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