Computers Geometry And Einstein Jason Lotay

TITLE: Computers, Geometry and Einstein - Jason Lotay CHANNEL: Oxford Mathematics DATE: 2026-03-25 ---TRANSCRIPT--- [music] [applause] Hello everyone. So today in this talk I want to draw a link between three different topics. Computers which are an integral part of our everyday lives now geometry which is the study of shapes and that’s what I do dayto-day and the work of Albert Einstein. So, if you’ve looked at the news recently about news in maths and computers, you’ll probably seen quite a few developments, maybe a little bit of hype as well. And the first thing I want to talk about is the International Mathematical Olympiad or IMO. So, just a quick show of hands, who’s heard of the IMO? All right, most people, but not everyone. So the international mathematical Olympiad is seen as the premier competition for young mathematicians and achieving a gold medal is seen as a really fantastic achievement. And last year at the IMO, Open AI and Deep Mind both successfully created AI tools that won the gold medal at the IMO, which was certainly uh impressive. Another thing that’s been an interesting, you could say, well, that’s very nice, but of course, IMO problems, they’re like puzzles, right? that they’re like little puzzles. So, can it do real maths like the real research maths? And there’s been some interesting developments in that direction related to the Fields Medal. So, I’m not going to ask who’s heard of the Fields Medal because I hope you’ve all heard of the Fields Medal. So, this is the, you know, maths equivalent to the Nobel Prize in some sense. And here are three winners of the Fields Medal. Peter Schultzer, Marino Viovska and Terry Tao who each have and there are others too who each have advocated the use of computers in one way or another in research mathematics. So I want to take one of these examples. Peter Schultzer who’s the first one who works in arithmetic geometry and he posed a challenge to the world of computational and computing mathematics and specifically to the formalization of mathematics. So what formalization means is that you you send your mathematical proof into a computer and the computer says whether your proof is correct or not. Okay, that’s the idea. And he said, right, I want you to try to check that the proof of the main theorem on liquid vector spaces. I have no idea what that is, but it sounds cool. And say, you know, is it is it correct or not? And this was done. This was done. So now a computer has verified that this theorem is true. The proof is correct. And similarly, Marina Viosvsk’s work, so famous work on exceptional sphere packings in dimension 8 has also now been formally checked by a computer. So now we have perhaps so this was you know addressing the issue of trust in mathematics. Do we believe that maths is correct or not? And maybe if a computer checks it rather than just a human checks it maybe we believe it more. Maybe so. Terry Tao on the other hand has been thinking more about AI in mathematics and in particular one of the things where AI and mathematics have collided most recently has been related to the Millennium Prize problems. So this is a set of seven problems posed in 2000 by the Clay Mass Institute and if you solve one of these problems you win a million dollars. Only one of these problems has been solved so far. But there has been very exciting progress recently about the Navia Stokes problem. This is a problem in fluid mechanics. In particular, again, Google Deep Mind in collaboration with mathematicians has shown that AI can generate research, new mathematics, which is towards potentially resolving this problem. And so a million dollars may go to AI at some point. Now I should say that there has been progress on this problem without AI also still using computers though but using more numerical methods and also without computers at all. So it’s sort of a race to see who’s going to win first. So that’s some of the recent news on mass and computers. But actually mass and computers this goes back a long way. It’s really old news. In particular, you could think back to the really the beginnings of computation and algorithms and the work of Alan Turing in the 1940s. So here is Turing here with the bomb the the computer or computing device computational device which helped crack the Enigma code. And of course maths and cryptography and and computing is something that’s still very much on our mind with the developments in quantum computers. quantum technology. Again, can we ask about research maths and computers? And again, this goes back a long way. In fact, this year is the 50th anniversary of the fourcolor theorem. So, proved in 1976 by Apple and Harkin. And what does this theorem say? So, again, a quick show of hands, who’s heard of the fourcolor theorem? Basically everyone. Okay. Fantastic. So says if you’ve got a map and I want to color the map so that no two neighboring countries or regions have the same color, then I only need four colors. Okay? And the only proof that we know uses computers. It’s the only one we know. There is no purely human proof of the fourcolor theorem. But I should say there are proposals to try to prove the four color theorem without using computers. There’s an interesting proposal that uses mathematical gauge