Either January 14th or February the 11th, I’ll be leading a discussion on the Poincare Conjecture. It will be the first of a series based on the Millennium problems, a group of seven problems which the Clay Institute put a million dollar bounty on, in the hopes that they would be solved by the end of the century. Here’s something to listen to while you follow along:
The Poincare Conjecture has the distention of being the only problem solved. The conjecture asks whether a compact connected and simply connected 3 manifold is the same as a 3-sphere, and proof that it is was submitted to arXiv.org in 2003, and was deemed correct by the mathematics world in 2006. Grigori Perelman, the Russian mathematician who solved it, quickly became a celebrity for solving it and gained more recognition when he turned down the the million dollar prize. This project combines my interest in Russian Mathematics and n-dimensional space, so of course this is the one I would choose. Tonight, lets start to take a look at reaching a basic understanding of the topic.
The first question I have is about things the conjecture says. What does “Compact connected and simply connected” mean? What’s a 3-manifold? How about a 3-sphere? When I went looking for some help understanding the proof, I found that “Perelman’s proof of the Poincare Conjecture extends the Hamilton (1982) program using the Ricci Flow to prove not just some special cases of the conjecture, but the entire statement.” (Quora) Hamilton program? Ricci Flow? What is any of this? These are questions to start with. On top of that, I must starting finding some problems that will help me think about the problems deeper.