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'''The Poincaré Conjecture'''

The Poincaré conjecture is about the three-dimensional sphere (also called the 3-sphere - in the hypersphere, this bounds the unit ball in four-dimensional space).  In particular, the conjecture states that every simply connected closed 3-manifold is homeomorphic to the 3-sphere.
It is easier to understand the two-dimensional version of this problem.  Consider a compact, two-dimensional surface (without boundary).  If every loop on this surface can be continuously shrunk until it is a single point, then this surface is homeomorphic to the 2-sphere, which is just the basketball that everyone is familiar with.  This process of shrinking a loop is illustrated in the picture below.  The Poincaré conjecture states that the same is true for three-dimensional spaces.
 
To understand the three-dimensional case Poincaré was interested in, imagine that you have a blob of something like play-doh.  You have a piece of string, but no matter how you tie up, it is always able to ooze out of your trap.  Then, according to the Poincaré conjecture, what you really have (instead of a random blob of play-doh) is just an out-of-shape ball. 
The Poincaré conjecture is the only Clay Millennium Problems that has been solved to date.  Before going into ''how'' it was solved, I’ll give some of the history behind it.

'''Henri Poincaré'''

Henri Poincaré, the man who first proposed this famous problem, was born in France in April 1854.  As a child, he was ill for quite a while with diphtheria; during this time, his mother Eugénie Launois tutored him.  When he finally did go to school, he excelled in nearly every subject, but especially in math.  His math teacher referred to him as a “monster of mathematics”.  He earned his Bachelor’s degree in 1871, then served in the Ambulance Corp during the Franco-Prussian War.  Finally, he returned to school at Ecole Polytechnique, where he continued to study mathematics and published his first paper.  After graduating, he worked as a mine inspector before finishing his doctorate at the University of Paris in 1879.  His doctoral thesis was about differential equations.  From 1881 on, he was a professor at the University of Paris.

'''The Question'''

By the turn of the 20th century, Poincaré had developed, based on the work of Enrico Betti, a system known as homology.  At first, in 1900, Poincaré claimed that homology was sufficient evidence to know that a 3-manifold was a 3-sphere.  But four years later, he wrote a paper describing a counterexample to his own idea, a space that is now known as the Poincaré homology sphere.  As their name suggests, homology spheres are manifolds that have the same homology as a sphere; since Poincaré’s initial example, many more have been constructed.  The key point here is that the Poincaré sphere is ''different'' from the 3-sphere, even though they have the same homology, and therefore homology is ''not'' enough to say that a given manifold is a sphere.  To show that this difference is true, Poincaré introduced a new topological invariant – the fundamental group – and showed that while the Poincaré sphere has a fundamental group order of 120, the 3-sphere itself had a trivial fundamental group.  

It was in this same paper that the Poincaré conjecture was born, if somewhat accidentally.  Poincaré wondered whether a 3-manifold with the homology of a 3-sphere ''and'' a trivial fundamental group had to be a 3-sphere.  Essentially, this is his original claim from 1900 corrected to include the extra condition about the fundamental group order being trivial.  The idea of “trivial fundamental group” is the same thing as saying “every loop can be shrunk to a point” (from the two dimensional case discussed earlier).  Poincaré only wondered about this, he never actually said that he believed this new condition was enough to fully characterize the 3-sphere.  However, the idea that it was enough nonetheless became known as the Poincaré conjecture, and it remained an open question for almost a century.

'''Attempts to Find a Solution'''

In the 1930’s, J.H.C. Whitehead claimed that he had a proof of Poincaré’s conjecture, but he soon discovered that he had made a mistake in his work and had to withdraw his claim.  Several other mathematicians did the same thing during the 1950s and 1960s, believing they had found a proof only to later realize they hadn’t.  In 1958, R.H. Bing successfully proved a special case of the conjecture; he was able to show that if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then that manifold is homeomorphic to the 3-sphere.  However, the general case remained unsolved.  During the 20th century, a lot of other work went into trying to solve the question; while no complete proof resulted, the work done did increase our understanding of 3-manifolds themselves.

In 1982, Richard Hamilton wrote a paper introducing the idea of a Ricci flow on a manifold.  In particular, Hamilton was able to use this concept to prove several more special cases of Poincaré’s conjecture (but still not the general case).  The basic idea behind his program involves a process in which a given 3-manifold is distorted; this distortion is controlled by a differential equation analogous to the heat equation.  Commonly discussed in physics, the heat equation quantifies the behavior of certain scalar values, such as temperature.  Specifically, it tells us that heat will flow from areas of high heat to areas of low heat until the whole space is the same temperature.  In a similar manner, Ricci flow describes curvature.  When he developed the program, Hamilton was hoping that would be able to prove that concentrations of greater curvature spread out until the entire 3-manifold has a uniform curvature (much like how a heater is able to heat the whole room to a single temperature).  In that case, you can start with a 3-manifold, let Ricci flow occur, and you should eventually get a “normal form”.  Over time he extended this concept, but was still unable to find a complete proof of the problem.  Specifically, no one could prove that the Ricci flow would occur unimpeded until the normal form was reached.  

'''The Solution at Last'''

Finally, in 2002 and 2003, Gigori Perelman posted three papers in an electronic archive, which together outlined a proof of the Poincaré conjecture.  His work was based on (and in fact completed) the Ricci flow program Hamilton had developed earlier.  In particular, he found a controlled, systematic way to remove any impediments to the Ricci flow.  To understand what this means, think back to the ball of play-doh.  The Poincaré conjecture said that if it could escape any way you tried to trap it, then it was in fact just a ball.  Ricci flow begins to prove this fact by allowing us to show that the play-doh can be stretched, cut, and otherwise distorted (until it becomes a ball or something impedes the process), as in the picture below.  Perelman’s work tells us for certain that we can ''finish'' the job of molding our play-doh into a ball.
 
The problem was that Perelman’s papers only gave a sketch of the proof.  It lacked some of the formality and rigor that is so important in mathematics today, and few people were capable understanding it anyway.    This ultimately led to controversy over whether or not his proof was actually correct.  In an effort to determine the answer once and for all, other mathematicians began to try to flesh out his proof, to make it complete rather than just an outline.  After several years, they had found some gaps in the proof, but were able to fill them in (using his methods, so it was still consistent).
  
On August 22, 2006, Perelman was awarded the Fields Medal for his proof of the conjecture, which he refused to accept.  On March 18, 2010, the Clay Mathematics Institute awarded him the $1 million dollar prize for solving one of the Millennium Problems, but he refused that as well.  His reasons for rejecting these awards are somewhat of a mystery to many people – who wouldn’t want a million dollars – but he says that he is not the only one who deserves credit.  His work may have been the last contribution to the proof of the Poincaré conjecture, but it certainly was not the first; his solution was built on the work of his predecessors, especially Hamilton, whose Ricci flow program provided a strong foundation.  Perelman has also stated repeatedly that he has a problem with the organized mathematical community as a whole.  At this point, the methods of choosing award winners in mathematics can be considered somewhat unfair – the only people with enough knowledge to understand ''why'' someone is deserving of such an award are in fact the very people who deserve such awards.  Perelman chose to stand his ground and make a statement rather than change his mind just because he was the one getting a prize.

'''References/Links'''

http://www.slate.com/articles/life/do_the_math/2006/08/who_cares_about_poincar.html

http://en.wikipedia.org/wiki/Poincare_conjecture

https://www.youtube.com/watch?v=AUoaTrQTM5o 


[Category:MA375Spring2012Walther] Stephanie Wicke