theory which is an idea that comes out of particle physics in fact originally and electromagnetism. So maybe that’s one future you could imagine where maybe we have a computer proof and then people keep striving to do it without computers afterwards. We never quite accept the computer only proof. It depends who you ask maybe about this whether this is proved or not or proved to their satisfaction if it’s just proved by a computer. Another interesting advance and the final one that I want to mention in terms of old news to do with mass computers involves the work of Mandelro in the 1980s. So this came out of the advance in computer graphics in particular by IBM which enabled you to produce beautiful pictures like this one of the Mandlerought set. So this actually had a genuine impact on mathematics because these sort of sets that come out of complex dynamics were studied much earlier like in 1910 or so by Julia and Fatu. But of course they couldn’t generate beautiful pictures like this. They just drew things by hand. It wasn’t quite nearly as impressive. And so that work sort of fell by the wayside somewhat. But when Mandelro and others were able to produce these fantastic pictures, it totally reinvigorated the subject. So it shouldn’t be underestimated just this very simple thing of computer graphics actually has had a genuine impact on mathematics. So now I want to draw a link between computers and geometry. Okay, that’s what I want to do. You’ve already seen a little bit of geometry here with the fractal geometry, but I want to tell you about the ways in which I like to use computers to study geometry. So I think there are three main ways. So the first one is related to what I just talked about which is visualization. So geometry as I said is the study of shapes right so that means you want to try to picture what the object is but sometimes it’s very hard to picture especially if you’re doing geometry in higher dimensions like how do you see these things so for example this is a very beautiful object that many many people have studied including myself it’s called the Clifford Taurus and this is a way to imagine this object which actually lives in four dimensions inside our three-dimensional place. This is a way to visualize what it looks like. There are many ways to visualize it, but this is one of the ways. And in general, thinking in it in four dimensions is kind of tricky. And so, here is another way to think about a very nice object that I like a lot in four dimensions, which is called the tesseract or hyper cube. So, it’s the four-dimensional version of the cube. It’s not just something in the Marvel movies. And what you’re seeing here is how the tesseract can rotate in four dimensions. So you see the sort of inside cube is rotating to the outside, right? That’s what’s happening. So this is actually a rotation in four dimensions that’s projected into three-dimensional space. And again, it shouldn’t be underestimated that, you know, seeing something like this really helps you to understand what’s happening. Of course, one of the simple things that you you just have to do if you’re a working mathematician, you have to calculate things. You can’t escape calculation. So, I’m afraid I’ll have to take away the tesseract, otherwise you’ll be hypnotized by it. So, you know, you have to just do long calculations, maybe not quite as long as this one. This isn’t an AI generated image. This is a genuine person, a physicist in Stanford writing out the equation. He’s partway through. He’s not even finished the equation yet. But imagine having to literally work with that equation every day and trying to simplify it. I mean, this would be ridiculous. I mean, you shouldn’t be doing this. But for a computer, this is simple. You just plug it in and off it goes. So, actually, this relates back to the IMO. So when the idea to use AI to compete at the IMO, it was thought that the problems that the AI would find most difficult would be the geometry problems because that’s sort of a visual thing. And computers, I mean, how would they understand this visual problem? What would they do? It turned out exactly the opposite was true. In fact, AI is extremely good at the geometry problems. It solves those the easiest. And the reason is that what it does is not what a human would do. It takes the picture, converts it all into coordinates, and then just solves an algebra problem. Right? It and that’s actually, of course, no human would do that in the IMO. But as a research mathematician, that is often what you do. You start with some shape, you convert it into some kind of algebra or differential equation or something, some kind of equation that you can solve, then you solve that one. And that is the kind of thing that computers can do and what I’ve used computers to do myself. As well as doing exact calculations like this, what computers are extremely good at is numerical calculations. They can get very good numerical approximations to things. So for example, suppose I wanted to compute the surface area of a very irregular shape like this one. like how where would you even start to do this? Like how would you even attempt to compute its surface area? But what we could do is as you can see in the picture, suppose I I I break up the surface into lots of triangles. And