Poincaré Conjecture - Numberphile

The Poincaré

conjecture

is what I would like to talk about and it is one of the seven types of mathematics questions, mathematics problems that were chosen by the Clay Institute in 2000 as one of the most important problems in mathematics. of truly massive unanswered questions. And they chose 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 has so far been solved. And it was Poincaré's

conjecture

. So the Poincaré conjecture which was put forward by Poincaré and I forgot at the beginning of the last century, around 1900, was a particular conjecture like a guess and we were and he stated that there were certain relations between Geometry and Topology, so I can try to explain a little what the Poincaré conjecture is.

It's from an area of ​​mathematics called topology, which has to do with the type of shape and how things fit together in space. And these are essentially spheres, so a common idea in topology is the idea of ​​deforming things, so here I have a cube. Which is made of mass and in topology we can move things a little. It is sometimes called rubber sheet geometry, which is why I was able to grab this cube. And I could play with that. And I could round it like this using my hands and I can get a close enough sphere.

So that's the sphere, the theory is that you're allowed to crush things, stretch things and all that kind of stuff, 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 them. a hole So, because of this, the number of holes something has is very important, so this sphere has no holes. I guess you could also have something like this donut, which has exactly one hole in it? And you can't go from a sphere to a donut by warping it without closing this hole or cutting it here and straightening it and squashing that sausage into a sphere.

And you can't cut and no sticking is allowed, so the kind of things that are different from each other, spheres are different from Donuts, which are different again from a pretzel with two holes or something with three holes or whatever number of holes, this is the kind of mathematics that I studied in college for a long time and did some very difficult ones? Mathematics Poincaré was a French mathematician who put forward this conjecture and suggested that if you have an object that does not have holes, the first condition is that it does not have holes. The second condition is that it be a little small.

It's not, it doesn't work. always in any direction, so it's a finite thing. Maybe you could put it in a box and close the lid and then it's a sphere or at least it can be made into a sphere and two-dimensional while three-dimensional. In vivo, this seems like a pretty sensible statement, but Poincaré suggested that this would be valid in any number of dimensions. So you can take that this is a three-dimensional sphere. But you could have a two-dimensional sphere, it would be a circle, you can go there too. above and have a four-dimensional sphere that you would have to imagine, but it is a kind of sphere with an extra dimension, it has a fixed radius, this one has a fixed radius in three different directions, but the four-dimensional sphere has another direction in the one that still has this size and any number of dimensions above that.

And then you know it's counterproductive to try to imagine those things because it really damages your brain, but this conjecture exists as a mathematical statement, and he put this forward in the 1900s, but he didn't have a proof for it and the actual proof of this took For quite a while, so in the 1960s and 70s there was a lot of topology going on, people were developing new tools to do topology and they achieved the five-dimensional case and more, but there were still four-dimensional spheres in the what to think And this was still unsolved and a lot of people tried it and it turns out that it took different approaches, I guess, to solve it.

This particular thing, so in the 1980s there was some work done and a guy named William Thurston made a completely separate conjecture that was about four-dimensional things and shapes and it turned out to be very closely related to that. But that wasn't proven either and I think it took until 2003, at which point some mysterious bugger appeared on the Internet that had proof for the Thurston conjecture, which also proved the Poincaré conjecture for this unknown four-dimensional case and it was very exciting, everyone was friendly. of struggling to try and try to find whether this was a real proof or not because the Poincaré conjecture has been around for so long It was one of the most falsely proven results in the history of mathematics, as if it had more incorrect 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 good things about. I'm thinking about the good things about this file that instantly made his work spread to everyone, I guess in the old days. Maybe these things would be sent as preprints, which of course I remember. Those old days, well, but I don't know if I would have had the resources to spread it, it would have taken a lot longer for people to realize that this was out there, the mathematics community dove into this paper, reviewed it, and verified that In fact, It was proof of what he was saying to me.

And it was, but I remember when Ben Chow said this is something incredible. Now we are going to have a seminar and we are going to meet several hours a week, and we are going to study this. and each chapter. This was a seven or eight chapter document, we were each assigned a chapter, but it turned out that understanding a chapter required a lot of work. The mathematician who had written this document was called Grigori Perelman. He was from Russia and received a field medal. Which is like the math equivalent of a Nobel Prize. Of course, he also received the million dollar millennium prize and for various reasons, I guess he turned down both things, he didn't necessarily want them, the fact that it did make an absolutely huge media storm everyone was interested.

Suddenly And He could have had any job anywhere in the world after doing this But he didn't want to accept it, and it's a bit like, you know, Wearing hair is funny, and he lives with his mother and sent him to Petersburg or something. I never actually met him. A mathematician solving a difficult question was not news. But a mathematician who solved a difficult question and doesn't want a million dollars was suddenly big news for about a year afterward. I published these documents. I remember it was November of 2002 because I remember I was with a group of people who were really interested in RicCi Flow in San Diego and those documents came out.

And we instantly focused on trying to understand those those documents. It was very, very difficult for Pearlman the following year. Did you do a kind of world tour? I know some friends in Berlin, did he come to visit them? Near Berlin he came to MIT and gave some talks, and did a little tour. And if I wasn't busy teaching and didn't have time to do it, well, maybe I should have, but I didn't go to see him at any of these talks if we knew he was going to come back immediately, say after a year, and come back to Stay away from the hype, it would be fun to meet him.

I guess I've simplified it here and it's a little more complicated to specifically define what I mean by each of these things. But I hope that gives you a rough idea of ​​the type of math this is about, there are many more types of difficult Blackboards full of scribbled numbers. You know that there are different ways to do mathematics. There's a way you can approach a big problem like that and say, I'm going to focus on that. The danger of that is that you don't do it. I don't solve it and then? You know that's not so good or you can see a lot of different problems. "I like to look at a lot of different problems.

I'm not focusing on one in particular, but he would take that problem in fact from what I understand from friends who knew him," he announced at one point. I forget exactly when, many years before he published these articles. I'm going to use Ritchie Flow to try to solve the Poincaré conjecture, and he just without I don't know if he was teaching at the time, but he lived in St. Petersburg, and then he did. If he hasn't seen it yet, we have a video about Ricci flow and RicCi flow surgery, which is like the golden bullet that helped them solve the Poincaré conjecture.

If you haven't seen them yet, maybe go and take a look. I'll have links in all the usual places. We also have a video about another of the millennium problems, the Riemann hypothesis. So maybe you can go and see it. video that hasn't been solved yet So there's a million dollars there for you if you can figure it out