#472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI

26:40I mean, that's a key qualitative feature that distinguishes some equations for being nice and predictable and like planetary motion. I mean, there's certain equations that you can predict for millions of years or thousands at least. Not really a problem. But there's a reason why we can't predict the weather past two weeks into the future because it's a supercritical equation. Lots of really strange things are going on at very fine scales. So whenever there is some huge source of nonlinearity that can create a huge problem for predicting what's going to happen. Yeah. And if nonlinearity is somehow more and more featured and interesting at small scales.

27:18I mean, there's many equations that are nonlinear, but in many equations, you can approximate things by the bulk. So for example, planetary motion, if you wanted to understand the orbit of the moon or Mars or something, you don't really need the microstructure of the seismology of the moon or exactly how the mass is distributed. You can almost approximate these planets by point masses. Just the aggregate behavior is important. But if you want to model a fluid like the weather, you can't just say in Los Angeles, the temperature is this, the wind speed is this. For supercritical equations, the fine -scan formation is really important.

27:54If we can just linger on the Navier -Stokes equations a little bit. So you've suggested maybe you can describe it that one of the ways to solve it or to negatively resolve it would be to sort of to construct a liquid, a kind of liquid computer. Right. And then show that the halting problem from competition theory has consequences for fluid dynamics. So show it in that way. Can you describe this? Right. Yeah. So this came out of this work of constructing this average equation that blew up. So as part of how I had to do this, so this is the naive way to do it. You just keep pushing every time you get energy at one scale, you push it immediately to the next scale as fast as possible.

28:41This is the naive way to force pull up. In terms of in five and high dimensions, this works. But in three dimensions, there was this funny phenomenon that I discovered that if you keep, if you change loads of physics, you just always keep trying to push the energy into small, small scales. What happens is that the energy starts getting spread out into many scales at once. So you have energy at one scale, you're pushing it into the next scale and then as soon as it enters that scale, you also push it to the next scale, but there's still some energy left over from the previous scale. You're trying to do everything at once.

29:17And this spreads out the energy too much. And then it turns out that that it makes it vulnerable for viscosity to come in and actually just damp out everything. So so it turns out this this directive or just doesn't actually work. There was a separate paper by some other authors that I actually showed this in three dimensions. So what I needed was to program at the lathe, so kind of like air locks. So I needed an equation which would start with a fluid doing something at one scale. It would push this energy into the next scale, but it would stay there until all the energy from the larger scale got transferred.

29:54And only after you pushed all the energy in, then you sort of open the next gate and then you push that in as well. So by doing that, the energy enters forward scale by scale in such a way that it's always localized at one scale at a time. And then it can resist the effect of viscosity because it's not dispersed. So in order to make that happen, yeah, I had to construct a rather complicated nonlinearity. And it was basically like, you know, like it was constructing like an other kind of circuit. So I actually thanked my wife for this because she was trained as an electro engineer. And you know, she talked about, you know, he had to design circuits and so forth.

30:34And you know, if you want to circuit, it does a certain thing like maybe I didn't have a light that flashes on and then turns off and then on and off. You can build it from from more primitive components, you know, capacitors and resistors and so forth. And you have to board a diagram. And these diagrams, you can sort of follow up your eyeballs and say, Oh, yeah, the current will build up here and it will stop and then it will do that. So I knew how to build the analog of basic electronic components. You're like, resistors and capacitors and so forth. And I would stack them together and start in such a way that I'd create something that would open one gate and then there would be a clock that would and then what's the clock hits us in threshold, it would close it.

31:10They're kind of a root go bird type machine, but described mathematically. And this ended up working. So what I realized is that if you could pull the same thing off for the actual equations. So if the equations of water support a computation. So like if you can imagine kind of a steam punk, but it's really water punk type of thing where, you know, so modern computers are electronic, you know, they're powered by electrons passing through very tiny wires and interacting with other electrons and so forth. But instead of electrons, you can imagine these pulses of water moving in a certain velocity.

31:44And maybe it's the two different configurations corresponding to a bit being up or down. Probably if you had two of these moving bodies of water collide, they would come out with some new configuration, which is which would be something like an AND gate or OR gate. You know, that the output would depend on a very predictable way on the inputs. And like you could chain these together and maybe create a Turing machine and then you have computers which are made completely out of water. And if you have computers, then maybe you could do robotics. So I, you know, hydraulics and so forth. And so you could create some machine, which is basically a fluid analog was called a VONOMIN machine.

32:25So VONOMIN proposed, if you want to colonize Mars, the sheer cost of transporting people machines, Mars is just ridiculous. But if you could transport one machine to Mars, and this machine had the ability to mine the planet, create some more materials, to smell them, and build more copies of the same machine, then you could colonize the whole planet all the time. So if you could build a fluid machine, which, yeah, so it's a fluid robot, okay. And what they would do, it's purpose in life, it's programmed so that it would create a smaller version of itself in some sort of cold state, it wouldn't start just yet.

33:05Once it's ready, the big robot configuration of it would transport all the energy into the smaller configuration and then power down, okay. And then like clean it up. And then what's left is this newest set, which would then turn on and do the same thing, but smaller and faster. And then the equation has a certain scaling symmetry. Once you do that, it can just keep iterating. So this, in principle, would create a blur for the actual Navier Stokes. And this is what I managed to accomplish for this average Navier Stokes. So it provided this sort of roadmap to solve the problem. Now this is a pipe dream because there are so many things that are missing for this to actually be a reality.

33:40So I can't create these basic logic gates. I don't have these in these special conductors of water. I mean, there's candidates that think would vortex rings that might possibly work, but also, you know, analog computing is really nasty. It can be digital computing. I mean, because there's always errors, you have to do a lot of error correction along the way. I don't know how to completely power down the big machine, so it doesn't interfere with the writing of the smaller machine. But everything in principle can happen, like it doesn't contradict any of the laws of physics. So it's sort of evidence that this thing is possible.

34:19There are other groups who are now pursuing ways to make Navier Stokes blow up, which are no one here as ridiculously complicated as this. They actually are pursuing much closer to the direct self -similar model, which can... It doesn't quite work as this, but there could be some simpler scheme than what I just described to make this work. There is a real leap of genius here to go from Navier Stokes to this touring machine. So it goes from what the self -similar blob scenario that you're trying to get the smaller, smaller blob to now having a liquid touring machine gets smaller, smaller, smaller, and somehow seeing how that could be used to say something about a blow -up.

35:06I mean, that's a big leap. So there's precedent. I mean, so the thing about mathematics is that it's really good at spotting connections between what you think of what you might think of as completely different problems. But if the mathematical form is the same, you can draw a connection. So there's a lot of previously or more of course, cellular automata. The most famous of which is Conway's Game of Life. This is infinite to speak grid, and any given time the grid is occupied by a cell or is empty. And there's a very simple rule that tells you how these cells evolve. So sometimes cells live, and sometimes they die.

