Nash Embedding Theorem - Numberphile

well there was a film a few years ago with the Russell Crowe starring as John Nash it was called A Beautiful Mind there's also a book a beautiful mind that it was based on by Sylvia Nasar I enjoyed the book more than the film that filmed a little bit fictionalized but it's a good good portrayal of his life story you may have heard of him because of his work in economics he won a Nobel Prize for economics and that was for game theory so he analyzed what are called non zero sum games and that was a big deal back in the 1950s when people were worried about nuclear conflict he used a topological theorem there Brau a fixed-point theorem to prove the existence of something called a Nash equilibrium in a game for a pure mathematician that was quite a simple application of those things but the really big deal was that he brought those mathematical ideas into economics but he's really well-known among mathematicians as well for for some serious pure math work that he did in a subject called differential geometry well the mathematician Mikhail Gromov said that he thought that Nash's work in differential geometry and on elliptic partial differential equations was orders of magnitude more important than his work in game theory but of course that's the point of view of a pure mathematician so I'd like to talk about something that's not so well known which is the National so it's about how can you realize abstract curved spaces as as subsets of euclidean space the space that we live in we can think first about the torus so I've got a bicycle energy up here this wasn't out of my bike it's actually in fresh one because the previous one bursts this is a doughnut shape if you like you can imagine that the surface is quite squidgy it's it's the same shape as as the surface of a doughnut so topologically it's equivalent but we can imagine that it's ER it's pretty stretchy it's got some distances already marked out on it so there's a grid marked out on it and we can imagine that you can already measure the distance between two points on the surface by tracing along this grid and seeing how many squares you go along so we can imagine a stretchy surface but it's got its own intrinsic idea of what the distance is if I twist it here you can see that the rectangles are getting at to be longer and thinner if I twist it the other way they get long and thin the other way so what Nash was interested in was given one of these surfaces with an idea of distance already marked on it can we find a way to put it in three-dimensional space in such a way that the distances agree with the distance traced around the surface in three-dimensional space so if I have a curved surface in three-dimensional space it's already equipped with a distance which is just the distance let's say between two points I could I can trace lots of different curves on the surface and I'll just take the shortest one of those and that gives me a notion of distance and that's called a Riemannian metric on the surface but you can also have an abstract surface which is equipped already with a Riemannian metric and then we want to know does it come from an embedding does it come from a way that you can put the surface into three-dimensional space just to talk about what embedding is this is an object that you might be familiar with it's an attempt at making an embedding of a of a Klein bottle so the Klein bottle is it is a surface it's a closed surface it's actually just got one side so if you imagine going around through the tunnel then you'd be on the inside of this tunnel and so you've got through to the inside of the bottle then so it's a rather unusual surface and it's got no boundary but there's a problem with this glass one which is forced on it by being in three-dimensional space which is that it intersects itself you can see this tube here is passing through a piece of the surface here and what we'd like actually is to realize the surface in let's say four dimensional space in such a way that there's no intersection with itself this is what's called an immersion so it's everywhere it's smooth nicely curved but it intersects itself so that's kind of like a floor that means it's not reversible that's a floor so this isn't really a client bottle we need to go into four dimensions to have a properly embedded client bottle so embedding means that any two different points on this abstract surface have to be placed at different point in in space it turns out that it's actually impossible to embed a Klein bottle in three-dimensional space oh I can't do it that's where the failure is that's right no matter how you try to fit it into three-dimensional space there are always going to be two different points on the klein bottle on this abstract surface which have to be at the same place so here we've got something else if we if we cut a bit out of the Klein bottle we end up with something called a Mobius band or maybe a strip and you've probably seen one of these before so I just took a long rectangle of paper and I gave it a half twist and I sellotaped the ends together this is getting a bit closer to what Nash proved because it's made of paper and paper already knows what the distance is between any two points when you curve paper it only likes to curve in one direction as always straight in the other direction so it's bent this way and it's straight this way and you can't bend it so that it looks like a piece of a sphere paper doesn't like to bend that way so so any any way that we embed this Mobius band made of paper into three-dimensional space it's always going to be isometric so isometric is is exactly the property that the distances on the paper are the ones that you would get from finding the shortest path along the surface in three-dimensional space whereas with the inner tube I could imagine if I have a justice sort of square grid going around it and going in meridians and longitudes around this around this torus then going around the inside I'm going to get a slightly shorter distance than if I go around the outside there's a natural metric on a torus called a flat torus where it's difficult to embed it in a smooth way into three-dimensional space but Nash proved that if you were allowed to embed it in a slightly irregular way then you could even embed the flat torus so that particular distance structure on the torus into three-dimensional space so that the flat torus is the one that I could get from a piece of paper if I started with a rectangle and I bent