So today I want to talk about John Nash and John Nash is slightly unusual as a mathematician because he's a mathematician that is better known to the public in general ; mathematicians don't tend to be known in the public eye. He was the subject of a film called A Beautiful Mind, the film had Russell Crowe in it starring as John Nash. There's also a book a beautiful mind that it was based on by Sylvia Nasar. I enjoyed the book more than the film. The film is a little bit fictionalized. I have to confess I haven't seen the film yet, I know I'm such a big film fan I'm such a massive film nerd. I imagine it's absolutely accurate in all respects because that's how Hollywood works. 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 he's known as like a father of game theory one of the pioneers of game theory. 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 some serious pure math work that he did in a subject called "Differential geometry". Mathematicians do think that actually his better work is his work in geometry and partial differential equations and so it was that work which was then finally recognized when he got the Abel Prize in 2015. 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 want to talk about why he won the Abel prize for his work in geometry. So if we cut off a bit of square brown paper - and this is something that is possibly well-known to some of you - but this square of brown paper, another way of thinking about this square, if we add some rules, it's called a flat torus. What are the rules ? The rules are : if I'm traveling along this paper here and I go off the right-hand side like that I then reappear on the left-hand side. If instead I was traveling up the paper here and go off the top, then I will reappear at the bottom of the square piece of paper (Brady) - It's like a game of asteroids. It's exactly like that, it's like a game of asteroids. The old computer games used to get, particularly the game of asteroids, where you'd go off the edge of the universe and come out the other side. So this is a flat torus it's called. Now there's something you can do with this you can embed this - they call this - you can embed this in three dimensions. This is two dimensions, this is a two-dimensional flat surface but we can embed it in three dimensions by doing this. So if I go off the right hand side and reappear on the left then the right and the left is the same - we're calling that the same - so let's glue them together and we make a cylinder. Can you see me there ? So we make a tube. In the same way if I go off the top of this torus I reappear at the bottom so the top and the bottom are the same we can join them together which is going to be difficult to do with this piece of paper. If we join them together they would make a doughnut shape like a classic Homer Simpson style doughnut. It is going to be difficult to do with this piece of paper to show you then I've got something I can use. 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 play in the game of asteroids this is actually the universe that you're living on you're actually living on the surface of this universe. If in my universe I travel let's say from top to bottom like that or I compare that with traveling from left to right like that then if I wrap this up... That's not quite perfect there - so if I wrap this up this is me going round my cylinder universe and if I can join the ends of this tube I would make my torus shape. And on the actual torus itself the green line would be me traveling around like this through the hole and then back out here joining up like that the red line however would be traveling around this universe like this. The squeaks are extra. So what I've done is I've turned this flat two-dimensional shape into sit in three dimensions. So this is occupying a three dimensional space. There is one problem with this. You see in my square torus this red line and this green line they're the same length, right, because it's square they're the same length. When I've stretched it into this shape the green line and the red line are no longer the same length so there's the green line going around you see this red line is much longer now so I had to stretch it to stretch around the tube. Is there a way to embed a square torus like that into our 3-dimensional world - maybe it will be a different shape, it might not look like this but can we do it in such a way so that we preserve distances so these lengths don't get stretched and they all get the same distances ? That's the question and it looked like you couldn't do that. And why can't you do that ? Because this is flat and the curvature - let's talk about curvature - it's a flat object, it has zero curvature. Something like this is a curved object, it has "positive curvature" we call it. So it looks like it's impossible, like we can't embed something because it would have a positive curvature and we want it to keep its flat curvature. Problem. John Nash found a way to do it. Maybe he's slightly breaking the rules, but he found a way to do it. And the method he found was, he was able to embed the object and deform the object in such a way that some points curvature had no meaning. Here's an example of the sort of thing I mean. So imagine you're traveling along the motorway at 100 miles per hour You deaccelerate and then you continue your journey at 30 miles per hour. So, that might be something that could happen. So this is speed, this is your speed against time. Imagine if I drew that graph in a slightly different way, very similar graph but I'm going to draw it slightly differently. I still want the speed up here, This is me traveling at 100 miles per hour and the end result is I'm going to be ending on 30 miles per hour and then I'm going to join up these. So they're very similar graphs. There is a difference between them. If we look at our acceleration, here there's zero acceleration you're travelling at a constant speed 100 miles per hour there's zero acceleration or the gradient of the graph is zero. Here, the acceleration might be let's call it minus 10 and then here it's it's zero again You're at a constant speed. Now with that graph at this point here and at this point here acceleration isn't defined, acceleration has no meaning here. What you've done - you've gone from zero acceleration suddenly, at an instant to minus ten and then suddenly again to zero And at that point here and at that point there there is no meaning to what the acceleration is. You've gone from 0 to -10 to 0. Now speed is defined, we know what the speed is here that's no problem, the speed is 100 miles per hour there and 30 miles per hour there. No problem with the speed but the acceleration isn't defined. With this curve - very similar that's not a problem here : acceleration is defined for every point on this curve. (Brady) - This one's the real world one isn't it ? - Yeah exactly that's real world , I agree And so this is maybe... you're taking some more abstract look at what you're doing. It works the same in geometry : if you have something like this then curvature is defined everywhere here. If we instead made that sort of shape like a hook, if you made that shape with a straight edge followed by a semicircle, like that, then here the curvature is flat flat flat flat suddenly here it's positive and at this point where you join on, curvature has no meaning There's no definition for curvature there. What Nash was able to do was : he was able to find a way to embed your square flat torus into the 3d world and there are some points like that where the curvature isn't defined. You can take your flat torus and then you can embed it in the 3d world and you embed it in such a way so maybe something like that you can embed it in such a way that the distances are a bit short so that your map, your deformation, your distances have become a bit shortened. All the distances have become a bit shortened. Then we're going to lengthen a little bit, we're going to hit this object with waves this is going to ripple across the surface of this object, waves, and those waves will ripple across - they're called corrugations - and they will lengthen all those distances slightly. And you hit it with another set of waves, those waves are a bit smaller, so smaller ripples. But then we're getting closer to the lengths again we're getting a bit closer to what the correct lengths are. And then you hit them with another set of waves and we get a little bit closer again to what the correct distances should be and you keep doing that and then you get an object which is full of these corrugations but all the lengths are the correct lengths as they were in the original flat torus. (Brady) - Are you saying by making the surface bumpy this green line has to travel further closer to what the red line is traveling ? - Yes a shortened distance like this green line would get lengthened so it becomes the same as this red line around the outside of the torus so they become the same length again like they where in the original flat torus, like they should be. What Nash was doing, he was studying this kind of thing and he was actually doing it more abstractly more general terms and he was do it in higher dimensions so much more abstract mathematics but this would be an application of what Nash was doing, this work in geometry. When he was doing this work, his solutions involved these things called partial differential equations. That's about the rate of change of things : the rate of change of curvature the rate of change of speed... Using these equations... His methods for solving these equations were so innovative that not only are they useful in geometry they turned out to be useful in the topic of partial differential equations itself as its own subject. That is why Nash got the Abel Prize for his work in geometry and partial differential equations. That's right of course his life story is interesting to all of us because of his of his mental illness and the fact 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 thought when he was going into the concept of discontinuity that the 3D shape was just going to be the revolution of an ellipse where the minor axis went to zero causing two sharp edges on the top and bottom of the shape.
I have never studied this particular work, but watching the video it smells like Fourier series.