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REVIEWS Contents
Mathematicians of the World, Unite! The ICM – A Human Endeavour
Mathematicians of the World, Unite! The ICM – A Human Endeavour by Guillermo P. Curbera, A.K. Peters, 2009, 344 pp, £42.50, $59.00, ISBN 978-1-56881-330-1. The International Congresses of Mathematicians began with the 1897 Zürich Congress. From that time on, the Congresses have taken place approximately at four-year intervals except for periods during World War I and World War II when no Congresses took place. In total twenty-five Congresses have been held, the most recent being in Madrid in 2006. The book under review looks at each of these twenty-five Congresses, particularly examining their social sides. The Congresses are divided into five sections: The Origins, Crisis in the Interwar Period, The Golden Era, On the Road and In a Global World. Each section has a short introduction which explains to the reader some of the difficulties which have been experienced by the Congresses as they tried to rise above the many political pressures and tensions which were inevitable given their world-wide nature. The atmosphere at each Congress is brought to life with many excellent pictures. Quotations from the opening address, details of music played, descriptions of the buildings in which the Congress was held, and details of exhibitions all help to paint a picture of the Congress. Numbers of mathematicians attending, together with their nationalities, show how the series of Congresses developed. Relatively little information is given about the mathematical content of the Congresses but the names of the plenary speakers, with titles of their talks, are given as are the mathematical sections into which each Congress was divided, and this information gives an indication of the changing trends in mathematical research. Between the chapters on the individual congresses, ’Interludes’ are included. ’Images of the ICM’ looks at logos, stamps and posters associated with the Congresses. ’Awards of the ICM’ relates the history of the Fields Medal, the Nevanlinna Prize and the Gauss Prize. ’Buildings of the ICM’ contains excellent pictures of many of the buildings in which the congresses took place. ’Social Life at the ICM’ is the longest and most important interlude in which the author examines how successful the Congresses have been in carrying out the aim set out at the 1897 Congress, namely to ’foster personal relations between mathematicians of different countries’. The pictures from over 100 years of Congresses form a special feature of the book. These include Jacques Hadamard showing a bit of his striped underwear while sitting on the beach at Ravenna during the 1928 Congress, a rare picture of G.H. Hardy on a boat on Lake Zürich during the 1932 Congress, and Élie Cartan on a boat on Oslo’s fjord during the 1936 Congress. The pictures succeed in bringing the Congresses to life and emphasise that the interactions of participants are the most important feature of these meetings. The book will bring back many memories for those who have attended the more recent congresses and will give those who have not attended an understanding of what they have missed. Edmund Robertson
The Mathematical Mechanic – Using Physical Reasoning to Solve Problems by Mark Levi, Princeton University Press, 2009, 196 pp, £13.95, $19.95, ISBN 978-0-69114-020-9. This is a most interesting book. It has its origins in a friendly argument with a school friend who had ’majored’ in physics, concerning the relative importance of the two subjects. Mark Levi also intended to specialise in physics eventually but only after mastering its main tool, mathematics. Essentially the book is a collection of examples taken from the physical world which provide interpretations of mathematical proofs. The author is quite clear that these examples are not in themselves proofs, but most applied mathematicians will be familiar with the insight that physics or perhaps biology gives when developing new models. This insight sometimes indicates ways to approach the construction of proofs of new theorems, including on the estimation of bounds etc. which are required to provide a robust and reliable solution to a problem situation. Anyone who has been involved in this process will certainly enjoy reading this book, no doubt with pencil and paper to hand! My favourite example concerns saving a drowning swimmer using Fermat’s principle; this involves building a mechanical analogue of the rescuing lifeguard’s time-optimal strategy as (s)he runs over the beach and swims in the sea. The result is that the equivalent of Snell’s law of refraction emerges. A close second is the use of a model involving the closure of a switch in a certain electrical circuit. This serves to reduce resistance from 1 to 0 in a section of the circuit and so by Rayleigh’s monotonicity law (which is proved in the section), the overall resistance is the same or less. Choosing the circuit appropriately provides a demonstration that the arithmetic mean of two positive numbers is not less than their geometric mean. Mark Levi is professor of mathematics at Pennsylvania State University and as well as inventing and collecting physical solutions to mathematical problems he collects physical devices which illustrate mathematical ideas. One of these is a jigsaw which he uses to demonstrate the stabilisation of an inverted pendulum using vibration. In this connection his work on Kapitsa potentials was published in the Society’s journal Nonlinearity. The book has numerous references and an Appendix on the necessary physical background required. Some of the Chapters have a selection of problems at the end with some hints or solutions. As well as being an interesting book, many of the ideas in it could be used as motivational or illustrative examples to support the teaching of non-specialists, especially physicists and engineers. In conclusion – a thoroughly enjoyable and thought-provoking read. Nigel Steele
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