The Poincaré Conjecture

            For this installment I would like to talk about the Poincaré conjecture. Again I will probably get a comment saying that the Poincaré conjecture has already been solved. Even though the Poincaré conjecture has been solved, it is still a great problem that was not solved until recently as 2003. I would like to first talk about the history of the problem, then I want to continue with the contributors, the problem itself, and finally I would like to talk about the prize associated with solving the Poincaré conjecture.

            The Poincaré conjecture was formulated by Henri Poincaré, a French mathematician. He lived during the 19^th century and had many works in physics and math. The Poincaré conjecture has to do with a field of math know as topology. Topology is the study of geometric properties not concerning a continuous change in shape or size. It means you are a mathematician who plays with play-dough.  Topology is basically about manipulating shapes in all dimensions. For a long time no one could prove the conjecture, so in the 1960’s and 70’s when topology became popular many mathematicians took a swing at it. The problem concerned proving a property of a sphere in all special dimensions. The conjecture was proven in all dimensions except the fourth spacial dimension. It was not until the development of something called Ricci Flow that the fourth dimensional property was finally proven in 2003.

            The man who had solved it was Grigori Perelman. There is not a whole lot of information out there about him. He essentially came, solved, and left. After using Ricci Flow to prove the conjecture, he went on a tour giving lectures to universities about his work, then he stopped after a year, and went back into isolation. When he had posted his proof no one had ever seen him before, and he did not want the publicity with solving it.

            Now to get to the actual problem. The conjecture is closely related to understanding the shape of our universe. So the universe could be a sphere, or it could not be a sphere. Instead it could be a more tube kind of shape with twists and turns and holes. Poincaré said if someone traveled around the edge of the universe and left a rope behind them as they went, once they have completed their journey then there is a close loop of some kind. At this point, pull on either end and synched up the rope.  Continue to pull and pull and if the rope can not be pulled anymore then the universe is not a sphere. If they can pull all the way, then it is. That is what the conjecture entails. Another perspective on this is if a have a single piece of clay with no holes in it I can make it a sphere. Simply by rolling the clay between my hands it can be done. If I have a doughnut shape or a shape with more holes I cannot turn it into a sphere without pinches or tears in the surface. That is simple to see in 3 dimensions, but Poincaré suggested that, that is the case in all dimensions. The conjecture could be proven in every dimension except the fourth special dimension, until Perelman did in 2003.

            Perelman’s discovery was a big deal in the math community. This problem is known as a millennial problem. In the year 2000 the Clay Institute of Mathematics realeased seven unsolved problems of math. These were considered the greatest unsolved problems at the time. The Poincaré conjecture is the only one to be solved since then. Each problem has a million dollar set aside for the winner. In a turn of events Perelman refused the money. Perelman was also awarded the fields medal. The fields medal is the equivalent of a Nobel prize in math. However, he refused both the medal and the money. It goes to show you that Perelman was not interested in the glory or the fame, but rather in the math itself.  The man is considered a hero in topology, if he wanted to he could get a job at any university he wanted. All he had to do is put on his resume that he solved this problem, and he would instantly be hired. Like mentioned before, Perlman was not about the fame, so he went back to his isolation. Hopefully this can show you that mathematicians do math for the sake of math not anything else, or it can also show you what math means to them.