it right into a cylinder include it together and then tried to bend it around to the two ends of the cylinder were glued together and you just can't do that with paper paper doesn't like to be embedded in that way in space whereas I've got this torus made a rubber that's easy to do because it's bendy now now showed that actually if you if you fold well folding is not quite right so folding is not allowed you've got to have you've got to have a direction at every point in his embedding so with a rubber tube there's this valve sticking out of it at right angles from the surface you could imagine that on on this mobius strip that you could also have little sticks pointing out at right angles from the surface everywhere those are called normal vectors now his condition was that he wanted the normal vectors to vary continuously as you move around the surface so if I make a fold then the normal vector here is sticking out this way and then as you go around the corner suddenly it flips to being over here so that's not allowed you're not allowed a fold so mash was interested in embedding this flat torus made of paper so that everywhere you embed it the normal vector exists and it moves continuously and he showed that actually you can do that which is quite a counterintuitive result the catch is that the the normal vector moves continuously but not differentiable so if you know what that means so it's like it's a bit like a fractal embedding in fact it was it was any Riemannian manifold so it doesn't even have to be a surface it could be three dimensional or four dimensional you just have a manifold so it's a curved space if you like an abstract space which you imagine as being made of rubber and you're trying to embed it in Euclidean space but maybe 10 dimensional or 20 dimensional Euclidean space so we've got let's say a five dimensional if you can imagine five dimensional rubber we've got this five dimensional object let's say it's it's like this torus there's no boundary and it's just a finite bounded thing mathematicians call that compact he showed that if you have an M dimensional manifold so a Riemannian manifold that means one of these curved spaces with the distance already prescribed on it it can be embedded isometrically so that means that the distance is exactly the one that you get from the Ophidian space you preserve its grid that's right in now how many dimensions do we need we need M times 3 m plus 11 over 2 that's a lot of dimensions there's an extra condition about the about there the type of the embedding so how regular is it can you take the coordinate functions and differentiate them lots of times for this theorem with this many dimensions this embedding can be as smooth as you possibly could imagine so the technical thing for that is C infinity it means you can differentiate all the coordinate functions as many times as you like so for the tire we've got we've got M equals to attitude is a surface so it's got two dimensions at any point on on the tire you can go in two different directions at right angles and so if we plug 2 in here what are we going to get 3 times 2 plus 11 that's 17 times 2 divided by 2 so we need 17 dimensions 17 dimensional Euclidean space that's hard to imagine but Nash's theorem says that no matter what structure you have on that on the surface to start with with the distances as long as there are as long as there suitably smooth so suitably differentiable then you can embed it in 17 dimensional space in such a way that you recover the distance from the embedding but does it have to be that hard it doesn't necessarily have to be that high that's what he proved you could do for any any possible distance that you put on the surface so this is an upper bound on this on the number of dimensions that you need so in the case of a torus it may be that three dimensions is enough and it may not be that three dimensions is enough so in particular if you want to embed the flat torus so that's the one that you get from taking a rectangle of paper and gluing the opposite sides if you want to embed that even so that you can differentiate the coordinate functions twice so just a little bit smooth then you can't embed it in three-dimensional space and that's because there's always going to be a point somewhere on the extreme of the torus where where the surface is curving in the same way in both directions so like a little piece of the sphere and the curvature of the surface is something that the surface knows about so just just from knowing the distances on the surface between two points I can tell you what the curvature is there and so a point of of positive curvature so that means like a sphere that's not going to be the same as a flat torus well before he came along there was a question about whether surfaces that you get in as sub surfaces of high dimensional spaces are really just the same category of objects as as as these abstract surfaces with roumanian metrics on them and so he answered that but also the techniques that he used were very important so so his his proof was solving a system of differential equations called partial differential equations and he actually did a lot of important work in proving the existence of solutions and also the regularity of the solutions so can you differentiate the solutions he proved that in many physical situations where you have a partial differential equation which is called elliptic so that's a particular condition that comes up a lot in in physics for example then the solutions are actually analytic and that was one of Hilbert's problems if you know about Hilbert's list of problems at the beginning of the 20th century that was Hilbert's 19 it's problem of course his life story is interesting to all of us because of his of his mental illness and the facts the remarkable fact that he recovered from it to become an active mathematician again but really his mathematics is what's going to last I've got something I can use how about what about that we have the torus shape and for my next trick here's a poodle so there's our torus shape and that's what the shape you want to make so if you were playing the game of asteroids this is actually the universe that you're living on you're actually living on the surface of this universe

Channel: Numberphile2

Views: 371,309

Keywords: John Forbes Nash Jr. (Academic), Nash Embedding Theorem, Topology (Field Of Study), torus, mobius strip, klein bottle

Id: 5xQ4ePN79-M

Length: 13min 42sec (822 seconds)

Published: Sun May 31 2015