Wednesday, June 04, 2008

When is a sphere not a sphere?

kw: book reviews, nonfiction, mathematics

I have been reading a rather difficult book. Though George G. Szpiro vowed to eschew equations in his book, I don't think of Topology as a subject that needs equations as I understand them. Anyway, I can handle equations, lots of them, but I have yet to wrap my mind around Topology beyond the "coffee-cup-equals-donut" stage.

The subject of Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles is Topology. Necessarily, in order to help a reader understand the point of Poincaré's Conjecture, now Poincaré's Theorem, author Szpiro needed to present oodles of topological concepts. It is a tribute to his skill that on occasion, I almost thought I understood what he was writing about.

To a mathematician of sufficient skill, the math itself is beautiful and fascinating. I am a working mathematician, but not at that level. I found greater interest in the many mini-biographies and the stories of the squabbling that surrounded the successive discoveries that were eventually used by Grigori Perelman to solve the Conjecture, just six or seven years ago now.

One aspect came through clearly: Perelman's final breakthrough combined Topology with the Differential Equations devised by Richard Hamilton (150 years after the more-familiar "Hamiltonian mechanics" were propounded by a different Hamilton). As my own best work has come about from combining seemingly contradictory techniques, I find Perelman and Hamilton a refreshing boost compared to the overly-narrow work of most practitioners.

I found it surprising how many mathematicians have caught "Poincaritis" over the years. They number in the hundreds, of whom dozens are limned by Szpiro . I reckon, though, that they do not outnumber those who still fall prey to "Riemannitis" and "Goldbachitis", to mention but two of some twenty major mathematical conjectures still to be proved or disproved (or proven unprovable), including at least twelve of Hilbert's list of 23, proposed in 1900, that have driven much of 20th Century mathematical progress. The Poincaré result is the first of these to be resolved in the 21st Century.

The great collection of mini-biographies in the book shows that mathematicians are a varied as the rest of us; they are not all ivory-tower dwellers with their nose in the air. In fact, few fit that bill. They range from playboys to surfers to family men to recluses. Of the latter category, Grigori Perelman is typical. He finds contention for priority so distasteful, and the "messy, human" mathematicians so prone to unethical behavior that, after posting his proof and explaining it to numerous workers who sought to verify it, he resigned his post and retreated to St. Petersburg. He declined to accept a Field Medal (the Math equivalent of the Nobel Prize), and will likely decline the million-dollar Millennium Prize that is likely to be (attempted to be) conferred.

I really can't state Poincaré's Conjecture, nor understand the statements thereof found in the book. It seems to require equating things to spheres (as generalized into any number of dimensions), and makes some case for the way to tell spheres from non-spheres with less work. If that sounds abstruse, the author notes that one worker became famous for proving that a circle divides a plane into two regions, an inside and an outside...famous for realizing this seemingly obvious result needed to be proved, rather than for devising the proof itself!

Considering that we still don't have a result for the number of hyperspheres (spheres of four dimensions) that can closest-pack a single hypersphere, there is lots of interesting math waiting to be done, by those who can understand the statements of the puzzles!

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