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Story of a Favourite Mathematical Tool

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Story of a Favourite Mathematical Tool book reviews pen. I do not think that Harvard University Press has a book of epochal importance in The Triple Helix. Most of the points, and the examples Lewontin uses to make them, will already be familiar to biologists. He has used the same examples in Suzuki’s genetics text (of which he is a co-author) An Introduction to Genetic Analysis (Freeman), in anti-adap- tationist papers and in some other books. However, both these new books could HULTON-DEUTSCH COLLECTION/CORBIS HULTON-DEUTSCH provide an easy way for biological readers to refresh their knowledge of, or argument with, Lewontin’s views. The books’ main value will probably be for biology students who have not yet met those views. And I can- not recommend them too highly for the many commentators and headline-writers who think that DNA is the blueprint for the organism. I Mark Ridley is in the Department of Zoology, University of Oxford, Oxford OX1 3RS, UK. Story of a favourite Founding figure: Poincaré’s visionary work opened up the huge subject of topology to investigation. more than 40 authors for this account. Some in such areas as mirror symmetry, black mathematical tool are eminent mathematicians, others are his- holes and gravitational collapse. History of Topology torians of mathematics. Each group displays Algebraic topology is a sophisticated edited by I. M. James many of the skills of the other, so the reader machine that students find hard to master, Elsevier: 2000. 1,056 pp. £125, $190.50 enjoys a consistently rich but varied diet. and many of its main tools are considered Jeremy Gray Inevitably, and rightly, some essays are here. There is a fine account by John for mathematicians only. But it is possible to McCleary of the early history of spectral As topology may well claim to be the twenti- find others that show how topology acquired sequences, one of topology’s most compli- eth century’s most original and successful its present central position. A founding fig- cated aspects, and another by M. Zisman on development in mathematics, a comprehen- ure is the French mathematician and scien- fibre bundles (loosely, parametrized families sive history of it is most welcome. In its alge- tist Henri Poincaré. His visionary but often of groups or manifolds). braic form, on which this volume focuses, inaccurate work is well described by Karan- The book contains a welcome emphasis topology provides a way of formulating and bir Sarkaria, who is a careful and well- of some 180 pages on topologists themselves, often answering questions about the geome- informed guide. Poincaré’s achievement was either singly or in schools. Much of the infor- try of spaces in any number of dimensions. in opening up a huge subject to investigation. mation here is new, and opens up fresh per- But it reaches further than that, and, like any As Erhard Scholz nicely describes, the theory spectives on some of the material described major branch of mathematics, has its own of manifolds (generalizations of smooth in earlier essays in the volume. The handy 24- charm and its own difficulties, attracting surfaces to any number of dimensions) links page index is another bonus. some of the world’s best mathematicians. Poincaré’s work to the more general study It is no criticism of a book of more than Topology owes its range and depth to the of the calculus of several variables. Such 1,000 pages to say that much has been left primitive nature of the mathematical fea- generalization goes back before Poincaré to out. There is nothing here on the Russian tures it studies. These features are preserved Riemann, but Poincaré promoted it, even school of topology, in which Lev Sem- even when a figure continuously changes its though he left it to others to find the right yonovich Pontrjagin and Pavel Sergeevich size or shape, including bending and stretch- tools for the job. And there is a fine essay by Alexandrov figured so prominently. Many ing, or even undergoes certain kinds of cut- Moritz Epple showing how problems in knot branches of physics draw on the methods ting and re-glueing. So whenever a property theory, which is rooted in nineteenth-centu- and ideas of algebraic topology, but some of is shown to be topological in nature, even ry physics, led to the creation of rigorous these are perhaps too recent to be included in though it might seem to need more sophisti- methods in the study of three-dimensional a history. Surprisingly, analysis and elliptic cated concepts for its definition, mathemati- manifolds. operators are mentioned only in passing. cians infer that they have got closer to the Some of the spectacular applications of This is a pity, for the subject reaches from the heart of the matter. Two classic examples of topology are also discussed. The study of Lie work of Riemann to the Atiyah–Singer index this are the number of independent integrals groups may have been started by the Nor- theorem and modern quantum field theory. on a surface and the study of differential wegian mathematician Sophus Lie, after Faced with such a huge story, however, James equations on a surface. The nineteenth- whom they are named, but, as Thomas is to be thanked for organizing so much century mathematicians Bernhard Riemann Hawkins lucidly describes, the modern for- information and solving the editorial prob- and Henri Poincaré showed that these mulation owes much to Hermann Weyl, who lems so well. This book is not meant to be the calculus-type properties of a surface depend showed how crucial topology is to that sub- last word on anything. It does much to on a single number, called the genus of the ject. Currently, topology is a favoured topic enable the next words to be said. I surface, which is a true topological invariant. among physicists, and Charles Nash’s essay Jeremy Gray is in the Department of The book’s editor, I. M. James, himself a moves quickly over the period covered by Mathematics, Open University, Milton Keynes distinguished topologist, has drawn together Epple to describe the exciting developments MK7 6AA, UK. © 2000 Macmillan Magazines Ltd 348 NATURE | VOL 406 | 27 JULY 2000 | www.nature.com.
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