The mathematical shapeshifter

 

We've had a lot to write about recently, what with the fun of Wimbledon and the Olympics, and the much less fun Australian Curriculum. And with all the activity we somehow missed the centenary celebrations for a great mathematician. We are referring, of course, to the brilliant Henri Poincaré.

Henri who? Perhaps you were expecting something on Alan Turing, whose 100th birthday we're celebrating this year. Turing has been much honoured in the press recently, and he definitely deserves it. Alan Turing was a great mathematician and logician, and his work, and his tragic life, should be remembered. We shall also write about Alan Turing, later this year.

The French mathematician Henri Poincaré has received much less attention. The centenary of Poincaré's death was on 17 July but, excepting a few French reports, we have found just one blog entry honoring the occasion. Nonetheless, Henri Poincaré was by far the more brilliant and more influential mathematician.

It is practically impossible to summarise Poincaré's contributions to mathematics. Poincaré made fundamental contributions to number theory, differential equations, complex analysis, algebraic geometry and many more difficult and distinct mathematical fields. As a mathematical physicist, Poincaré did groundbreaking work on the three-body problem, the notoriously difficult analysis of three planetary bodies pulling each other around under the force of gravity; Poincaré's work on the problem gave birth to chaos theory. And, Poincaré has a reasonable claim to have beaten Einstein to the theory of relativity.

Poincaré's name appears everywhere in mathematics, like graffiti absent mindedly left during his wanderings. There's the Poincare Lemma, Poincaré maps, the Poincaré disk, Poincaré duality, the Poincaré metric, the Poincaré group, the Poincaré recurrence theorem and endless more.

But Poincaré's name is most famously associated with a problem that he couldn't master: the Poincaré Conjecture. This problem, stunningly solved in 2003 by the enigmatic Grigory Perelman, arose in Poincaré's study of algebraic topology, a field that Poincaré essentially invented.

Algebraic topology involves the beautiful and powerful idea of using algebra to analyse the shapes of surfaces. To give an almost clichéd example, compare the surface of a donut to the surface of a sphere. (In what follows, it is important to keep in mind that it is the surfaces we are discussing here: no digging!) These surfaces appear to be fundamentally different, in that no amount of twisting or bending of the donut will morph it into the sphere.

One approach to proving that the donut and sphere are different is indicated in the picture above. Imagine a rubber band, a loop, around the ring of the donut. The loop is allowed to slide and to stretch and to shrink. However, as long as the loop stays within the donut's surface, it will be stuck around the ring of the donut: the loop cannot be shrunk to a point. By contrast, any loop on the surface of the sphere can easily be shrunk to a single point at, for instance, the North Pole.

None of that may seem very algebraic. However, Poincaré showed that the loops on a surface can be "added" in a certain sense, turning the collection of all such loops into an abstract world known as a group. It is then the algebra of groups that enabled Poincaré to transform the intuitive ideas above into rigorous proofs.

By considering loops on surfaces, Poincaré was able to determine when surfaces were fundamentally different, and when they could be morphed into one another. In particular he was able to prove that, amongst finite surfaces without edges, the sphere is the only surface in which all loops are shrinkable to a point.

Poincaré went on to study three-dimensional (and higher dimensional) worlds. It may be difficult to even imagine what there is to study, since we usually imagine 3D space as simply being space-shaped: it goes on and on, and that's it. But in fact there are many (mathematical) 3D worlds, and they can loop around in exactly the same manner as the donut surface above.

We won't go into the details. (We discuss it briefly here.) Suffice to say, there is a 3D analogue of the sphere, and Poincaré conjectured that this was the only 3D world in which all loops are shrinkable. This was the conjecture that Grigori Perelman finally proved, as past of a general classification of 3D worlds.

In his later years, Henri Poincaré lectured and wrote upon the nature of mathematical discovery. In particular, he was interested in the separate and often conflicting roles played by logic and intuition. The 19th century had been a revolutionary time for mathematics, when the techniques had become too powerful to be handled by intuition alone and more solid foundations were demanded. However, the early 20th century took things much further, with attempts to reduce mathematics to a logical game. Poincaré fought against these attempts with all his might, determined to keep intuition as a central and respected part of mathematics.

It's hardly surprising that Poincaré would argue that way, since he was one of the all time great intuitionists. Poincaré could leap fearlessly from mathematical peak to mathematical peak, while we mortals have to painfully climb our way up, step by logical step. 

But of course Poincaré valued logic and proof as well. He famously wrote:

Logic, which alone can give certainty, is the instrument of demonstration; intuition is the instrument of invention.

Henri Poincaré was a true master of both. 

 

Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathematical juggler, origami expert, bubble-master, shoelace charmer, and Count von Count impersonator.

Marty Ross is a mathematical nomad. His hobby is smashing calculators with a hammer.

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