Algebra, Logic, and Combinatorics

About 45 years ago, when I was an undergraduate in my senior year, a professor named George W. Booth, the best teacher I ever had from kindergarten through graduate school, pointed out to me a set of books titled Lectures on Modern Mathematics. Edited by Thomas Saaty, this is a three-volume set of expository articles on various branches of mathematics, written by experts in those areas: Coxeter on geometry, Halmos on Hilbert spaces, Kaplansky on Lie algebras, Nirenberg on partial differential equations, Brauer on group representation theory, and many others. I spent a large part of the following summer reading these articles, and even in those (fairly common, I must admit) situations where I wasn’t at all sure I understood what was going on, found them fascinating. This was my first opportunity to really get a sense of the mathematics that lie ahead, in graduate school; Kaplansky’s article, in particular, was wonderful, and quite likely played a role in my subsequent decision to write my dissertation on Lie algebras.

I engage in this little stroll down memory lane for several purposes: to point out explicitly that ever since these days, I have been a sucker for a nicely written expository article on an area of mathematics that I am not that familiar with, and to point out my firm belief that a good expository article can be both informative and influential. The book now under review, though much shorter than Saaty’s omnibus (five articles instead of a total of 18), is very much in the spirit of the latter, and reading it was a pleasure.

This book is the third in the LTCC (London Taught Course Centre) Advanced Mathematics Series. Like its two counterparts in this series, this one consists of expository articles in related mathematical areas, all done at a fairly sophisticated (graduate school and non-specialist professional) level; the other two books cover Fluid and Solid Mechanics and Advanced Techniques in Applied Mathematics. (Future volumes will include analysis and mathematical physics, geometry in advanced pure mathematics, and dynamical systems.) As the name of this volume implies, the topics covered here are algebra, combinatorics and logic. In more detail, here is a list, and summary, of the five articles:

• Peter Cameron surveys enumerative combinatorics, covering many familiar topics: generating functions and recurrence relations, posets, orbit-counting, inclusion-exclusion, etc. Other topics covered include the theory of species and asymptotic analysis. Along with standard facts, original work of the author (and others) is mentioned. The chapter ends with a selection of exercises, solutions to a small subset of which are provided.
• A survey of finite simple groups is given by Robert A. Wilson, author of a Springer text, The Finite Simple Groups, that was favorably reviewed in this column. This article gives a statement of the classification theorem for finite simple groups, and then proceeds to explain just what the various groups mentioned in that theorem are. Along the way the reader is exposed to Lie algebras, Jordan algebras, quadratic forms and other fun things.
• Anton Cox writes on the representation theory of algebras and quivers (directed graphs). The chapter includes a nice survey of the Weddeburn-Artin theory of semisimple algebras (all algebras in this chapter are assumed to be associative and have an identity) as well as topics in module theory such as projective and injective modules. A number of theorems are given at least proof sketches, and the chapter ends with a set of exercises, to which sketch solutions are provided.
• The next chapter, by Peter Fleischmann and James Shank, offers an introduction to invariant theory of finite groups. Perhaps the most demanding of the articles in the book, but one that should still be quite accessible to students with a good background in graduate-level algebra, this provided an interesting mix of commutative algebra and group theory. The chapter ends with four exercises, solutions to which are provided.
• The final chapter is on logic, specifically model theory. Written by Ivan Tomasic, this article begins from scratch with the definitions of first order logic and models, discusses some famous results in the area, and then addresses some more specialized work connecting model theory with algebra and Diophantine geometry.

The articles are all quite well-written and informative. None of these expository essays will turn you into an expert in any of these areas, but they do provide some sense of what the words mean and what the important ideas are. Graduate students might well find themselves motivated to study one of these topics in more detail.

The book, unfortunately, does do one thing badly: there is no index, either at the end of the book or the end of the articles. This is rapidly becoming a pet peeve of mine, and unfortunately I am noticing it more and more frequently these days in textbooks.

But let me not end on a negative note. In these days of increasing mathematical specialization, it’s always nice to find something that helps bridge the gap between disciplines. It’s even nicer when the contents are of the quality of these articles. This is an interesting, enjoyable and valuable book.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.