if I make the triangle small enough, then they’ll be almost flat, all of those triangles. And I can compute the area of a flat triangle easily. That’s something I know how to do. And I just add them up. And this idea of what’s called triangulating an object is actually a very important one in geometry and topology. So this isn’t just a numerical tool. This is actually a useful tool when we want to think about this shape. So for example, if you give me a shape and you just tell me how many corners are there in this shape, how many edges and how many triangular faces, I can tell you whether there’s a hole in that shape or not. So I’ll be able to tell the difference between that shape and the sphere just by knowing the number of corners, the number of edges, and the number of faces. Okay, so these are some ways in which computers can help do research geometry. So that’s the two of the things I promised you. The third one was the work of Albert Einstein. So where does he come into the story? And he comes in now with the study of gravity. So in 1916, Einstein published his work on general relativity. Here he is. And this gives a way to describe how gravity works. It updates Newtonian gravity substantially. So how does this connect to geometry? Well, one of the main ideas that Einstein had was that you should think of an object. If you want to think about the world around us, you want to describe gravity, then what you should do is you should think about spaceime. So this is a fourdimensional object which is the three dimensions of space and the one dimension of time. But you think of them together. You don’t think of them as three space and one time separately. you think of them as one four-dimensional object. Okay, so this isn’t actually a new idea. This is an idea that possibly goes back even to Galileo. But what Einstein did was really push this much further and thought about it in terms of the geometry of spacetime. And to do that, he needed some maths that had only been invented relatively recently. And that was the work of Reman who is here. So you might ask why has Reman got three dates? Seems a bit weird. What’s going on? So the story goes that Gaus asked Reman to write a lecture or prepare a lecture on geometry for his habilitatons. This is a bit like getting tenure. Okay. So this is what and he said okay. And so Reeman being Reman thought, well, you know, I won’t just write a lecture. I’ll just invent an entire new subject same time. So he invented what we now call Remanian geometry in this lecture in 1854. But no one seems to have paid much attention. Even though he just totally revolutionized maths, no one really paid attention. And so it took 14 years until after actually Reman had already died before they published the lecture. And that was done by Dedic who published it. And still nobody paid attention until 1900 around then when Gregorio Richi and Tulio Leviva started to develop what we now know as Remanion geometry. And so that’s only 16 years before Einstein’s work. And so they really in fact were developed almost in parallel. Remanian geometry and general relativity. And what Reeman did is really amazing. I mean he did many amazing things but one thing that he did he was the first one to even think about geometry in dimensions beyond three. He was the first even to think about geometry in higher dimensions no one had done. And in particular he defined curvature in higher dimensions. What does it mean for a higher dimensional object to be curved? He made sense of that. And so what Einstein’s equations say, so the equations that govern gravity, what they say is that you should think about gravity as being related to the curvature of spaceime. This four-dimensional object where space and time are combined. Okay, that’s the idea. Sounds good. What does it mean? I can say it, but what does it mean? So let’s let’s try to unpack that. What does it mean? So this is a well-known analogy, but I’ll say it anyway. So suppose that you had a trampoline. You put something very heavy in the middle of the trampoline. It’ll dip down, right? So if I walk into the garden, I see a a trampoline dip down, I know that something heavy should be there, right? Even if I can’t see it. And the same idea goes for gravity. If I put something heavy in spaceime, say the earth, it’ll bend spaceime around it. So gravity will then curve the spacetime. Okay, that’s the that’s the idea. But it can do things that are even more drastic. So it it curves spaceime. And now what it does is it actually it changes what is the shortest path between points. It’s no longer a straight line because I’m following the curve of spaceime. So, for example, if I take a beam of light and it’s just in vacuum, it will go at a constant speed, the speed of light of course, in a straight line. But if I put something heavy nearby, say a star or the sun, then actually the light will bend round because it’s following the curvature of spaceime. So we always think of light just traveling always straight, but actually it doesn’t. It will curve round. So if you put something extremely heavy in the way, the light could bend a lot. What’s something extremely heavy? A black hole. So here is an animation which shows a galaxy passing a black hole and you see that as it goes around the black hole