Friday, March 29, 2013

Are all mathematicians crazy?

 (This post is an edited version of something I first published in 2007 on my original blog Lotsa 'Splainin' 2 Do.)

When answering the question which is the title of this post, there are two possible answers.

Answer 1: "All mathematicians? All? No, not all mathematicians are crazy."

Answer 2: "Define crazy."

This Rasputin lookin' individual is Grigori Perelman, known to his friends, if any, as Grisha. Grisha is currently unemployed and lives at home with his mom in Saint Petersburg, Russia. A few years back, Grisha gave a talk and published a paper that proved the Poincaré Conjecture was true, gaining worldwide fame in the math community, as well as some headlines out in the real world.

The Poincaré Conjecture is a big damn deal in math. Henri Poincaré, who would be a consensus pick among mathematicians as one of the ten greatest of all time, made this conjecture over a century ago. Lots of smart folks thought a long time about how to prove the statement true.

Grisha actually did it.

Here comes the crazy part. Solving the Poincaré Conjecture comes with a prize of... $1,000,000! (Put your pinky finger to your mouth like Dr. Evil if you feel so inclined.)

Grisha doesn't want it.

Separate from that cash, Grisha has been awarded the Fields Medal, equivalent to the Nobel Prize in math, which also comes with a nice clump of cash. (There is no Nobel Prize in math.)

Grisha doesn't want it.

Maybe his mama could talk some sense into this boy. But taking a good look at him, if she could talk sense into him, she'd probably start by not dressing him funny anymore.

Here is a vague explanation of the Poincaré Conjecture.

In this picture, we have three different objects, a sphere, a torus and a Klein bottle. We are going to consider only the surface of each, which we can think of as a two dimensional thing in a three dimensional world.

The sphere is the easiest of these. It splits the three dimensional world into three parts: the inside of the sphere, (known as a ball), the skin of the sphere and the outside.

A torus is the next easiest. There is an inside, the skin and the outside, but there's the "hole in the middle", which makes a torus different from a sphere in mathematically important ways.

Then we have the physically impossible model that is the Klein bottle. It can be thought of as two Möbius strips glued together along their respective edges. It has to pass through itself in three dimensions without their actually being a hole, which is the impossible part. It has no inside or outside, just like a Möbius strip doesn't have two sides. Mathematicians call a shape like the Klein bottle non-orientable.

Now we get back to the conjecture. All the things up there are two dimensional things embedded in three dimensions. Poincaré was looking at a category of three dimensional objects embedded in four dimensions, and he speculated that all the things in the category were "like" the sphere is in three dimensions. Nice and orientable, no holes like a torus. As simple as things can get in four dimensions.

Lots of people tried to prove it.

Grisha did it.

Now if we just get him to pick up that pile of money with his name on it.

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