35:43And when I was a student, it was a very public screen saber to actually just have these animations going on. And they look very chaotic. In fact, they look a little bit like turbulent flow sometimes. But at some point, people discovered more and more interesting structures within this Game of Life. So for example, they discovered a single glider. So a glider is a very tiny configuration of like four or five cells, which evolves and it just moves at a certain direction. And that's like these vortex rings. So this is an analogy. The Game of Life is kind of a discrete equation, and the fluid Navi -Sokes is a continuous equation.

36:16But mathematically, they have some similar features. And so all the time, people discovered more and more interesting things that you could build within the Game of Life. The Game of Life is a very simple system. It only has like three or four rules to do it, but you can design all kinds of interesting configurations inside it. There's some called a glider gun that does nothing of spit out gliders one at a time. And then after a lot of effort, people managed to create and gates and wall gates for gliders. Like this is massive ridiculous structure, which if you have a stream of gliders coming in here and a stream of gliders coming in here, then you may produce a stream of gliders coming out.

36:58If maybe if both of the streams have gliders, then there we are and output stream. But if only one of them does, then nothing comes out. So they could build something like that. And once you could build and these basic gates, then just from software engineering, you can build almost anything. You can build a touring machine. I mean, it's a kind of enormous steam pump type things. They look ridiculous. But then people also generated self -replicating objects in the Game of Life, a massive machine, a bono machine, which over a lot of huge period of time and they were always little glider guns inside doing these very steam pump calculations.

37:36It would create another version of itself, which could replicate. It's so incredible. A lot of this was like community crowdsourced by amateur mathematicians, actually. So I knew about that work. And so that is part of what inspired me to propose the same thing whenever you're stoked. Now, if you just a much, as I said, analog is much worse in digital. Like it's going to be, you can't just directly take the constructions from the Game of Life, plump them in. But again, it shows as possible. You know, there's a kind of emergence that happens with these cellular automata. Local rules, maybe similar to fluids, I don't know.

38:16But local rules operating at scale can create these incredibly complex dynamic structures. Do you think any of that is amenable to mathematical analysis? Do we have the tools to say something profound about that? The thing is, you can get this emerging very complicated structures, but only with very carefully prepared initial conditions. So these glider guns and gates and sort of machines, if you just plant randomly, some cells, and you're not seeing any of these. And that's the analogous situation of Navier Stokes again. With typical initial conditions, you would not have any of this weird computation going on.

38:58But basically through engineering, especially designing things in a very special way, you can pick clever constructions. And one of the first possible to prove the negative of like, basically prove that only through engineering can you ever create something interesting. This is, if we're currently challenging mathematics that I call the dichotomy between structure and randomness. That most objects that you can generate in mathematics are random. They look like random. Like the ditches are pie. Well, we believe there's a good example. But there's a very small number of things that have patterns.

39:32But now, you can prove something as a pattern by just constructing something as a simple pattern and you have a proof that it does someone like repeat itself every so often. You can do that. And you can prove that, for example, you can prove that most sequences of digits have no pattern. So like if you just pick digits randomly, there's some of the large numbers that tells you you're going to get as many ones as twos in the long run. But we have a lot fewer tools to give you a specific pattern like the ditches are pie. How can I show that this doesn't have some weird pattern to it? Some other work that I have spent a lot of time on is to prove or construct your theorems or inverse theorems that give tests for when something is very structured.

40:15So some functions are what's going to add to it. Like if you have a function that makes the natural numbers, the natural numbers. So maybe you know two maps to four, three maps to six and so forth. Some functions of course go additive, which means that if you add if you add two inputs together, the output gets added as well. For example, a multiply by constant. If you multiply a number by 10, if you multiply a plus b by 10, that's the same as multiplying a by 10 and b by 10 and then adding them together. So some functions are additive. Some of them are kind of additive but not completely additive.

40:47So for example, if I take a number n, I multiply by the square root of two and I take the integer part of that. So 10 by square root of two is like 14 points something. So 10, up to 14, 20, up to 28. So in that case, add it to these two then. So 10 plus 10 is 20 and 14 plus 20 is 28. But because of this rounding, sometimes there's round off errors and sometimes when you add a plus b, this function doesn't quite give you the sum of the two individual outputs, but the sum plus minus one. So it's almost additive, but not quite additive. So there's a lot of useful results in mathematics and upwork to a lot of things like this, to the effect that if a function has an exhibit some structure like this, then it's basically, there's a reason for why it's true and the reason is because there's some other nearby function, which is actually completely structured, which is explaining this sort of partial pattern that you have.

41:43And so if you have these little inverse theorems, it creates this sort of dichotomy that either the objects that you study are either have no structure at all, or they are somehow related to something like the structure. And in either way, in either case, you can make progress. A good example of this is that there's this old theorem in mathematics called semi -radi's theorem, proven in the 1970s. It concerns trying to find a certain type of pattern in a set of numbers that the patterns have make progression, things like 35 and 7 or 10, 15 and 20. And some already on trees, some already proved that any set of numbers that are sufficiently big, what's called positive density, has ethnic progressions in it of any length you wish.

42:27So for example, the odd numbers have a set of density one -half, and they contain ethnic progressions of any length. So in that case, it's obvious because the odd numbers are really structured. I can just take 11, 13, 15, 17, I can easily find ethnic progressions in that set. But the memories of them also applies to random sets. If I take this set of odd numbers and I flip a coin for each number, and I only keep the numbers for which I got a heads. So I just flip coins, I just randomly take out half the numbers, I keep one half. So that's the set that has no patterns at but just from random fluctuations, you will still get a lot of ethnic progressions in that set.

43:10Can you prove that there's arithmetic progressions of arbitrary length within a random... Yes, I mean, we have the infinite monkey theorem. Usually, methodicians give boring names to theorists, but occasionally they give colorful names. The popular version of the infinite monkey theorem is that if you have an infinite number of monkeys in a room with each of a typewriter, they type out text randomly. Almost surely, one of them is going to generate the entire school of hamlet or any other finite sum of text. It will just take some time, quite a lot of time actually, but if you have an infinite number, then it happens.

43:43So basically, the theme says that if you take an infinite string of digits or whatever, eventually any finite pattern you wish or you merge, it may take a long time, but it will eventually happen. In particular, ethnic progressions of any length will eventually happen like a pergenial, an extremely long random sequence for this to happen. I suppose that's intuitive. It's just infinity. Yeah, infinity absorbs a lot of sense. Yeah, how are we humans supposed to deal with infinity? Well, you can think of infinity as as an abstraction of a finite number for which you do not have a bound for. That, you know, I mean, so nothing in real life is truly infinite, but you know, you can you can ask these old questions like, what about how much money is I wanted?