the the light bends a lot right in fact you can see it actually bends around so even before the galaxy gets there you can see the light on the other side called gravitational lensing. So this is the idea. Okay. So this mo motivates us to study geometry related to Einstein’s equations and these are called Einstein metrics. So now I’m going to do a bit of maths. This is going to be the math slide. Okay. So what we’re going to do is we’re going to go back to Reman. So what Reman introduced in 1854 was something called a metric called G. And what this metric gives you is it says you can define something called curvature. So I’m going to call that K for curvature. Okay. Now what you can ask is well what does the curvature mean? Okay. So if the curvature is zero what should that mean? What should curvature zero mean? That should mean that you’re flat. You’ve got no curvature. You should be flat. Now suppose that your curvature is constant and it’s positive. What does that mean? It means that you’re a sphere basically. I’m lying slightly, but let’s not worry about that. So basically, you’re a sphere. Okay, that’s what it means to have positive constant curvature. So constant curvature is a little bit boring because you’re just going to get spheres and flat things. So can we do something more interesting? And this is where Richi, I mentioned him a couple of slides ago in 1900, introduced this idea of the average curvature. So not the whole curvature, but an average of the curvature. And I’m going to call that Rick fori. And what an Einstein metric is is to say that the average curvature is constant. So the mathematical equation is very simple to write down. It says that Rick is a multiple of G where lambda is a constant. So this is what an Einstein metric is. So why is this interesting? Okay, you can write it down but why is it interesting? It’s interesting because on the one hand if I have constant curvature then I have to be Einstein. So it’s more general than the constant curvature. Another thing that’s interesting is that you can view this equation as related to Einstein’s theory of relativity. It’s related to vacuum solutions to Einstein’s equations. So you say there’s no matter around, then that’s what an Einstein metric is describing. So for example, if you take a black hole again and you look at the exterior region of the black hole. So you look everything outside the black hole. Pretend there’s nothing else there, just a black hole, nothing else. Then you can describe what the curvature looks like outside the black hole because that’s a vacuum solution to Einstein’s equations. It’ll satisfy an equation like I wrote up there. Now, you may ask, what’s this lambda all about? What’s this constant all about? Has this got anything to do with physics? Or is it just a mathematical device? This lambda you might have heard of. It’s called the cosmological constant. So that’s what it is. It’s something that physicists are trying to measure. And there’s some evidence that maybe it’s a small positive constant. If it is, that’s one of the reasons why people propose the idea of dark energy is because of this measurement of the cosmological constant which showed that the universe was expanding at a certain rate. All right. It’s all well and good coming up with a definition, defining something. I can define Einstein metrics. I can say they do all these things. But are there any examples? Can we actually find any Einstein metrics? So are there any examples? Let’s go to the simplest case just two dimensions. Two dimensions simplest case. So suppose we take a sphere. Can we find an Einstein metric on the sphere? Yes, because we already know that if I have a constant curvature metric, that’s an Einstein metric. So I get one with lambda positive. How about if I want lambda to be zero? Can I find such a thing? Can I find it? Cosmological constant zero. Yes, I can because I can take the Taurus. So that’s like a ring donut shape or inner tube. If I take this then I can find an Einstein metric with zero cosmological constant. Okay, so we’ve got two out of three. Can I find the negative Einstein constant? That’s what I want to find next. Can I do that? Yes, I can take a double Turus like this. And on that thing, I can find a negative Einstein metric. So I can find them all. So I’m happy can find all the different types and I can see that I have to change my surface but that’s okay. I can find different different answers. So now I can tell you what is the key question when you want to study Einstein metrics. The key question and one of the questions that I think about a lot is a main part of my research which is to say when do Einstein metrics exist? So suppose that I give you some object some n-dimensional object let me call it M. Are there any Einstein metrics on that M? Can I find any at all? That’s the like the key question in the field and the answer in general is we have no idea. But suppose that you found one. Okay, suppose you know for some reason there’s at least one. How many are there? Again, no idea in general. Really no clue. And so this is really the key key questions that I now want to talk to you about. So there’s an interesting fact that I want to tell you before I get on to the main question. And that’s if if I look in low dimensions, if I look in