NOTES de la SMC SEPTEMBER/SEPTEMBRE

A Miscellany for Mathematicians and Others Book Review by David Borwein, University of Western Ontario

They view The Mathematical Intel- outspoken very left-wing political ac- ligencer to be such a journal, and their tivism at the time of his receipt of Mathematical Conversations – selection of articles is their personal the Fields medal in Moscow in 1966. Selections from The Mathematical choice of materials to fulfill their aim A story is told about Wiener’s absent Intelligencer of reaching and interesting a wide au- mindedness: His family moved house compiled by Robin Wilson and dience. This is, of course, an extremely and Wiener was given a key to the new Jeremy Gray difficult task since the mathematical house when he left for work on the day Springer-Verlag New York, 2000 knowledge of the target group ranges of the move. After work he returned to vii+ 488 pages from almost complete ignorance to spe- his old house and was dismayed when cialist expertise. The choice of topics he found that his key did not open the reflects very much the tastes of the se- door and the house appeared empty. He lectors. What they consider to be ele- ran up to a child coming down the street gant mathematics may appeal to like- and cried “Little girl, I’m very upset. minded experts but is usually gibber- My family has disappeared and my key ish to the mathematically “challenged” won’t fit the lock.” She replied, “Yes, and may even appear to be dull and te- daddy. Mommy sent me for you.” dious to experts in disparate mathemat- Part seven, entitled History of ical fields. Mathematics, deals inter alia with the Wilson and Gray have divided the life of Kurt Godel,¨ Louis de Branges’ book into seven parts and attempted to proof of the Bieberbach conjecture, make the first and last accessible to the the bitter controversy between David non-mathematical. In this they have Hilbert and L.E.J. Brouwer around the been only partially successful. For ob- editorship of Mathematische Annalen, vious reasons, I’ll not review the mate- and the present state of Hilbert’s 23 rial in all seven parts in depth or com- problems. Much of this part would This book is a diverse collection of prehensively, but will cherry pick to certainly be inaccessible to the non- forty articles which appeared in The convey some sense of what the book mathematical public. Mathematical Intelligencer over the covers. The other five parts of the book past twenty years. They were chosen Part one, entitled Interviews and are addressed to the mathematically by Robin Wilson and Jeremy Gray to Reminiscences, contains lots of inter- knowledgeable. present to a general mathematical au- esting material concerning the views, Part two, entitled Algebra and dience and interested others what it is and in some cases the activities, of Number Theory, has six articles, in- in mathematics that makes it so useful the mathematicians Michael Atiyah, cluding one on Faltings’ proof of the and intriguing. Wilson and Gray ex- Stephen Smale, Jean-Pierre Serre, Julia Mordell Conjecture. Another is an es- press their own views as follows: Robinson, and the physicist C.N. Yang: say by M. Ram Murty on Artin’s Con- “...there is a need for journals that it also includes anecdotes about Ste- jecture for Primitive Roots in which display in an accessible way what fan Bergman, Abram Besicovitch, Kurt Murty quotes the following insightful mathematics is and what it can do. Pop- Godel,¨ Solomon Lefschetz, and Nor- comment by Artin: ularization wins friends for the subject. bert Wiener. Atiyah is reported to have “We all believe that mathematics It brings in students, it informs col- said “I don’t pay very much attention is an art. The author of a book, the leagues in other fields, and it helps as- to the importance of proofs. I think it lecturer in a classroom tries to convey sure politicians and taxpayers that pub- is more important to understand some- the structural beauty of mathematics to lic money is being well and responsibly thing." But then he admits “a proof is his readers, to his listeners. In this at- spent; it can refresh professional math- important to check your understanding, tempt, he must always fail. Mathemat- ematicians who necessarily dig deep that’s all. It is the last stage in the oper- ics is logical to be sure, each conclu- into their own studies but still wish to ation - an ultimate check - but it isn’t the sion is drawn from previously derived know what their neighbours are doing, primary thing at all.” Many mathemati- statements. Yet the whole of it, the and it helps locate and affirm the place cians, myself included, would disagree real piece of art, is not linear; worse of mathematics in the collective cul- with this view of the place of proofs than that, its perception should be in- tural activity of humanity.” in mathematics. Smale describes his stantaneous. We have all experienced