44:33You know, what if I could go as fast as I wanted? And a way in which mathematicians formalize that is mathematics has found a formalism to idealize instead of something being extremely large, extremely small to actually be exactly infinite or zero. And often the mathematics becomes a lot cleaner when you do that. I mean, in physics, we joke about assuming spherical cows. We all have problems, I've got all kinds of real artifacts, but you can idealize certain things to infinity, certain things to zero. And the mathematics becomes a lot simpler to work with it. I wonder how often using infinity forces us to deviate from the physics of reality.

45:16Yeah, so there's a lot of pitfalls. So, you know, we spend a lot of time, you know, undergraduate math classes, teaching analysis, and analysis is often about how to take limits and and whether, you know, so for example, A plus B is always B plus A. So when you have a finite number of terms, you add them, you can swap them and there's no problem. But when you have an infinite number of terms, they're these sort of show games you can play where you can have a series which converges to one value, but you rearrange it and suddenly converges to another value. And so you can make mistakes. You have to know what you're doing when you allow infinity.

45:49You have to introduce these epsilons and deltas and there's a certain type of wave of reasoning that helps you avoid mistakes. In more recent years, people have started taking results that are true in infinite limits and also are finalizing them. So you know that's something true eventually, but you don't know when. Now give me a rate. So such a thing, if I don't have an infinite number of monkeys, but a large finite number of monkeys, how long do you have to wait for him to come out? And that's a more quantitative question. And this is something that you can attack by purely finite methods and you can use your finite intuition.

46:29And in this case, it turns out to be exponential in the length of the text that you're trying to generate. So, and so this is why you never see the monkeys create hamlet. You can maybe see them create a fuller word, but nothing that big. And so I personally find once you find it high, say infinite statement, it's just a much more intuitive and it's no longer so weird. So even if you're working with infinity, it's good to find it out so that you can have some intuition. Yeah, the downside is that the finite times proves are just much, much messier. And so the infinite ones are found first, usually, decades earlier.

47:05And then later on, people find it high. So since we mentioned a lot of math and a lot of physics, what is the difference between mathematics and physics as disciplines, as ways of understanding of seeing the world? Maybe we can throw an engineering in there. You mentioned your wife as an engineer and give it new perspective on circuits. So this different way of looking in the world, given that you've done mathematical physics, you've worn all the hats. Right. So I think science in general is interaction between three things. There's the real world. There's what we observe over the real world, observations.

47:39And then our mental models as to how we think the world works. So we can't directly access reality. Okay. All we have are the observations, which are incomplete and they have errors. And there are many, many cases where we would want to know, for example, what is the weather like tomorrow? Have we had the observation? And we'd like to predict. And then we have these simplified models, sometimes making unrealistic assumptions, you know, spherical cow type things. Those are the mathematical models. Mathematics is concerned with the models. Science collects the observations and it proposes the models that might explain these observations.

48:20What mathematics does it, we stay within the model and ask what are the consequences of that model? What observations, what predictions will the model make of the future observations or past observations to test it fit, observe data. So there's definitely a symbiosis. I guess mathematics is unusual among other disciplines is that we start from hypotheses, like the axioms of a model and ask what conclusions come out from that model. In almost any other discipline, you start with conclusions. I want to do this. I want to board a bridge. I want to make money. I want to do this. Okay. Then you find the path to get there.

49:04There's a lot less sort of speculation about it. Suppose I did this. What would happen? Planning and modeling. Specular fiction maybe is one other place. But that's about it actually. Most of the things we do in life is conclusions driven, including physics and science. I mean, they want to know where is this asteroid going to go? What is the weather going to be tomorrow? But that also has this other direction of going from the axioms. What do you think there is this tension in physics between theory and experiment? What do you think is the more powerful way of discovering truly novel ideas about reality?

49:41Well, you need both. Top down and bottom up. It's a really an interaction with all these things. Over time, the observations and the theory and the modeling should both get closer to reality. Initially, there's always a case out there that they're always far apart to begin with. But you need one to figure out where to push the other. If your model is predicting anomalies that are not picked up by experiment, that was experimenters were to look to find more data to refine the models. It goes back and forth. Within mathematics itself, there's also a theory and experimental component. It's just that until very recently theory has dominated almost completely 99 % of mathematics is theoretical mathematics.

50:31There's a very tiny amount of experimental mathematics. I mean, people do do it. If they want to study prime numbers or whatever, they can just generate large data sets. Once we had a computer, we'd be able to do it a little bit. Although even before like Gals, for example, he discovered he conjectured the most basic theory in number theory to call the prime number theorem, which predicts how many primes that are up to a million, up to a trillion. It's not obvious question. Basically, what he did was he computed, mostly by himself, but also hired human computers, people whose professional job it was to do arithmetic.

51:09To compute the first 100 ,000 primes or something and made tables and made a prediction, that was an early example of experimental mathematics. But until very recently, it was not, I mean, theoretical mathematics was just much more successful. Because doing complicated mathematical computations was just not feasible until very recently. And even nowadays, even though we have powerful computers, only some mathematical things can be explored numerically. There's some called the combinatorial explosion. If you want to study, for example, Xero -Madeus theory, we want to study all possible subsets of numbers 1 to 1000.

51:44There's only 1000 numbers. How bad could it be? It turns out that the number of different subsets of 1 to 1000 is 2 to the power 1000, which is way bigger than any computer can currently continue, in fact, any computer ever, or whatever can be enumerated. So you have to be, there are certain math problems that very quickly become just intractable to attack by direct brute force computation. Chess is another famous example. The number of chess positions, we can't get a computer to fully explore. But now we have AI. We have tools to explore this space, not with 100 % guarantees of success, but with experiment.

52:24So we can empirically solve chess now, for example. We have a very, very good AI that they don't explore every single position in the game tree, but they have found some very good approximation. And people are using actually these chess engines to make, to do experimental chess. They're revisiting old chess theories about, oh, you know, when you, this type of opening, this is a good type of move, this is not. And they can use these chess engines to actually refine, in some cases overturn, conventional wisdom about chess. And I do hope that mathematics will have a larger experimental building in the future, perhaps powered by AI.

53:04Well, of course, talk about that. But in the case of chess, and there's a similar thing in mathematics, that I don't believe is providing a kind of formal explanation of the different positions. But it's just saying which position is better or not, that you can intuit as a human being. And then from that, we humans can construct a theory of the matter. You've mentioned the Plato's cave algorithm. So it gets people to know it's where people are observing shadows of reality, not reality itself. And they believe what they're observing to be reality. Is that in some sense what mathematicians and maybe all humans are doing is looking at shadows of reality?

