I know I promised an entry on group actions today, and we'll get to that soon enough, but I think for a couple of posts I'm going to go in a slightly different direction - partially because group actions are hard to explain, partially because I'm getting a little bored with writing about group theory, and mostly because opportunity has struck in the form of Dr. Perelman's idiocy.
Grigori Perelman is a Russian topologist (from what I can tell) and quite the eccentric to say the least. In 2002 he solved a very old and very difficult math problem called the Poincare Conjecture which happens to be one of the seven Millennium Problems. In 2000 the Clay Institute of Mathematics commissioned a $1 million award for a solution to any of seven of the most difficult and important problems that mathematicians were facing at the time, which they called the seven Millennium Problems. Then, two years later, Perelman presented a solution to the Poincare conjecture which has taken until now to verify. As promised, the Clay Institute awarded Perelman the $1 million prize. Also, in 2006 he was awarded the Fields Medal, which is the most prestigious award offered to mathematicians and comes with a $15,000 prize.
Now, the most interesting part of these accomplishments is that Perelman refused both the $15,000 Fields Medal award as well as the $1 million Millennium Prize award. He says its because he hates the intellectual and moral shortcomings of the mathematical community and does not want to be seen as its figurehead. Since his proof in 2002, Perelman tried teaching for a brief while at American universities, but has ultimately given up math completely. He now lives in a flat with his mother and enjoys playing table tennis against his wall in his basement. When a journalist asked him to comment he replied, "You are disturbing me. I am picking mushrooms."
The problem that Perelman solved, the Poincare Conjecture, is extremely difficult, but surprisingly accessible. Suppose you stretch a rubber band over the circumference of a sphere - or really around a sphere in any way. You can then slowly and carefully (the math terms are continuously and uniformly) move the rubber band along the surface of the sphere such that the rubber band shrinks down into a single point. Now, suppose that we try to do the same thing with a doughnut (mathematically a torus) instead of a sphere. If we can imagine that the rubber band has been stretched around the donut in an appropriate way, then the rubber band cannot be reduced to a point without either tearing the rubber band or breaking the doughnut. In topological terms, the sphere is "simply connected" and the doughnut is not. This is considered the three dimensional case and the Poincare Conjecture, in essence, asked for a way to classify four-dimensional simply connected objects.
This brings us to a nice way of introducing topology. The difference between a doughnut and a sphere is more than simple-connectedness. If you imagine that a sphere were made of clay, then there is no way to deform that sphere into a doughnut without poking a hole in the clay. However, if you had a clay coffee cup, you can imagine how it is possible to deform the coffee cup into a doughnut because the coffee cup already has a hole. If you're having trouble visualizing that, wikipedia has a nice animation of this transformation. In topological terms, we call the coffee cup and the doughnut homeomorphic (note that the words homomorphism and homeomorphism are not the same). Topology, from my point of view, is basically an in-depth study of what it means and how to tell when two things are homeomorphic. That being said, though, if you took a first semester in topology or read an introductory topology text, you would find yourself pretty far removed from that idea of homeomorphism and, in fact, the definition of a topological space does not really resemble anything I've mentioned here.
Topology is extremely interesting and it often forms a gateway to other ideas in other branches of mathematics. I think I might start to work though some topology during some breaks in the group theory stuff even though topology does not help me in my character theory goals. Topology will most likely be in a lower capacity since I do not understand it, myself, as well as I would like, but sometimes I feel like the barrage of group theory has been pretty intense lately and topology might provide a nice break from that at times.
