Making problems look like a million dollars

Stephen Smale
September 22, 2000

An exciting title for an exciting mathematician, but this is not, alas, an exciting book. It comprises an acolyte's account of the psyche and personal life of one of this century's most influential mathematicians, intertwined with a valiant attempt to get some of Smale's ideas across to the layman. If only some of the space devoted to the former had been used to convey the breathtaking wider significance of Smale's horseshoe, not to mention his other great contributions to topology and geometry.

However, in the author's defence, it must be said that for Smale's spectacular "sphere eversion" (turning a rubber ball inside out allowing interpenetration but keeping the surface smooth everywhere), even a video scarcely reveals the geometric complexity that lies behind the mathematical jargon.

The title is not quite as glamorous as it sounds. It has been known for centuries that it can sometimes be easier to solve a mathematical problem in a certain number of variables, ie in a certain dimension, by first generalising to a higher dimension and then specialising back.

Famous examples are the use of complex (two-dimensional) numbers to help understand problems involving real (one-dimensional) numbers only, and the representation of waves in two space dimensions and time, which can be far simpler if use is made of three-space dimensional solutions.

It was Smale's realisation that the so-called Poincaré conjecture could be attacked in this way that shot him to fame in the mathematical world. The conjecture was framed at the beginning of the 20th century by Henri Poincaré, one of the founding fathers of the then-emerging subject of topology. It deals with the fundamental nature of three-dimensional space, and remains, to this day, one of the most important unsolved problems in the subject. Indeed, a prize of $1 million has recently been offered for its solution.

Smale discovered that the analogous problem in dimensions five and above is actually easier than the three-dimensional version. His novel solution earned him a Fields medal, the highest award in mathematics. It sparked a flurry of activity in topology, but Smale took little part in this: he had already switched tracks to other areas. What distinguishes Smale from many theoreticians is his contribution to a wide range of scientific activity, from economics to numerical analysis. Perhaps this breadth can be traced back to the influence of the great Raoul Bott, an electrical engineer turned topologist/geometer, but this book fails to do Smale intellectual justice in this respect.

The horseshoe with which he will forever be associated is described in some detail, but there is little mention of the revolution it engendered. The idea that even evolutionary systems as simple as ordinary differential equations could do other than eventually tending to quiescent states or to easily describable oscillations was quite alien in applied science in the 1960s, despite the warnings of pioneers such as Mary Cartwright, Norman Levinson and John Littlewood.

The unexpected behaviour is the mathematical manifestation of what is now called chaos, a situation where the evolution locks into an infinitely fine network of oscillations without ever staying very close to any one of them for long. The fact that relatively simple nonlinear difference and differential equations can readily possess such complicated "attracting sets" has changed so much of mathematical modelling in mechanics, biology, chemistry, finance and, especially, turbulence; the reader must look elsewhere for this story, but here is at least a helpful springboard.

John Ockendon and Marc Lackenby are fellows of St Catherine's College, Oxford.

Stephen Smale: The Mathematician who Broke the Dimension Barrier

Author - Steve Batterson
ISBN - 0 8218 2045
Publisher - Open University Press
Price - £25.00
Pages - 306

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