53:50Is it possible for us to truly access reality? Well, there are these three ontological things. There's actual reality, there's observations and models. And technically they are distinct, and I think they will always be distinct. But they can get closer over time. And the process of getting closer often means that you have to discard your initial intuitions. Astronomy provides great examples. An initial model of the world is flat because it looks flat. And it's big. And the rest of the universe, the sky is not like the sun, for example, looks really tiny. And so you start off with a model which is really far from reality.

54:41But it fits the kind of the observations that you have. So things look good. But over time, as you make more and more observations, bring it closer to reality. The model gets dragged along with it. And so over time, we had to realize that the Earth was round, that it spins. It goes around the solar system, so it goes on the galaxy, and so on and so forth. And the guys part of the universe, you have this expanding. The expansion is self -expanding, accelerating. And in fact, very recently in this year or so, there's even the explosion of the universe itself is so this evidence that is non -constant.

55:11And the explanation behind why that is is catching up. It's catching up. I mean, it's still the dog mounted dog energy. This kind of thing. We have a model that sort of explains that fits the day really well. It just has a few parameters that you have to specify. But so people say all that's fudge factors with enough fudge factors, you can explain anything. But the mathematical point with the model is that you want to have fewer parameters in your model than data points in your observational set. So if you have a model with 10 parameters that explains 10 -up -tenile observations, that is completely useless model.

55:48It's what's called overfitted. But if you have a model with two parameters, and it explains a trillion observations, which is basically, so the dark matter model, I think, has 14 parameters, and it explains petabytes of data that the other one was have. You can think of a theory. One way to think about physical mathematical theory is a compression of the universe and a data compression. So you have these petabytes of observations. You'd like to compress it to a model which you can describe in five pages and specify a certain number of parameters and it can fit to reasonable accuracy, almost all of your observations.

56:29The more compression that you make, the better your theory. In fact, one of the great surprises of our universe and of everything in it is that it's compressible at all. It's the unreasonable effect of mathematics. I'm not a quote like that. The most incomprehensible thing about the universe is that it is comprehensible. And not just comprehensible. You can do an equation like E equals M, C squared. There is actually some mathematical possible explanation for that. So there's this phenomenon in mathematical universality. So many complex systems at the macro scale are coming out of lots of tiny new interactions at the macro scale.

57:01And normally because of the common form of explosion, you would think that the macro scale equations must be infinitely exponentially more complicated than the macro scale ones. And they are, if you want to solve them completely exactly. If you want to model all the atoms in a box of air, I have a guide with numbers. You're mungus. There's a huge number of particles. If you actually have to track each one, it will be ridiculous. But certain laws emerge at the macro scale that almost don't depend on what's going on at the macro scale. Only they've been on a very small number of parameters. So if you want to model a gas of, you know, quintillion particles in a box, you just need to know temperature and pressure and volume a few parameters, like five or six.

57:45And it models almost everything you need to know about these 10th or 23 or whatever particles. So we have, we don't understand universality anywhere new as we would like mathematically. But there are much simpler 20 models where we do have a good understanding of why universality occurs. Most basic one is the central limit theorem that explains why the bell curve shows up everywhere in nature. But so many things are distributed by, I was called a gas distribution famous bell curve. There's now even a meme with this curve. And even the meme applies broadly. The universality to the meme. Yes, you can go matter if you like.

58:25But there are many, many processes. For example, you can take a lot of independent random variables and average them together in various ways. You can take a simple average or more complicated average. We can prove in various cases that these bell curves, these calcium's emerge. It is a satisfactory explanation. Sometimes they don't. So if you have many different inputs and they will correlate it in some systemic way, then you can get something very far from a bell curve. Sure. And this is also important to know when the situation is really wrong. Fails. So universality is not a 100 % reliable thing to rely on.

58:59That global financial crisis was a famous example of this. People thought that mortgage defaults had this sort of calcium type behavior. If you ask if a population of 100 ,000 Americans with mortgages, that's what proportion that would default. If everything was decarolated, it would be an ass -bell curve. And you can manage risk of options and derivatives and so forth. And it is a very beautiful theory. But if there are systemic shocks in the economy that can push everybody to default at the same time, that's very non -gassing behavior. And this wasn't fully accounted for in 2008. Now I think there's some more awareness that this is a systemic risk is a huge issue.

59:48And just because the model is pretty and nice, it may not match reality. So the mathematics of working out what models do is really important. But also the science of validating when the models fit reality and when they don't. I mean, that you need both. And but mathematics can help because for example, the central limit is it tells you that if you have certain axioms like non -correlation, that if all the inputs were not correlated to each other, then you have these class things are fine. It tells you where to look for weaknesses in the model. So if you have a mathematical understanding of central limit theorem and someone proposes to use these Gaussian Copylers or whatever to model default risk, if you're mathematically trained, you would say, okay, but what is the systemic correlation between all your inputs?

1:00:41And so then you can ask the economists, you know, how much risk is that? And then you can co -look for that. So there's always this this synergy between science and mathematics. A little bit on the top of universality. You're known and celebrated for working across an incredible breadth of mathematics. So I'm an instant of Hilbert a century ago. In fact, the great fields metal winning mathematician Tim Gowers has said that you are the closest thing we get to Hilbert. He's a colleague of yours. Good friend. But anyway, so you are known for this ability to go both deep and broad in mathematics. So you're the perfect person to ask.

1:01:24Do you think there are threads that connect all the disparate areas of mathematics? Is there kind of deep underlying structure to all of mathematics? This certainly a lot of connecting threads and a lot of the progress of mathematics can be represented by taking by stories of two fields of mathematics that were previously not connected and finding connections. An ancient example is geometry and number theory. So in the times of ancient Greeks, these were considered different subjects. I mean, mathematicians worked on both. You know, you could work both on geometry most famously, but also on numbers.

1:02:05But they were not really considered related. I mean, a little bit like, you know, you could say that this length was five times this length because you could take five copies of this length and so forth. But it wasn't until they carved, you really realized that you could develop and that geometry that you can you can parameterize the plane a geometric object by two real numbers. Every point can be and so geometric problems can be turned into into problems about numbers. And today, this feels almost trivial. Like there's no content to list. Of course, the plane is xx and y because that's what we teach and it's internalized.

1:02:46But it was an important development that these two fields are unified. And this process has just gone on throughout mathematics over and over again. Algebra and geometry were separated and now we have a suitable algebraic geometry that connects them and over and over again. And that's certainly the type of mathematics that I enjoy the most. So I think there's sort of different styles to being a mathematician. I think hedgehogs and fox. A fox knows many things a little bit, but a hedgehog knows one thing very, very well. And in mathematics, there's definitely both hedgehogs and foxes. And then there's people who are kind of who can play both roles.

