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.

Fermat's Last Theorem

            Today’s blog post will about Fermat’s Last Theorem. I know right off the bat I am going to get a comment saying that Fermat’s last theorem has been solved, and you would be correct. However, it is still a great mathematical problem that was unsolved for a couple hundred years. So no this is not an unsolved problem, but it is one of the simpler problems to explain, that was once an unsolved problem. First I would like to talk about some background information, such as the origin of the problem, the problems contributors, etc. Then I want to talk about the problem itself, and how it works. Then I want to talk about how common it comes up in society. I will not be talking about the proof in this post because, it is too long for a single post, in the future I might post another, or maybe two more posts to explain it in depth but that will not be happening in this post.

            It seems only right to start with Fermat himself. Pierre de Fermat was a French man who lived during the 17^th century in France. He himself was not a mathematician, in fact he was a judge. He was a judge that had an interest in math during his spare time. He would read these published mathematical texts and write notes in the margin. On one occasion he was reading a book called Arimetica by Diophantus. He stumbled on an equation that was very similar to that of the Pythagoras, that being X^2+Y^2=Z^2. He was curious to an alteration of the problem he wanted to know if the equation had any other solutions that would satisfy a higher order of this equation. He thought about it, and then in the margin wrote something to the extent of, “I can prove that there is no solution to a higher order, but there is not room for me to write it in the margin.” His book was littered with these little marginal notes. He would write,” I have a proof to this, but I am too busy to write it out.” Fermat then without proving anything died. His son found his annotated copy of the book and had it republished with his father’s notes added to it. Because of this, mathematicians started working on these proofs that Fermat claimed to have been able to prove. Now proofs are flying in left and right, one after another, the marginal notes are being ratified. That was all good until, they got to this one involving Pythagoras’ equation. The theorem could not be solved.  This problem remained un solved for centuries afterward. The problem itself is not that hard to explain, but by the 20^th century it is understood that this problem does not have a simple solution. Many mathematicians have tried a crack at the problem, but to no avail, until Andrew Wiles comes along. Andrew Wiles grows up learning and loving the problem. So after going through college getting a PHD and going on with life. Someone proves that if you can solve The Taniyama Shimura conjecture, you would also be proving Fermat’s last Theorem as well in doing so. Now it is not important to know about the Conjecture other than by solving it Wile’s would get what he wanted. He spent several years working on it alone, then gave a lecture solving the conjecture. Unfortunately, he had made a mistake. It took him another year, but then he fixed his mistake. After 8 years he had gotten what had been trying to do since the age of ten.

            The problem itself is rather straight forward. If you have had junior high math you know Pythagoras’ equation X^2+Y^2=Z^2. Well Fermat thought to himself, "I wonder if there are solutions to equations like X^3+Y^3=Z^3 or to the fourth power." He could not find any whole number solutions. He said that there were no whole number solutions to any power greater than two so in math terms, there is no solution to X^N+Y^N=Z^N where X,Y and Z are not equal, and N is greater than two. That is all the problem is. I said it was a simple problem. If you do not believe me, go ahead try to find a solution.

            Now I mentioned that this problem actually comes up in the media. There are a few famous instances of this. The main one being the Simpsons. In one the episodes when Homer is writing on the chalk board doing calculations it flashes an apparent solution to Fermat’s unsolvable equation. The equation shown on the board is obviously not correct, but rather to pay tribute to the mathematician. Another example in a book called, The Devil and Simon Flagg, where Simon makes a deal with the devil that he gets to keep his soul if the devil can prove Fermat’s Last theorem. Also the problem crops up in star trek. The point is, the problem is more common than you think it is. So maybe next time someone makes reference to the problem you will be able to spot it.