dimensions less than or equal to three and I take an Einstein metric G, then if you remember a couple of slides ago, I said if you had constant curvature, then you were Einstein, right? I said that if you’re constant curvature, then you’re Einstein. In dimensions less than to three, if you’re Einstein, you have to have constant curvature. So that means that all those examples that I showed you of surfaces, in fact the the dimensions you find all have to have constant curvature. So they’re kind of boring. But that says that actually if you want to study Einstein’s equations, they’re only interesting for dimensions greater to four. Four dimensions was exactly the dimension that we were most interested in. Right? Maybe it’s a coincidence, maybe not, but it certainly is why it’s mathematically interesting that exactly the dimension where Einstein metrics become interesting is the one that we care about the most in some sense. So, as I said, in general, we have no idea how to answer these yellow and red questions. The, you know, if there are any Einstein metrics on a given space, and if so, how many? So I’m going to try to make the question simpler by telling you what space I want to think about. And the space I want to think about is the sphere. Okay, but the n-dimensional sphere. So that’s what I want to talk about now. So suppose I’ve got an n-dimensional sphere. Okay, SN. And the question that I want to ask is are there any known Einstein metrics on the sphere which have positive cosmological constant but which are not the constant curvature one because we know we have the constant curvature one. So that’s I want to remove that from my list. I want to say are there any more? That’s my question. Okay. So now there’s going to be some audience participation. I’m going to ask you some questions and we’re going to see how we go. So first of all, so the it’s going to be a yes no question. So you got 50-50 chance of being right. Okay. So we’re going to start with dimension two. So I’m going to tell you, you know, in dimension two there can’t be any because I already said in dimensions two and three there can’t be any. Four is the first interesting one, right? So you know the question the question is if I have a fourdimensional sphere do we know of any Einstein metrics on that sphere positive cosmological constant which aren’t the constant curvature one okay so we’re going to have a vote so hands up who thinks the answer is yes uh hands up to people who think it’s no much fewer Fewer. Okay. Much fewer. Now you can be smug. The few people. [clears throat] You got the answer right. We don’t know of any. I’m not claiming that there aren’t any. So, let me just be clear. I’m not claiming that there aren’t any. I’m just claiming we don’t know of any. All right. There could be some. I just don’t know them. And it’s not just me being ignorant. Nobody knows. Five. All right. Let’s go. Five. Who’s willing to stick with their yes? Who’s Who’s the yes people? Oh, much fewer this time. Oh, there’s a few more people feeling a bit more confident. Okay. Yes. Now, how about no? Oh, it’s actually pretty split. It’s pretty split. The answer is yes. The answer is yes. Okay. So, there are some but we have but now remember I had that other question. How many? How many? So, we know there are none in these cases. So, now I’m going to make it three options. Okay, you’ve got three options. There’s one between two and 10. More than 10. That’s going to be the options. One. Two to 10. More than 10. So, who thinks there’s only one? Okay. Okay. A few between two and 10 are more popular and then more than 10. Oh, that’s the most popular. Yeah, you can be smug. In fact, it’s infinite. It’s a bit more than 10. It’s infinite. So, that was proved by um Kristoff Burm in 1998. So I mean this is this took a long time from the definition of Einstein metrics you know 1900 or so you know 1916 and to there it takes a long time and in fact when he did this he showed it’s also true all the way up to dimension nine so all the way five up to nine you get infinitely many 10 doesn’t say Who thinks yes? No one’s confident. No. Way more. Way more. It’s Yes. Okay. But how many? So you have a chance of redemption. Those of you got wrong at this one. So So who thinks it’s Remember it’s one two to 10. More than 10. So who thinks it’s one? A lot. Yeah. Between two and 10. Not so many. More than 10. Yeah. Not so many. So, everyone thinks it’s one. Two to 10 people are smug. It’s three. But this is way more recent. In fact, it was just published last year by Nin House and Wink. So, we’re we’re it’s a really hard problem. So, I So, at 11, the answer turns out to be yes. And it’s infinite. So this was also only it the most recent result for this was Lu Sano and Tasin again 2024. It shows it’s really infinite. Like it’s infinitely infinite. It’s really loads. There’s loads of them. All right. I’m just going to do one more with you. 12. Okay. So let’s let’s go yes or no one more time. This is the last one. So, who thinks yes for 12? Okay. Okay. No. Oh, it’s it’s pretty split. It’s pretty split. The answer is yes. How many do we think? I mean, it’s not much of a pattern. How many do we think? So, who thinks it’s one? Okay. Who thinks it’s between two and 10? more than 10. Yeah, one was the most popular. And