1:03:21And I think ideal collaboration between mathematicians involves very, you need some diversity. For fox working with many hedgehogs all vice versa. So yeah, but I identify mostly as a fox. So I like arbitrage somehow. Like learning how one field works, learning the tricks of that wheel and then going to another field which people don't think it is related, but I can adapt the tricks. So see the connections between the fields. So there are other mathematicians who are far deeper, but I am like who they really hedgehogs. They know everything about one field and they're much faster and more effective in that field, but I can I can give them these extra tools.

1:04:04I mean, you said that you can be both a hedgehog and the fox depending on the context and depending on the collaboration. So what can you if it's at all possible speak to the difference between those two ways of thinking about a problem. Say you're monitoring your problem, you know, searching for the connections versus like very singular focus. I'm much more comfortable with the fox paradigm. Yeah, so yeah, I like looking for analogies, narratives. I spend a lot of time if it is a result I see it in one field and I like the result. It's a cool result, but I don't like the proof. Like it uses types of mathematics that I'm not super familiar with.

1:04:47I often try to reproof it myself using the tools that I favor. Of my proof is worse, but by the exercise of doing so, I can say, oh, now I can see what the other proof was trying to do. And from that, I can get some understanding of the tools that are used in that field. So it's very exploratory, very doing crazy things and crazy fields and like reinventing the wheel a lot. Whereas the hedgehog style is, I think, much more scholarly. You know, you're very knowledge -based. You stay up to speed on all the developments in this field. You know, all the history. You have a very good understanding of exactly the strength and weaknesses of each particular technique.

1:05:32Yeah, I think you'd rely a lot more on calculation than sort of trying to find narratives. So yeah, I mean, I could do that too, but there are other things extremely good at that. Let's step back and maybe look at a bit of a romanticized version of mathematics. So I think you've said that early on in your life, math was more like a puzzle -solving activity when you were young. When did you first encounter a problem or proof where you realize math can have a kind of elegance and beauty to it? That's a good question. When I came to graduate school in Princeton, so John Conway was there at the time.

1:06:20He passed away a few years ago. But I remember one of the very first research talks I went to was a talk by Conway on what he called extreme proof. So Conway had just said this is an amazing way of thinking about all kinds of things in a way that you would normally think of. So he thought of proofs themselves as occupying some sort of space. So if you want to prove something, let's say that there's infinitely many primes, okay? You're all of the different proofs, but you could rank them in different axes. Like some proofs are elegance, who's along, some proofs are elementary and so forth. And so this is cloud.

1:06:52So the space of all proofs itself has some sort of shape. And so he was interested in extreme points of the shape. Like all these proofs, what is one of those? These shortest at the the extensor, everything else or the most elementary or whatever. And so he gave some examples of well -known theorems and then he would give what he thought was the extreme proof in these different aspects. I just found out really eye -opening that it's not just getting a proof for Zal, it was interesting, but once you have that proof trying to optimize it in various ways, that proofing itself had some craftsmanship to it.

1:07:39It's something for my writing style that, like when you do your math assignments and undergraduate, your homework and so forth, you're sort of encouraged to just write down any proof that works, okay? And the hand is in, as long as it gets a tick mark, you move on. But if you want, it was out to actually be influential, it'd be read by people. It can't just be correct. It should also be a pleasure to read, you know, motivated, be adaptable to generalize to other things. It's the same in many other disciplines, like coding. There's a lot of analogies between math and coding. I like analogies, if you haven't noticed.

1:08:17But you can code something spaghetti code that works for a certain task, and it's quick and dirty, and it works. But there's lots of good principles for writing code well, so that other people can use it, board upon it, and so on, that has fewer bugs and whatever. There's some of the things with mathematics. Yeah, first of all, there's so many beautiful things there, and Kama is one of the great minds in mathematics ever and computer science. Just even considering the space of proofs. And saying, okay, what does this space look like? And what are the extremes? Like you mentioned, coding is an analogies, interesting, because there's also this activity called code golf, which I also find beautiful and fun, where people use different programming languages to try to write the shortest possible program that accomplishes a particular task.

1:09:11Then I believe there's even competitions on this. It's also a nice way to stress tests, not just the sort of the programs, or in this case the proofs, but also the different languages. Maybe that's a different notation, or whatever, to use the catalog of the different tasks. Yeah, you learn a lot. I mean, it may seem like a frivolous exercise, but it can generate all these insights, which if you didn't have this artificial objective to pursue, you might not see. What to use the most beautiful or elegant equation in mathematics? I mean, one of the things that people often look to in beauty is the simplicity.

1:09:53So if you look at e equals of c squared. So when a few concepts come together, that's why the oil or identity is often considered the most beautiful equation in mathematics. Do you find beauty in that one in the oil identity? Yeah, well, as I said, I mean, what I find most appealing is connections between different things that you like. So if you eat the pi i equals minus one. So yeah, people, I use all the fundamental constants. That's cute. But to me, so the exponential function was interesting, to measure exponential growth. So compound interest or decay or anything which is continuously growing continuously decreasing growth and decay or dilation or contraction is modeled by the exponential function.

1:10:38Whereas pi comes around from circles and rotation. If you want to rotate a needle, for example, 100 degrees, you need to rotate by pi radians. And i, complex numbers, represents this problem, which we want to imagine, axes of a negative rotation. So a change in direction. So the exponential function represents growth and decay in the direction that you already are. When you stick an eye in the exponential, now it's instead of motion in the same direction as your composition, it's the motion as a right angle to your composition. So rotation. And then so if the pi i equals minus one, tells you that if you rotate for time pi, you end up at the other direction.

1:11:17So it unifies geometry through dilation and exponential growth, dynamics, through this act of of complexification, rotation by i. So it connects together all these two as mathematics, yeah, that I was geometry and complex and complex and the complex numbers, they were considered almost their own vectoral neighbors in mathematics because of this identity. Did you think the thing you mentioned is cute. The collision of notations from these disparate fields is just a frivolous side effect or do you think there is legitimate like value in one notation, although our old friends come together at night.

1:11:54Well, it's confirmation that you have the right concepts. So when you first study anything, you have to measure things to give them names. And initially sometimes because your model is again too far off from reality, you give the wrong things the best names and you only find out later what's really important. Physicians can do this sometimes. I mean, but it turns out okay. So actually, physics of so it equals empty squared. Okay, so one of the big things was the eat. So when our start all first came up with his laws of motion and then Galileo and Newton and so forth, they saw the things they could measure.

1:12:32They could measure mass and acceleration and force and so forth. So Newtonian mechanics, for example, ethical's MA was the famous Newton's second law of motion. So those were the primary objects. So as they gave them the central building in the theory, it was only later after people started analyzing these equations that they always seem to be these quantities that were conserved. So a particular momentum and energy. And it's not obvious that things happen energy. Like it's not something you can directly measure the same way you can measure mass and velocity and so forth. But over time, people realized that this was actually a really fundamental concept.

