## Friday, November 23, 2007

### Are all mathematicians crazy?

Yay, Central America!
Yay, Costa Rica!
Yay, Flags of many lands™!

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."

This Rasputin lookin' mofo 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, and gained 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 look at this Rasputin lookin' mofo, if she could talk sense into him, she'd probably start by not dressin' him funny anymore.

Matty Boy, can you 'splain the Poincaré Conjecture to your gentle readers, some of whom have serious issues with the math?

Let's give it a shot.

In the picture above, 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.

In honor of Grisha, today's Random 10 starts from a non-random point on a song by Tom Lehrer, and ends with a non-random bonus track by Prince.

Enjoy.

Lobachevsky Tom Lehrer
Stardust Hoagy Carmichael
Just One Cornetto Pookiesnackenburger
The Weight The Band
Only Love Can Break a Heart Gene Pitney
After You’ve Gone Django Reinhardt
Po’ Lazarus James Carter & The Prisoners
Sweet Dreams Eurythmics
It Didn’t Turn Out That Way Mose Allison
non random bonus track: Money Don't Matter 2 Night Prince

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Now playing: Prince - Money Don't Matter 2 Night
via FoxyTunes

Matty Boy said...

You might ask why mathematicians worry about impossible things in three dimensions or shapes in four dimensions.

The short answer is: The shapes of sets of solutions of differential equations.

And now you're sorry you asked, aren't you?

Splotchy said...

The Klein bottle looks more like a messed-up bugle. Is it too late to change the name?

Matty Boy said...

At that angle, you're right, it looks like a bugle. I turned it 90 degrees to put it in this picture. Standing up, it looks like a messed up vase.

Math folk are kind of fond of Klein, we won't get rid of his name. And given that there isn't a place to put your lips, the bugle name could easily confuse the literal minded.

Just one opinion.

FranIAm said...

Oh Matty, how much I have learned from you.

I may never be good at "maths", but as a result of our online acquaintance I can say I know a lot about "maths", which is good enough for me.

Today I heard -oh what is her name - the woman who was a girl on The Wonder Years, you wrote about her recently.

She spoke of math and her love of it and how many girls are discouraged from it.

Anyway, I listened with a different ear than I might have in the PLSD. (That would be Pre Lotsa Splainin Days).

Hope you had a good Thanksgiving.

dguzman said...

Maybe if he'd pick up that prize money, Grisha could stop wearing zip-up sweat hoodies under blazers.

Matty Boy said...

Money does not buy taste, dg. I think even with oodles of cash, Grisha is not going to be escorting the Russian supermodels and dressing like George Clooney.