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BOOK REVIEWS Contents
Selected Papers of Alan Hoffman Selected Papers of Alan Hoffman with commentary, edited by Charles A. Micchelli, 2003, World Scientific Publishing, 446 pp, £78, ISBN 981-02-4198-4. For over forty years Alan Hoffman worked as a mathematician at the IBM Research Center in Yorktown Heights. His mathematical career began with studies at Columbia and Princeton, where he worked on the foundations of geometry. In 1951 he moved to the National Bureau of Standards, where he was introduced to a new subject, linear programming. From these roots there grew a lifetime's fascination with problems about linear equations and inequalities, matrices, and algorithms. The Selected Papers cover a wide range of topics, some motivated by challenges from a purely theoretical perspective, others clearly inspired by practical problems. There are about 40 papers, in seven sections: Geometry, Combinatorics, Matrix Inequalities and Eigenvalues, Linear Inequalities and Linear Programming, Combinatorial Optimization, Greedy Algorithms, Graph Spectra. A few of the highlights can be picked out. In the Geometry section there is a 1956 paper, written with Newman, Straus, and Taussky, that uses facts about eigenvalues of the incidence matrix of a projective plane to prove a highly nontrivial combinatorial theorem. In the Section on Linear Programming there are two classics: one with Hoffman's bound for the approximate solutions of linear inequalities, and one that gave the first example of cycling in the simplex algorithm. In the section on Matrix Inequalities there are three papers on 'Gerschgorin variations' that vividly illustrate the aspects of matrix theory that are not simply part of abstract linear algebra. There is also a paper on the spectra of normal matrices, partly inspired by the work of Alfred Horn, the originator of the famous conjectures about the spectrum of the sum of hermitian matrices. In the section on Combinatorial Optimization, there is the Hoffman-Kruskal paper on integral solutions and totally unimodular matrices, first published in 1956 in the famous book on Linear Systems edited by Kuhn and Tucker. And finally, in the section on Graph Spectra there is the Hoffman-Singleton paper on Moore graphs with diameter 2, containing the simple proof that the size must be 5, 10, 50 or 3250, and the construction of the unique graph with 50 vertices. This is a result that raised unfulfilled expectations. It turned out the graph with 50 vertices is associated with a classical simple group, and a few years later there was great excitement when several new simple groups were discovered by similar methods. But if there is a graph with 3250 vertices (and to this day no one knows), then its group must be almost trivial. So Hoffman and Singleton did not join the small band of those with sporadic groups named after them. Also in this section is a classic paper giving lower bounds for the chromatic number of a graph in terms of its largest and smallest eigenvalues. The papers themselves are the heart of the book, and each one has a brief introduction that explains its origins and motivation. In addition to the major contributions mentioned above there are several little treasures. For example, there is Hoffman's paper on spectrally bounded graphs, proving that a graph with least eigenvalue that is large in absolute value must have an induced subgraph of a very special kind. The whole is rounded off by a twenty-page autobiography, notable for its insights into the diverse aspects of the life of a professional mathematician, and anecdotes about the many interesting people whom Hoffman encountered. As a mathematician conditioned by circumstances to adopt a utilitarian view, but nevertheless alive to the intellectual challenges of the subject, he has a refreshing outlook on life in general and mathematics in particular. Most important, he has the happy knack of remembering the good times, and it is noticeable that the only person who comes in for serious criticism is himself. Norman Biggs A Brief History of Infinity The Quest to Think the Unthinkable by Brian Clegg, 2003, Robinson, 255 pp, £8.99, ISBN 1-84119-650-9. A Brief History of Infinity by Brian Clegg is a lively and informal account of the way the very small, the very large and the concept of infinity have been treated over the centuries. The narrative starts with a selection of Ancient Greek mathematics: Zeno’s paradoxes, Archimedes’ ‘Sand Reckoner’ and the irrationality of the square root of 2. Moving forwards in time, we have a discussion of Galileo's paradoxes, the problems surrounding 'infinitely small' quantities, and an account of the machinations around the development of