1:13:04Hamilton, eventually in 19th century, reformulated Newton's laws of physics into what is called Hamiltonian mechanics, where the energy which is now called the Hamiltonian was the dominant object. Once you know how to measure the Hamiltonian of any system, you can just completely detect the dynamics like what happens to it or to all the states like it's it really was a central actor, which was not obvious initially. And this helped actually this change of perspective really helped when quantum mechanics came along. Because the early physicists who studied quantum mechanics, they had a lot of trouble trying to adapt in Newtonian thinking because the other event was a particle and so forth to quantum mechanics.

1:13:46Because I think people was a way, but it just looked really, really weird. Like you ask what is the quantum version of ethical's MA? And it's really, really hard to to give an answer to that. But it turns out that the Hamiltonian, which was so secretly behind the scenes in classical mechanics, also is the key object in quantum mechanics that there's also an object called Hamiltonian. It's a different type of object. It's what's called an operator rather than a function, but again, once you specify it, you specify the entire dynamics. So there's some controomous equation that tells you exactly how quantum systems evolve once you have the Hamiltonian.

1:14:22So side by side, they look completely different to objects. You know, like one involves particles, one involves waves and so forth. But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics. So for example, in classical mechanics, there's this single notice theorem. Every time there's a symmetry in a physical system, there was a conservation law. So the laws of physics are translation invariant. Like if I move ten steps left, I experience the same laws of physics as I was here. And that corresponds to conservation momentum.

1:14:53If I turn around by some angle, again, I experience the same laws of physics. This corresponds to the conservation of angular momentum. If I wait for ten minutes, I still have the same laws of physics. So there's time transition invariance, this corresponds to the law of conservation energy. So there's this fundamental connection between symmetry and conservation. And that's also true in quantum mechanics, even though the equations are completely different. But because they're both coming from Hamiltonian, Hamiltonian controls everything. Every time the Hamiltonian has a symmetry, the equations will have a conservation law.

1:15:25So it's, it's, it's, it's, it's, once you have the right language, it actually makes things a lot cleaner. One of the problems is why we can't unify quantum mechanics and general relativity yet. We haven't figured out what the fundamental object is. Like, for example, we have to give up the notion of space and time being these almost cleaning type spaces. And it has to be, you know, and, you know, we kind of know that at very tiny scales, there's going to be quantum fluctuations. There's a space, space time foam. And trying to, to use Cartesian coordinates XYZ is going to be, it's a, it's, it's a non -starter.

1:15:57But we don't know how to, what to replace it with. We don't actually have the mathematical, with, um, concepts. Yeah, the analog of Hamiltonian, but sort of organized everything. Does your gut say that there is a theory of everything? So this is even possible to unify, to find this language that unifies general relativity and quantum mechanics? I believe so. I mean, the history of physics has been that of unification, much like mathematics over the years. You know, electricity in a magnetism was separate theories and then backs will unify them. You know, you can unify the motions of heavens for the motions of objects on the earth and so forth.

1:16:33So it should happen. It's just that the, um, again, to go back to this model of the observations and theory. Part of our problem is that physics is a victim of its own success that of two big theories of, of physics, general relativity and quantum mechanics are so, so good now. So together, they cover 99 .9 % of sort of all the observations we can make. Um, and you have to like either go to extremely insane particle celebrations or, or the early universe or, or things that are really hard to measure, um, in order to get any deviation from either of these two theories to the point where you can face that we, you know, we've, we've been doing this for centuries.

1:17:15We've made progress before. And there's no reason why we should stop. Do you think you will be a mathematician that develops a theory of everything? What often happens is that when the physicists need, um, some theory of mathematics, there's often some precursor that the mathematicians, um, worked out earlier. So when Einstein started realizing that space was curved, he went to some mathematician and asked, is there, is there some theory of curved space that the mathematicians already came up with that could be useful? And he said, oh, yeah, there's, I think, a, we, we might have developed a, we might have developed a, we might be in geometry, um, which is precisely, um, you know, a, a theory of spaces that are curved in various general ways, which turn out to be almost exactly what was needed, um, or Einstein's theory.

1:18:01This has gone back to, to a big, this unreasonable effectiveness of mathematics. I think the theories that work well, they say in the universe, tend to also involve the same mathematical objects that work well to solve mathematical problems. Ultimately, there's just both ways of organizing data, um, in, in, in, in useful ways. It just feels like you might need to go some weird land that's very hard to, to intuit, like, you know, you have like, string theory. Yeah, that, that's, that was, that was a leading candidate for many decades. It's, I think, is slowly pulling out of fashion because it's, it's not matching experiment.

1:18:33So one of the big challenges, of course, like you said, is experiment is very tough. Yes, because of the, how effective both theories are. But the other is like, just, you know, you're talking about, you're not just deviating from space time. You're going until like some crazy number of dimensions, you're doing all kinds of weird stuff that, to us, we've gone so far from this flat earth that we started that, like, you mentioned, yeah, yeah, yeah. We're just, it's, it's very hard to use our limited, a descendants of a, uh, cognition to intuit what that reality really is like. This is why analogies are so important.

1:19:11You know, I mean, so, yeah, I mean, the round earth is not intuitive because we're stuck on it. Um, but, you know, but you, you know, but round objects in general, we have pretty good intuition. Oh, but, uh, and we've been through about light works and so forth. And like, it's, it's actually a good exercise to actually work out how eclipses and phases of the sun and the moon and so forth. It can be really easy to explain by, by, by, by round earth and round moon, you know, um, and models. And, and you can just take, you know, a basketball, and a golf ball, and a, and a, and a light source, and actually do these things yourself.

1:19:42Um, so the intuition is there. Um, but, yeah, you have to transfer it. That is a big leap into lecture for us to go from flat to round earth because, you know, our life is mostly lived in flat land. Yeah. To load that information and we're all taking it for granted. We take so many things for granted because science has established a lot of evidence for this kind of thing, but, you know, we're in a, we're out rock. Yeah. Like through space. Yeah. Yeah. That's a big leap. And you have to take a chain of those leaps, the more and more and more leap progress. Right. Yeah. So modern science is, maybe again, a victim with his own success is that, you know, in order to be more accurate, it has to move further and further away from your initial intuition.

1:20:24And so, um, for someone who hasn't gone through the whole process of science education, it looks more more suspicious because of that. So, you know, we need, we need more grounding. I mean, I think, um, I mean, you know, there are scientists who do excellent outreach. Um, but there's this, this, this, this, lots of science things that you can do at home. I have this lots of YouTube videos. I did a YouTube video recently of Grand Sanison, we talked about earlier, that, you know, how the ancient Greeks were able to measure things like the distance of the moon, distance of the earth. And, you know, using techniques that you could also replicate yourself.

