rules were quite strict: it was not permitted to recycle Masters level courses; the standard must be demostrably higher than Masters level.
Most of the successful bids went for a distance-learning, technology-based solution. But London was a bit different. Since most mathematics departments in the colloquium were in or close to London, the London Taught Course Centre (LTCC) went for face-to-face lectures and interaction between students and lecturer (and among the students).
The business plan was that EPSRC would provide “pump-priming funds” but expected the Centres to become self-supporting within a few years. In practice this meant that universities in the consortium would be asked for contributions to fund the Centre. In the case of London, it was decided to look into the possibility of an extra funding stream by publishing volumes containing material from our portfolio of courses. World Scientific agreed to publish the LTCC Advanced Mathematics Series, and the first three volumes in the series (edited by the directors of the Centre, Shaun Bullett, Tom Fearn and Frank Smith) have just appeared. They are on Advanced Techniques in Applied Mathematics, Fluid and Solid Mechanics, and Algebra, Logic and Combinatorics. Three more volumes are in preparation and should be out shortly.
I wrote a chapter for volume 3, Algebra, Logic and Combinatorics: my free copies arrived yesterday.
There are five chapters, each of about 40 pages. Each has exercises, with solutions or hints to some of them, and suggestions for further reading, and most have an introduction. Below is a list of the other sections.
- My chapter is on “Enumerative Combinatorics”, and covers formal power series; subsets, partitions and permutations; recurrence relations; inclusion–exclusion; posets and Möbius inversion; orbit counting; species; and asymptotic analysis. (Similar to part one of my St Andrews notes.)
- Robert Wilson gives an “Introduction to the Finite Simple Groups”, with detail on alternating groups; subgroups of symmetric groups (O’Nan–Scott); linear groups; subgroups of general linear groups (Aschbacher); forms; classical groups; Lie theory; octonions and G2; exceptional Jordan algebras and F4; Mathieu groups; the Leech lattice and Conway groups.
- Anton Cox gives an “Introduction to Representations of Algebras and Quivers”. He treats algebras and modules; quivers and their representations; basic structure theorems (Jordan–Hölder, Artin–Wedderburn, Krull–Schmidt); projective and injective modules. I haven’t thought about quivers since I first heard the word “categorification”; there seems to be a close connection.
- Peter Fleischmann and James Shank describe “The Invariant Theory of Finite Groups”. They do finite generation and Noether normalisation; Hilbert series; integral extensions and integral closure; polynomial invariants (and reflection groups); the depth of modular rings of invariants; Castelnuovo regularity and finite decomposition type.
- Ivan Tomašić’s chapter on “Model Theory” starts with a gentle introduction to first-order model theory and then accelerates to reach recent developments in diophantine geometry. He covers first-order logic; basic model theory; applications in algebra; dimension, rank, stability; classification theory; geometric model theory; and model theory and diophantine geometry.