3 SEPTEMBER/SEPTEMBRE CMS NOTES on some rare occasions the feeling of ent to S, and various constructions of tions of the proofs of Michael Fried- elation in realizing that we have en- space filling curves due to Peano, Os- man and Simon Donaldson which led abled our listeners to see at a moment’s good, Sierpinski,´ Lebesgue and oth- to the verification of Poincare’s´ con- glance the whole architecture and all its ers; there is also a description of Jeff jecture concerning smooth 4-manifolds ramifications.” Xia’s proof in 1988 of Painleve’s´ long- being homeomorphic to a 4-sphere, and Number theory of the analytical standing conjecture concerning an as- gained them Fields Medals in 1986; in kind is the subject of R.C. Vaughan’s pect of the n-body problem. addition, there are articles on minimal article, Adventures in Arithmetick, or: The applied mathematics part in- surfaces, braid and link theory, and hy- How to Make Good Use of a Fourier cludes articles on string theory, soli- perbolic 3-manifolds. Transform, which connects such seem- tons, the relevance of topology to the ingly disparate themes as the Riemann behaviour of yeast, and one by J. Lam- To sum up, the book contains a Hypothesis and Fermat’s Last Theorem bek describing how certain key results wealth of material with various parts via investigations into Goldbach’s con- in theoretical physics can be expressed accessible to differing groups of in- jecture. concisely in the language of quater- dividuals. It is not a book that one Parts three, four, five, six are en- nions. would read from cover to cover, but titled Analysis, Applied Mathematics, The part on arrangements and pat- anyone with curiosity and some basic Arrangements and Patterns, Geometry terns deals with both two-dimensional mathematical knowledge could browse and Topology respectively. Each part (tailings or tesselations) and three- through it and pick parts which would contains five or six articles covering a dimensional (crystallography and be interesting and instructive. I’m wide range of subjects. solid-state physics) patterns, and ends afraid, however, that politicians and bu- The analysis part deals to quite with an article by H. S. M. Coxeter on reaucrats with little or no mathematical an extent with counter-intuitive results, the work of two sculptors using hollow background are unlikely to be driven such as the Banach-Tarski Theorem triangles. to funnel more funds into mathematics which states that it is possible to de- The geometry and topology part as a result of trying to read the book. compose a sphere S into two spheres contains an article on the topology of The book would make an ideal gift for S1 and S2, each of which is congru- 4-manifolds which includes descrip- anyone mathematically inclined.

FROM THE INSTITUTES BIRS Call For Proposals Fields Institute Appoints New Director

Kenneth R. Davidson, professor of mathematics, Waterloo The proposal for the creation of the Banff International Re- University, has been appointed Director of the Fields Insti- search Station for Innovation and Discovery in the Mathe- tute for Research in Mathematical Sciences, beginning July matical Sciences (BIRS) is now fully in motion, according 1, 2001. Ken Davidson received his undergraduate degree to a news release from BIRS Scientific Director, Robert V. at the University of Waterloo in 1972 and his Ph.D. from the Moody. University of California at Berkeley in 1976. He was a C.L.E. A special joint PIMS-MSRI Newsletter devoted to BIRS Moore instructor at M.I.T. for two years before moving to the has been printed and distributed worldwide. It provides ex- University of Waterloo in 1978. His research interests are in tensive information about the modes of operation of BIRS operator theory and operator algebras, and he won the Israel as well as a Call for Proposals. Both the Newsletter and Halperin prize in this area in 1985. He was an E.W.R.Steacie the Call for Proposals are available on the BIRS webpage fellow 1988-90 and a Killam Research Fellow 1995- 97. He http://pims.math.ca/birs/ was elected a fellow of the Royal Society of Canada in 1992. Strong proposals, which if possible should be made on- He has been on the editorial boards of various journals in- line, are solicited for the year 2003 (March through Decem- cluding the CMS journals and Integral Equations and Opera- ber). The 2003 program at BIRS is expected to be vetted tor Theory. He has served on the CMS in various capacities by the BIRS Scientific Advisory Board in December 2001. including Vice President (Ontario) in 1995–97. He sat on the The deadline for submission of proposals is October 1, 2001. NSERC mathematics GSC in 1990-93, and served as chair; BIRS is committed to representing all aspects of the Mathe- was a member of the NSERC Strategy Implementation Task matical Sciences and we look forward to many exciting pro- Force in 1995, and on the Mathematics Steering Committee posals from the world-wide community. 1996-98. He served on the Fields Scientific Advisory Panel This is subject to funding decisions by NSERC, NSF, and 1991-96, and was a co-organizer of the C*-algebra year at the the Alberta Science Research Authority (ASRA). institute.

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