Friday, March 26, 2010

Poincaré Conjecture

Do you have to be crazy to turn down 1 million dollars handed to you at your front door? Does such an act of utter contentment with your life make you less human?  

Grigori Perelman would answer with an emphatic no.  After years of others' false claims and proofs, this Russian mathematician (big surprise, it's like they breed math whizzes over there) solved the mysterious Poincaré conjecture back in 2003 with all the kinks worked out by 2006.  He didn't bother to show up at the awarding of the coveted Fields Medal in 2006, one of the most prestigious awards given in the mathematical world. And now this month, he was awarded $1,000,000 by the Clay Mathematics Institute for solving Poincaré conjecture as one of the seven Millennium Prize problems.  However, this gifted man didn't even open the door of his rundown apartment in St. Petersburg for the prize but stated, "I am not interested in money or fame.  I don't want to be on display like an animal in a zoo."  

What a beautiful example of the power of the human mind and spirit, the disregard of all worldly prestige for simply challenging oneself toward an explanation, an answer to an intricate, natural puzzle.  News reports have been EATING this story up trying to wrap their minds around this seemingly inane mindset necessary to refuse the glamorous life of a millionaire.  I only see an enlightened man, brilliant in tracing the possibilities, the patterns.

Now let's take a look at this particular conjecture formulated by Henri Poincaré, another great  mind and theoretical physicist whose maps I learned to love in differential equations. Poincaré did awesome work in algebraic topology, particularly interested in topological characteristics of the sphere.

The standard phrasing of the conjecture is this:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

In other words, say we have a space that up-close locally looks like an ordinary 3-D space, but is connected, finite, and has no boundary.  If it also has the property that each loop in this space can be continuously tightened to a point, then the space is a sphere.  Sure, sounds good right? Well the methods in attempting to prove this in 3-space can be very complicated while dimensions 4 and higher were proved more easily.  
One of the attempts was Hamilton's program which put a Reimannian metric on this simply connected, closed 3-manifold.  Now we improve this metric by using the Ricci flow equations

R is the Ricci curvature, and the conjecture is easily proven if there is always positive Ricci curvature and the program points out singularities in the space if there is not.  Perelman's work found a way around these singularities by a procedure called Ricci flow with surgery which involved carefully cutting the manifold into pieces along the singularities (yes we can cut pieces of space in math, it's awesome like that).  A few of his theorems sum up in this fact about the well-defined Ricci flow on these manifolds: If the fundamental group is a free product of finite groups and cyclic groups then the Ricci flow with surgery becomes extinct in finite time, and at all times all components of the manifold are connected sums of S2 bundles over S1 and quotients of S3.  and TA-DAAAAAAAA the Poincaré conjecture follows yippie!!!!!!

Here's a pic of some Ricci flow (sounds like the newest R&B step)

SO challenge of the day/eternity: solve the next Millennium prize problem, win a million dollars, and donate it to kitty adoption centers worldwide mew mew!

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