1:20:55Um, it doesn't all have to be like fancy space telescopes and and very deep intimidating mathematics. Yeah. That's, uh, I had to recommend that. I believe you have a lecture and you also did an incredible video with Grant. It's a beautiful experience to try to put yourself in the mind of a person from that time and shrouded in a mystery. Right. You know, you're like on this planet, you don't know the shape of it, the size of it. You see some stars, you see some, you see some things and you try to like localize yourself in this world. Yeah, and try to make some kind of general statements about distance to places.

1:21:29Change of perspective is really important. You say travel board is the mind. This is intellectual travel. You know, put yourself in the mind of the ancient Greeks or some other persons from other time period, make hypotheses, spherical cows, whatever, you know, speculate. Um, and, you know, this is, this is what my physicians do and some other sort of artists do actually. It's just incredible that given the extreme constraints, you could still say very powerful things. That's why it's inspiring. Looking back in history, how much can be figured out? Right. We don't have much figure. How stuff would propose axioms, then the mathematics, let's you follow those axioms to it to their conclusions.

1:22:05And sometimes you can get quite a lot, quite a long way from, you know, initial hypothesis. If you're staying in the land of the weird, you mentioned general relativity, you've, uh, contributed to the mathematical understanding of Einstein's field equations. Can you explain this work? And from a sort of mathematical standpoint, what aspects of general relativity are intriguing to you, challenging to you? I have worked on some equations. There's something called the wave -mapsed equation, or the sigma field model, which is not quite the equation of space -time gravity itself, but of certain fields that might exist on top of space -time.

1:22:45So Einstein's equations of relativity just describe space and time itself. But then there's other fields that live on top of that. There's the electromagnetic field, there's things like Yang -Mills fields. And there's this whole hierarchy of different equations, of which Einstein's considered one of the most nonlinear and difficult. But relatively low on hierarchy was this thing called the wave -mapsed equation. So it's a wave, which at any given point is fixed to be like on a sphere. So I can think of a bunch of arrows in space and time, and yeah, it's pointing in different directions. But they propagate like waves.

1:23:19If you wiggle an arrow, it will propagate and make all the arrows move kind of like a sheep's or wheat in a wheat field. And I was interested in the global or clouded problem again for this question. Like, is it possible for the energy here to collect at a point? So the equation I considered was called a critical equation, where it's actually the behavior at all scales is roughly the same. And I was able barely to show that you couldn't actually force a scenario where all the energy concentrated at one point. But at the end, you have to dismiss a little bit and moment it was a little bit, it would stay regular.

1:23:54Yeah, this was back in 2000. That was part of why I got interested in Nari's socks afterwards. Actually, yeah, so I developed some techniques to solve that problem. So part of it was, it was, this one is really nonlinear because of the curvature of the sphere. There's, there was a certain nonlinear effect, which was a non perturbative effect. It was, when you sort of looked at it normally, it looked larger than the linear effects of the wave equation. And so it was hard to keep things under control, even when the energy was small. But I developed what's called a case transformation. So the equation is kind of like an evolution of hives of wheat and they're all bending back and forth.

1:24:30And so there's a lot of motion. But like if you imagine like stabilizing the flow by attaching little cameras at different points in space, which I'm trying to move in a way that captures most of the motion. And under this sort of stabilized flow, the flow becomes a lot more linear. I discovered a way to transform the equation to reduce the amount of nonlinear effects. And then I was able to, to, to, to solve the equation. I found this transformation while visiting my art in Australia. And I was trying to understand the dynamics of all these fields. And I, I couldn't do a pen and paper. And I had non -office -led different computers to do any computer simulations.

1:25:07So I ended up closing my eyes, people on the floor, and just imagining myself to actually be the specter field and rolling around to try to, to see how to change coordinates in such a way that somehow things in order of actions would behave in a reasonable linear fashion. And yeah, my aunt walked in and thought I was doing that. And she was asking, what are that? What am I doing? Doing this? It's complicated. Yeah, yeah. And okay, fine. You know, your young man, I don't ask questions. I have to ask about the, you know, how do you approach solving difficult problems? What, if it's possible to go inside your mind when you're thinking, are you visualizing in your mind the mathematical object symbols, maybe, what are you visualizing in your mind usually when you're thinking?

1:25:56A lot of pen and paper. One thing you pick up as mathematician is sort of, I call it cheating strategically. So the, the beauty of mathematics is that you get to change the world, change the problem, change the rules as you wish. You don't get to do this for any other field. Like, you know, if you're an engineer and someone says, put a bridge over this river, you can't say, I want to build this up pretty over here instead or I want to put out a paper instead of steel. But imagine you can do whatever you want. It's like trying to solve a computer game where you can, there's unlimited cheat codes available.

1:26:32And so, you know, you can, you can set this, there's a dimension that's large. I've set it to one. I'd solve the one version problem first. So there's a main term and error term. I'm going to make a spherical curl assumption. I'll assume the error term is zero. And so the way you solve these problems is not in sort of this iron man mode where you make things maximally difficult. But actually, the way you should approach any reasonable math problem is that you, if there are 10 things that are making it like difficult, find a version of the problem that turns off and dine the difficulties, but only keeps one of them.

1:27:04And so that. And then that just, so you, you, you, you install nine sheets. Okay, we saw 10 sheets then then the game is trivial. But you saw nine sheets. You saw one problem that, that, that, that, that teaches you how to do that to get difficulty. And then you turn that one off and you see someone else, someone else else on and then you saw that one. And after you, you know, how to solve the 10 problems, 10 difficulty separately, then you have to start merging them if you had a time. I, I was a kid. I watched a lot of these Hong Kong action movies. This is from a culture. And one thing is that every time it's the fight scene, you know, something like that, the hero gets swarmed by 100 bad guy goons or whatever.

1:27:41But it'll always be choreographed so that you'd always be only fighting one person at a time. And then you would defeat that person and move on. And because of that, they could defeat all of them. But whereas if they had fought a bit more intelligently and just swarmed the guy once, it would make for much, much worse. I'm cinema. But they would win. Are you usually pen and paper? Are you working with computer and late tech? I'm mostly pen and paper actually. So in my office, I have four giant blackboards. And sometimes I just have to write everything I know about the problem on the full blackboards and then sit my couch and just sort of see the whole thing.

1:28:19Is it all symbols like notation or is there some drawings? Oh, there's a lot of drawing and a lot of bespoke doodles that only makes sense to me. And this is a bit of a blackboard you raise. It's a very organic thing. I'm beginning to use more more computers partly because AI makes it much easier to do simple coding things. If I wanted to plot a function before which is moderately complicated as some iteration or something, I'd have to remember how to set up a Python program and how does a for loop work and debug it and it would take two hours and so forth. And now I can do it in 10, 15 minutes as much.