calculus. After this, we are told about Cantor's development of set theory and his problems with Kronecker. Interwoven with the mathematical threads are accounts of theological and philosophical ideas about infinity, biographical sketches of the personalities involved and tangential historical details. This sometimes makes the book feel like a series of anecdotes and a few of the detours left me rather confused. For example, after a discussion of the 'Sand Reckoner' we move via John Donne and William Blake into a discussion of Saint Augustine's views on infinity, before coming back to the proof of irrationality of root 2. The book is addressed very much to the non-mathematician, and mathematicians may find slightly off-putting the ways in which their subject and its heroes are sometimes portrayed. I found over-dramatic the claim that Cantor and Gödel were ‘driven ... over the edge into insanity’ by ‘contemplation of the infinite.’ When discussing the proof of the irrationality of the square root of 2, Clegg writes ‘In case, like me, your mind always switches off when faced with x’s and y's...’ and this strikes me as an unnecessary attempt at populism. Mathematical misdemeanours also creep into the discussion of set theory. Replacement and Foundation are missing from the list of Zermelo - Fraenkel axioms and Clegg gives the impression that the Russell Paradox is still an issue in ZF. These may seem to be technical quibbles about a popular science book (most authors would not have bothered to list the ZF axioms), but they slightly diminished my confidence in the accuracy of the rest of the text. Nevertheless there is a lot of enjoyment to be had from this book. Clegg tells a good story at a vigorous pace, and it is perhaps useful for mathematicians to be reminded of how they are seen by others. David Evans The Art of the Infinite: Our Lost Language of Numbers by Robert Kaplan and Ellen Kaplan, 2003, Allen Lane, 324 pp, £20, ISBN 07139-9629-3. According to the introduction (or ‘Invitation’) of The Art of the Infinite: our Lost Language of Numbers, Robert and Ellen Kaplan aim to show that ‘Anyone who can read ... can come to delight in the works of mathematical art, which are among our kind’s greatest glories.’ As we (and our students) know, the real appreciation of these works of art often requires a substantial effort. The Kaplans take their task seriously and include some reasonably difficult proofs, although some of these are tactfully consigned to the appendices. For the non-mathematician, if reading Clegg's book is like viewing the photographs from a strenuous adventure holiday, then the Kaplans are asking you to pack your bags and go trekking in the Andes. The main theme is again the development of the number systems: natural numbers, integers, rationals, real and complex numbers, then ordinals and cardinals. The approach is both algebraic (no fear of x's and y's here: the field axioms are proudly displayed) and geometric. The geometry appears formally as Euclidean geometry motivating the need to extend the natural numbers. It also appears informally in a very appealing feature of the book: the many hand-drawn diagrams and illustrations produced by Ellen Kaplan. Geometry also provides some of the most attractive set-pieces of the book: there is a nice discussion of Gauss' theorem on constructibility of regular polygons and I particularly enjoyed the account of the nine point circle of a triangle (for me an object of loathing, if not quite fear, at school). Projective geometry, Desargues’ and Pappus’ theorems also make an appearance. Most of the obligatory biographical and historical sketches are woven quite smoothly into the text, although I found an interlude on the disagreement between Brouwer and Hilbert somewhat obtrusive. This is a book that I would hope some of my first-year undergraduate students would want to read, and some of the more enthusiastic ones might already have read. There are some really delightful insights, explanations and examples here. However, I suspect that the book’s lyrical prose style might be too rich for many of their tastes. At the risk of straining my earlier metaphor, I sometimes felt that whilst I was happy to go trekking with the Kaplans, I didn't want to be stopped every few minutes to listen to their description of the views. The book is gentler and less ambitious in its scope than the classic of this genre: What is Mathematics? by Courant and Robbins. I would certainly recommend that you order a copy for your library and, particularly if you are teaching first-year undergraduates, take a look for yourselves: you may find some delights which you had forgotten you once knew. David Evans
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