you’re right, it’s one. And that was just published this year. And this is the one that I want to highlight. So this is where we draw the threads together because this is what uses computers. So this is the only way we know how to produce an Einstein metric, an interesting Einstein metric on a sphere in 12 dimensions, is to use computers. So that’s what I want to explain. Now you can ask what happens. So actually in 13 the answer is again yes and infinite. And that’s true for all the odd dimensions in fact. So for all the odd dimensions going on the same work shows that it’s it’s infinite for all the odd dimensions. But for 14 we have no idea. But the answer is no for 14. So if you want a challenge when you go home try to find an Einstein metric on a 14dimensional sphere. There might be one lying around that we haven’t found yet. Okay, so how can computers help us to solve a problem in 12dimensional spheres? I mean it seems ridiculous. How on earth is this going to work? So the idea is the following. So I’m going to tell you the strategy. The strategy goes like this. What I do is assume that I have lots and lots of symmetry. So symmetry just means if I you know rotate something around reflect it around it looks the same. So for example if I take a dandelion clock like this I rotate it around has lots of symmetry right it looks the same from many many different directions. Okay. Now if I impo so the if I I want to impose lots and lots of symmetry I want to look for an not just any Einstein metric but a very symmetric one. That’s what I want to find. So Einstein’s equations is really really difficult. Okay, very very hard to solve. But if I impose symmetry then if I impose enough these Einstein equations can be rewritten as what are called ordinary differential equations and these are very nice. So you have seen or maybe you’ve you know a little bit about ordinary differential equations perhaps but if you haven’t then what you can do is you can use them say in Newtonian gravity you can find an ordinary differential equation which tells you that the planetary orbits have to be ellipses. Okay so it’s the kind of thing we can literally solve it. Okay can actually solve. Now, it probably won’t surprise you that these ordinary differential equations you find in this case, we don’t know how to solve. They’re really hard. Okay, 12dimensional thing. It’s really tricky. Don’t know how to solve it. But what a computer can do is something that I mentioned a long time ago is it’s very good at doing numerical approximation. Right? It’s one of the things I said. It’s very good at numerically approximating things. And so what you can do is you can plug the differential equations, the ordinary differential equations into a computer, maybe a bit like this one, maybe slightly more modern maybe, but uh this is an original Apple computer that you’ve got here. But anyway, other computers are available. Anyway, the the computer will produce for you an approximate solution. So we call it H, okay? Because H is after G. So I I have h the approximate solution which a numerical solution and what the computer can do is it can get me so close. I know so much about this approximate solution. I know it so well that I can find an Einstein metric G near that approximate solution. That’s the strategy. That’s the strategy. Now, most of the time this strategy fails miserably, but it happens to work in dimension 12. It manages to work. And so, you’ve got yourself a very happy Einstein. Okay. So, that’s where we are in terms of studying Einstein metrics on spheres and studying Einstein metrics in general. So what does the future hold? What’s the future in in this subject? Well, as I said, you know, Einstein metrics are incredibly challenging. You know, it’s the it’s a beautiful theory. It’s very exciting. It has these connections to gravity. We’d like to know everything about it, but it’s just really hard. These problems are really hard. We just don’t know how to solve them. But you know computers are actually enabling us to solve many more problems. I know that you know in certain parts of mathematics and mathematicians are a bit resistant to using computers. But I’ve I hope I’ve convinced you that actually computers have been used in math for a very long time. It’s not a brand new thing that we suddenly use computers to solve math problems. And I think you know if we embrace the use of computers we can actually make a lot of progress. For example, one of my PhD students, Kushi Wang, has used exactly the strategy that I gave you on the previous slide to produce new Einstein metrics in four dimensions. That’s the one we really like, four dimensions, but with the negative cosmological constant. So, this is closer to the study of gravity because, of course, we’re in four dimensions. So what’s my final thought on this area? Well, my final thought is that actually this is a very exciting time. I think that as I said, if we if we use computers in the right way, we actually have a brand new toolbox that we have that can help us to solve many of these really hard open problems not just in Einstein metrics, but in geometry more general and maths as well and beyond into some of the most uh difficult and exciting problems that are out there. Thank you very much for your attention. [applause]