To celebrate such an excellent resource, this unremarkable little blog will now start featuring a new idea, in the parasitic hope of basking in its shadow. I’m on Wikipedia most days of the week, often link surfing through articles of only minor relevance to my degree. It’s a part of my highly ritualised procrastination routine. Very occasionally I’m going to make a note of the more interesting ideas here, in a feature called Wiki Rummage, where I’ll summarise an article that potentiates food for thought and add other information from sources online. It won’t be a cut and paste job, more of a re-communication in my own explanatory way. For those readers coming here to learn more about life in Korea, I’ll still be updating as normal and there will always be plenty of photos in the archives. If you find it boring, just bear with me while I indulge my shameless geekiness. And if you find it interesting, great.
The controversy doesn’t stop there. The Poincare Conjecture is such a big deal, that it was previously named by the Clay Mathematics Institute as one of the seven Millenium Prize Problems. If anyone solves any of them it means that the Institute will award them US$1,000,000. Perelman is now the only person to have solved one. He has not yet accepted this prize, despite living in poverty with his mother in an old apartment in St Petersburg. He has quit mathematics, avoids the media and apparently plays table tennis with himself against a wall.
All of this adds up to a very interesting biography called Perfect Rigor by Masha Gessen. I haven’t read it, but the reviews of it that I’ve found say that it provides an interesting insight into the mind of Perelman. Perelman wouldn’t grant an interview to the author, and so she gathered the information by talking to his friends and colleagues. I find all of this to be quite fascinating, but one thing the book apparently doesn’t address is what the Poincare Conjecture actually is. The Wikipedia article on it is also a little too technical for my liking. I guess one of the weaknesses of Wikipedia is that for certain topics, the most concise explanation may not be understood by the majority of the population.
So an ordinary sphere drawn on paper forms the boundary of a ball in three dimensions, even though it’s represented on a 2 dimensional surface. In the same way, a 3-sphere consists of an object in 3 dimensions that forms the boundary of a ball in four dimensions. There are many higher dimensions in theoretical mathematics that we ordinary folk don’t pay much attention to, but apparently they exist. I’ll take their word for it. Simple objects become highly complex when represented geometrically in four dimensions.
If you imagine a rubber band stretching over an apple, you could imagine shrinking or expanding it without ever having to tear it or allow it to leave the surface. This property is known as being ‘simply-connected’. On the other hand, if you think about a doughnut shape (a toroid), you could imagine that it’s possible to interlink the rubber band in such a way that the rubber band could not shrink past a certain point without cutting the doughnut. In terms of surface properties, this is the major difference between an apple and a doughnut. The Poincare Conjecture is basically asking whether a 3-sphere is simply-connected or not.
This is not as easy to prove as it sounds. For a start, we humans are physically incapable of observing an actual 3-sphere. The diagrams above are representations of various aspects of the 3-sphere, and the actual thing itself combines properties of all of them. The red lines represent the parallels of the shape, the blue lines are the meridians and the green lines are the hypermeridians. The yellow points are where the curves intersect. All curves are circles and the point where each curve intersects has an infinite radius, represented as a straight line. A real 3-sphere would be much more elaborate, but we’re incapable of comprehending the dimensions in which it exists. All we can do is acknowledge that the dimensions do exist, and try to imagine what might be going on. The crude representations above would be as inadequate as attempting to paint the Mona Lisa using a banana stuck in a donkey’s ear.
Take for example the Large Hardon Collider at CERN. At US$9 billion, it’s the most expensive science experiment in human history. And it’s all to find out whether the Higgs Boson is real and what the universe was like during the Big Bang. The very fact that the experiment has been approved shows that there are enough people in the world who think that the answer is worth more than 3 billion Sausage-and-Egg McMuffin Meals.
Outside of topological circles though, the significance of the Poincare Conjecture is due to the techniques Perelman used to solve it. Simply put, in order to solve such a complex question, Perelman had to invent methods that no one else had thought of. These breakthroughs can now be applied to other questions in mathematics.