Gregory Perelman

The New Yorker recently published an article about the Poincare Conjecture. This is a mathematical problem that has finally been solved after about a hundred years. Gregory Perelman is the man who accomplished that feat, and the The New Yorker article is about the tempest in the academic tea pot that his solution caused. I encourage everyone to read the article for the full story. (I’ve included the link at the bottom of this entry.) However, for those of you who aren’t so inclined, I’ll talk a little about Perelman and the conjecture.

In its essence, the Poincare Conjecture is easy to understand. In the field of mathematics called topology, shapes can be classified according to how many holes they have in them. On the basis of this scheme, a space in the shape of a doughnut, called a torus, is in a different classification from a sphere, because the torus has a single hole in it, while a sphere has none.

What are called continuous transformations can be applied to the spaces in topology. These are changes to the space that do not involve tearing the surface of the space. For example, a sphere can be transformed into something with the shape of a football. No problem there, but spaces with different classifications cannot be transformed into one another using a continuous transformation. You can see this for yourself. Make a ball of clay, then try to reshape that ball into a doughnut without breaking the surface of the clay. It can’t be done.

The Poincare Conjecture states that any space with no hole in it can be reshaped into a sphere using a continuous transformation. This seems obvious when one thinks about it, but it is not unusual for a mathematical conjecture to appear obvious, even to non-professionals, while being exceedingly difficult to prove. Many brilliant and accomplished mathematicians have had a go at the Poincare Conjecture, and it was finally proved for all spaces but three-dimensional spaces. However, these are the most important kind, especially to physicists, because the space that we know and live in is described using three dimensions. That’s why it was such a big deal when Perelman proved the conjecture for the three-dimensional case.

In many ways, Gregory Perelman is the stereotypical, eccentric mathematical genius. In 1992, he spent a semester at NYU. During his time there, he wore the same brown corduroy jacket every day. He claimed to live on nothing but bread, milk, and cheese. His fingernails were several inches long because he saw no reason to cut them. At age twenty-nine, in the mid-nineties, he returned to Russia to live as a recluse with his mother.

Odd enough, for sure, but it wasn’t mere eccentricity that led Perelman to become a recluse, It was professional politics. The mathematics profession is subject to the same sort of political machinations that are to be found in all academic disciplines. Many of the best mathematicians have enormous egos, and are much concerned over getting credit for their discoveries. Perelman apparently thought that they cared about their careers more than the mathematics, so he left the profession. As he vehemently states in the New Yorker piece, he’s not a politician, and obviously has no intention of playing at being one. Good for him!

It is for that reason that Perelman has become the first man to decline the Fields Medal. The Fields is the equivalent of the Nobel Prize in the field of mathematics. It is given out every four years, usually to three or four people. The winners of the Fields Medal invariably become figureheads for the profession, and that is evidently the last thing that Perelman wants. That’s why he turned it down, and it seems that there’s a question of integrity involved on Perelman’s part because he has also refused another prize of one million dollars to the first person to prove the Poincare Conjecture.

Characters like Perelman intrigue me. I was a bright kid, and scored in the low genius range on IQ tests and such. However, I was by no means a prodigy, though I certainly envied them. It seemed that the whole world of the intellect was open to them, and that’s what I wanted. It was only much later, when I was an adult that I realized what it really was about prodigies that I envied. The life of the mind is the only thing that truly matters to me and to be a genius like Perelman, or Einstein meant to be a demigod. And why not, if Plato is to be credited, and knowledge is virtue?

I was browsing some science blogs last week, and ended up on the blog of a math professor at James Madison University. He was talking about Perelman’s accomplishment, and pointed out that to most math professors it’s another world. As he said, to address the big, open problems requires a monkish devotion, and the relevant mathematics is only to be found in difficult articles in the journals, many of them hundreds of pages long, which leaves out most mathematicians. As this professor said, he stood no chance of understanding what Perelman was doing.

I’d love to write a novel about a character like Perelman, but I wouldn’t know how to go about it. How do you create an interesting book about someone who lives so completely in the mind? It would indeed be like trying to write a novel about a contemplative monk, a task that is well beyond my powers.

http://www.newyorker.com/fact/content/articles/060828fa_fact2