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10 years after the Proof of the First Millennium Problem, what have we learned?


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MarcZ

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Hello folks! (Warning if your a hardcore mathematician this article is written by a layman for other laymen, if you feel like being picky enlighten us but don't be condescending!)

To anyone who follows mathematics November 11, 2002 is an important date. On that day Grigori Perelman released the first of three papers which documented a solution on how to solve the Poincare Conjecture (also the Geometrization Conjecture which is not as famous), one of the seven millennial problems set out by the Clay Mathematics Institute. The solution which was quite literally out of the blue and almost completely unexpected propelled Perelman to international fame, and brought about all sorts of "mathematical pundits" trying to sell to us the layman all the miracles and insights this proof would bring us. Now ten years later we can take a bit of reflection and see actually how important an event this was to us.

What is the Poincare Conjecture?

Quite simply the Poincare Conjecture is a problem in topology which is the study of surfaces. The conjecture revolves around by a question posed by the great French mathematician Henri Poincare in 1904 that any loop on a 3 dimensional manifold that can be contracted to a point is a 3-sphere. (Please not a 1-D sphere is a circle which exists in 2 dimensions, a 2-D sphere is our classic sphere which exists in 3 dimensions, so the 3-sphere is thus the sphere which exists in 4 dimensions, this is a very abstract concept to most people, and not easy to visualize.) Topology basically classifies all surfaces into two possible shapes: spheres and torus. A sphere is practically any surface that is all connected (such as a shoe, a blanket, a book or so forth) that when stretched and morphed can be made into a sphere, a torus on the other hand is a special word for a donut and any object with a hole in it, when stretched will always turn into a donut. Poincare's conjecture is easy to visualize in three-dimensions because a loop being pulled on a surface can always be made smaller without ever coming out of contact with the surface, the same is not true for a torus as a loop cannot cross the hole. Mathematicians were then able to say based off this information that in four dimensional space the same must also be true. However, this turned out to be horribly difficult to prove and it's solution thus evaded mathematicians for almost 100 years.

So how was it solved?

Two mathematicians Thurston and Hamilton took seperate cracks at this problem in the 1980's: Thurston attempted to solve the problem by making a conjecture of his own about different types of geometry that exist in four dimensions and then proving that conjecture to prove Poincare's he actually ran aground and created another unsolved problem called the Thurston geometrization conjecture. Hamilton on the other hand formulated a new mathematical tool altogether (Calculus for example is a mathematical tool), called Ricci Flow which was modeled somewhat closely to the Heat Equation in physics, which basically attempted to address the problem by saying that any shape in three dimensions could ultimately be made to flow into a sphere by pushing in the curves of the surface, however unlike three dimensional space where it would be possible to push a fluid into a sphere (somewhat similar to playing with a bubble to make a sphere), four dimensional space had something up it's sleeve, it would often create singularities which would cause the surface to tear apart (in violation of topological rules) and thus his research also ran aground and remained so for twenty years.

Then along came Perelman who after living in recluse in Russia for some odd number of years out of nowhere released 3 papers to the arXiv that shocked everybody by bringing in several realizations that had never before been conceived to solve this conjecture. These included introducing two new concepts to the tool of Ricci Flow the concept of Perelman Entropy (somewhat based loosely off the physical concept of entropy, though Perelman was not a physicist and developed his theory separately of physical ideas), and surgery. This allowed Perelman to show how singularities could be resolved in an orderly fashion in four dimensions and thus allowed them to prove that in fact yes anything that can be contracted to a point in 4 dimensions is a 3-sphere, and at the same time the geometrization conjecture on how many types of geometries there is in 4-dimensions.

So what good is this knowledge?

In the immediate aftermath of the proof mathematicians were skeptical and had to do research and verification amongst each other until 2006 when they finally declared that Perelman's proof was correct. It was claimed to be 2006's scientific breakthrough of the year, and in 2010 the Clay Mathematics declared that the first of the seven millennium problems (and to date only one) was solved.

This had a lot of people asking, well this is great but what exactly does this mean, and why really does anybody care. Well mathematicians made the defense of saying that beyond new pure mathematical tools (often which have no implication to anything in the real world), it could lead to as of yet unknown discoveries and simply knowing this fact is what has mattered. Others tried to convince people who required more material uses of research that in fact this proof could help us understand the shape of the universe!

So what of these claims?

Having some years now to review research and claims we can kind of verify some of these elaborate claims and perhaps theorize about what good the Poincare conjecture's proof has actually been to us.

Firstly, the Ricci flow that was developed for this proof has become an extensively used topological tool. In fact it was used for another long-unsolved proof in 2007 for the Sphere Theorem which had been unsolved since the 1950's, this proof built of Perelman's work. So for topology the Poincare conjecture is a big deal. One must remember however the topology is a very heavy area of pure mathematics and to a lot of people this knowledge has really very few uses.

Secondly, Ricci flow, also has come to the attention of physicists for properties it shares with the Heat equation, but also for peculiarities that mark some pretty stark differences that may be useful for models in extreme environments, that at least for the near future will have little importance to us.

Thirdly, Perelman entropy has also come to the attention of mathematicians and physicists because of it's similarities to typical entropy that everything goes from order towards disorder, but also for some strange differences which could be of interest in understanding randomness, and differential equations both of which are important fields of applied mathematics with real implications in finance and physics.

Fourth and finally, there was a claim that the Poincare conjecture could help us understand the shape of the universe. This has largely been discredited, as although there was some belief in cosmology that the universe could in fact be modeled in four dimensions in the early to mid 2000s this idea has fallen into disfavor as scientists studying the cosmic microwave background have found that evidence contradicts this belief and thus the topological model is currently be reformulated meaning that the Poincare conjecture will likely have no bearing determining what the actual shape of our universe is.

Conclusion

There will undoubtedly be more research on this landmark paper in years to come as for now it is the most important mathematical work of the present century. And in the future as the typical mathematical ability increases I could see Ricci Flow or an Introduction to Perelman becoming possibly a standard undergraduate course in pure mathematics. On a personal note I hope to do enough pure mathematics to do some more self-interest research on this topic myself whilst doing more useful things like financial mathematics and computer science, simply because this development fascinates me.

I apologize for the lack of pictures. What do you guys think of this work? Which of you before reading this topic had never heard of this development? Do you think that this (at present) highly pure math concept will find it's way to be a useful topic to us or do you think it will always be doomed to just being a specialist topic that although was a millennium problem as one of the most important problems in mathematics will actually be of little use to us in the long run?

This is a concept for everyone to discuss, I'm sorry if I dumbed it down too much, or explained it poorly, I can only do as well as I can understand myself. :)




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