The Poincare's Conjecture is what I'd like to talk about and it is one of seven kind of maths questions, maths problems which were picked by the clay institute in well in the year 2000 as kind of the the biggest outstanding problems in maths the kind of Real Massive unanswered questions. And they they picked a list of seven things and for each one. There is a million-dollar prize That they literally have that money waiting for people and of these seven problems only one of them so far has been solved. And it was the poincare conjecture So the poincare conjecture which was put up by poincare and I forget around the turn of the last century around 1900 was a particular conjecture as a guess and we were and he claimed that there were these certain relationships between Geometry and Topology so I can try and explain a little bit about what the poincare conjecture is It's from an area of math called Topology which is to do with kind of shape and how things fit together in space. And but it's essentially about Spheres, so a common idea in topology is the idea of deforming things, so I've got here a cube. Which is made out of dough and in topology we can sort of kind of move things around a little bit. It's sometimes called rubber sheet geometry so I could take this Cube. And I could kind of mess around with it. And I could round it off like that using my hands and I can get a sphere pretty much near enough. So that's that the sphere the theory is that you're allowed to squash things stretch things And all that kind of thing But you're not allowed to punch holes in things you're not allowed to make a hole and you're not allowed to close up a hole So because of this the number of holes that something has is very important, so this sphere has zero holes I guess you could also have something like this donut Which has exactly one hole in it? And you can't get from a sphere to a donut by deforming it without kind of closing up this hole or by cutting it here and straightening it out and Squashing that sausage into a sphere And you're not allowed to cut and you're not allowed to stick so the kind of things that are different from each other spheres are different from Donuts which are different again from like a pretzel with two holes in it or something with three holes in it or or Any number of holes this is the kind of maths that I studied at university for a very long time and did some very difficult? Maths So on Poincare was a French mathematician who posed this conjecture he suggested that if you have an object which Doesn't have any holes in it so first condition no holes Second condition is that it is kind of small It's not it doesn't go on forever in any direction, so it's it's kind of finite You could maybe put it in a box and close the lid then It's a sphere or at least it can be made into a sphere and in two dimension while in three dimensions Which is where we live this seems like a fairly sensible statement, but poincare a suggested that this would hold in any number of dimensions So you can take this is a three-dimensional sphere But you could have a two-dimensional sphere would be a circle you can also go up and have a four-dimensional sphere which You'd have to pretty much imagine, but it's it's sort of a sphere with an extra dimension it's it's got a fixed radius in this one has a fixed radius in three different directions, but the Four-Dimensional sphere has another direction in which it's still this size and any number of dimensions above that And then you know it's it's counterproductive to try and imagine those things because it really does hurt your brain but this conjecture Exists as a mathematical statement, and he posed this in the 1900s but didn't have a proof for it and the the actual proof of this took quite a long time, so the 1960s and 70s Lots of Topology was happening what people were developing new tools for doing topology and they managed to prove During that time the case of five dimensions and above, but there was still four dimensional spheres to think about And this was sort of still unsolved and lots of people had a go at this and it turns out it took different approaches I guess to solve this particular thing so in the 1980s there was some work done and a Completely separate conjecture was made by a guy called William Thurston Which was about kind of four dimensional things and shapes and it turned out that that was quite closely related to it? But that wasn't proved either and it took until I think 2003 at which point a mysterious cumin appeared on the internet which had proof of Thurston's conjecture which also proved The poincare conjecture for this unknown four dimensional case and it was very exciting everyone was sort of scrambling to try and try and find Whether this was a real proof or not because the poincare conjecture having been around for so long Was one of the most falsely proved results in the history of maths like it's had more wrong proof attempts than any other Statement which is brilliant because it means people are trying really hard to solve. It actually this is one of the nice things about the I'm thinking the nice things about the this archive that instantly got his work Spread to everybody I guess in the old days Maybe these things would be sent out as preprints which of course I remember those old days well But I don't know if he would have had the resources to spread it would have taken a lot longer for people to realize that That this stuff was out there the mathematical community dived on this paper and checked it and verified that It was in fact a proof of what it claimed to me And it was but I remember when Ben chow said this is amazing stuff We're gonna now have a seminar and we met several hours a week, and we're going to study this and each chapter This was it was about a seven or eight chapter Paper, we're each assigned to chapter but it turned out that to understand a chapter took huge amounts of work, the Mathematician that had put this paper up was called grigori perelman. He was from Russia And he was awarded a fields medal. Which is like maths equivalent of a nobel prize He was also Of course awarded the the millennium prize of a million dollars and he for various reasons I guess declined both of those things he didn't necessarily Want them the fact that he did made an absolutely huge media storm everyone was interested Suddenly And he he could have had any job anywhere in the world after he did this But he didn't want to take it, and he's a bit of a you know He wear his hair is funny, and he lives with his mother and sent Petersburg or something I've actually never met him mathematician that solves difficult question wasn't news But mathematician that solved difficult question and doesn't want a million dollars was suddenly huge news for about a year after After he put these papers out I remember was november of 2002 because I remember I With a bunch of people who were really into RicCi flow in San Diego and those papers came out And we instantly just focused on trying to understand those those those papers And they're very very difficult about the for the year after that Pearlman Did sort of a world tour I know some friends in Berlin he came to visit them? Near Berlin he came to mit and gave some talks, and he did a bit of a tour And if I wasn't busy teaching and didn't just didn't have the time to really Well, maybe I should have done it, but any I didn't go to see him at any of these talks if we knew that he was going to just immediately let's say after a year go back and go back into Staying away from from publicity it would be fun to meet him. I guess I've simplified It here and is a little bit more complicated to define specifically what I mean by each of these things But hopefully that gives you a rough idea of the kind of maths that it's talking about there are a lot more kind of difficult blackboards full of squiggly numbers you know there are different ways of doing math one way you can take some big problem like that and Say, I'm going to do just focus on that the danger of that is you don't you don't solve it and then? You know that's not so good or you can look at many many different problems. I like to look at many different problems I'm not focusing on one particular one, but he would take that problem in fact from what I understand from friends who knew him He announced at some point. I forget exactly when many years before he actually came out with these papers I'm Gonna use ritchie flow to try to Solve to try to solve the poincare Conjecture, and he just without I don't know if he was doing any teaching at the time But living in St.. Petersburg, and then he did it If you haven't seen it already we've got a video about Ricci flow and RicCi flow Surgery which is kind of like the Golden bullet that helped them solve the poincare conjecture if you haven't seen them yet Maybe go and have a look I'll have links everywhere in all the usual places We've also got a video about another one of the millennium problems the Riemann hypothesis So maybe you could go and watch that video that one hasn't been solved yet So there's a million dollars there for you if you can crack it