1:28:57I'm using more and more computers to do simple extroations. Let's talk about AI a little bit if we could. So maybe a good entry point is just talking about computer assisted proofs in general. Can you describe the lean formal proof programming language in how it can help as a proof assistant and maybe how you started using it and how it has helped you? So lean is a computer language much like sort of standard languages like Python and C and so forth. Except in most languages the focus is on using executable code. Lines of code do things. They flip bits or they make a real one move or they deliver you text on even it or something.

1:29:44So lean is a language that can also do that. It can also be run as a standard traditional language. But it can also produce certificates. So a software like Python make do a computation and give you the answer is seven. Okay, that it does the sum of people's four is equal to seven. But lean can produce not just the answer but a proof that how it got the answer of seven as three plus four and all the steps involved in. So it creates these more complicated objects, not just statements, but statements were proofs attached to them. And every line of code is just a way of piecing together previous statements to create new ones.

1:30:22So the idea is not new. These things are called proof assistants and so they provide languages for which you can create quite complicated intricate mathematical proofs and they produce these certificates that give it 100 % guarantee that your arguments are correct. If you trust the compiler, obviously, but they made the compiler really small and you can have a several different files available for the same. Can you give people some intuition about the difference between writing on pen and paper versus using lean programming language? How hard is it to formalize statement? So lean a lot of mathematicians will invoke in the design of lean.

1:30:59So it's designed so that individual lines of code resemble individual lines that have an argument like you might want to introduce a variable. You want to want to prove there are various standard things that you can do and it's written so ideally it should like a one -twentil correspondence. In fact, it isn't because lean is like explaining a proof to extremely pedantic colleague who will point out, okay, did you really mean this? Like what happens if this is zero? Okay, how do you justify this? So lean has a lot of automation in it to try to to to be less annoying. So for example, every mathematical object has to come with a type.

1:31:38Like if I talk about x, is x a rule number or a natural number or a function or something. If you write things informally, it's not an informed context. You say clearly x is equal to let x be the sum of y and z and y and z were already rule numbers. So x should also be a rule number. So lean can do a lot of that, but every solvent it says, wait a minute, can you tell me more about what this object is? What type of object it is? You have to think more at a philosophical level, not just sort of computations you're doing, but sort of what each object actually is in some sense. Is it using something like LLM's to do the type inference or like you match with a real level?

1:32:22It's it's using much more traditional, good or fashion to AI. You can represent all these things as trees and there's always algorithm to match one tree to another tree. So it's actually doable to figure out if something is a real number or a natural number. Every object comes with a history of what it came from and you can kind of trace it. Oh, I see. Yeah, so it's designed for reliability. So modern AI's are not used in it's a disjoint technology. People are beginning to use AI's on top of lean. So when a mathematician tries to program a proof in lean, often there's a step. Okay, now I want to use the fundamental thing we'll the lean developers have built this massive project called methylib collection of tens of thousands of useful facts about mathematical objects.

1:33:09And somewhere in there is the fundamental calculus, but you need to find it. So a lot of the bottleneck now is actually LLM's search. There's a tool that you know is in there somewhere and you need to find it. And so you can there are various search engines specialized for methylib that you can do. But there's now these large language models that you can say, I need the fundamental calculus at this point. And I said, okay, for example, when I code, I have GitHub code pilot installed as a plugin to my IDE. And it scans my text and it sees what I need. It says, yeah, I'm not even typing. Now I need to use the fundamental thing with calculus.

1:33:43Okay. And it might suggest, okay, try this and maybe 25 % of the time it works exactly. And then another 10, 15 % of the time it doesn't quite work, but it's close enough that I can say, oh, I've just changed it here and here. It will work. And then like half the time it gives me complete rubbish. So, but people are beginning to use AI's a little bit on top. Most of them are level basically fancy auto complete. That you can type half of one line of approved and it will find it will tell you. Yeah, but a fancy, especially fans with the sort of capital that our app is removes some of the friction.

1:34:17Yeah. Mathematician might feel when they move from patterned paper to formalizing. Yes. Yeah. So right now I estimate that the effort, time and effort taken to formalize the proof is about 10 times the amount taken to write it out. Yeah. So it's doable, but you don't, it's annoying. But this is like kill the whole vibe of being a mathematician. Yeah. So, I mean, having a pedantic worker. Right. Yeah. If that was the only aspect of it. Okay. But, um, okay. There's something, because it was actually more pleasant to do this formally. So those are those are the formalized and those are certain constant 12 that came out of it in the final statement.

1:34:56And so this 12 had be carried all through the proof. And like everything had to be checked. That it goes all the, all these other numbers that had be consistent with this final number 12. And then so we want a paper through this theorem with this number 12. And then a few weeks later, so I said, oh, we can actually improve this 12 to an 11 by we working some of these steps. And when this happens with pen and paper, um, like every time you change your parameter, you have to check line by line that every single line of your proof still works. And there can be subtle things that you didn't quite realize.

1:35:22Some probably is not a mature, but you didn't even realize that you were taking advantage of. And so a proof can break down at a subtle place. So we had formalized the proof of this constant 12. And then when this new paper came out, uh, we said, oh, okay, let's, so that took like three weeks to formalize and like 20 people to formalize this, this is original proof. I said, oh, but now let's, let's um, uh, let's update the total 11. And what you can do with lean, so you just, in your headline theorem, you change your 12 to an 11, you run the compiler. And like of the thousands of lines that code you have, 90 percent of them still work.

1:35:55And there's a couple that are line and red. Now I can't just buy these steps, but it meets the isolates which steps you need to change. But you can skip over everything which works just fine. Um, and if you program things correctly, um, with, with good programming practices, most of your lines will not be read. Um, and there'll just be a few places where you, I mean, if you don't hard code your constants, but you sort of, um, you, you smart tactics and so forth, yeah, you can, you can localize, um, the things you need to change to a very small, um, period of time. So it's like it wouldn't a day or two we had updated our proof to, uh, because this is a very quick process here.

1:36:29Um, you make a change. There are 10 things now that don't work for each one. You make a change. And now there's five more things that don't work, but, but the process converges much more smoothly than with pen and paper. So that's for writing. Are you able to read it? Like if somebody else has a proof, they're able to like, how, what's, what's the, uh, versus paper and yeah, so the proof is a longer, but each individual piece is easier to read. So, um, if you take a math paper and you jump to page 27 and you look at paragraph six and you have a line of, of, of text or math, I often can't read it immediately because it assumes various definitions, which I have to, to go back and, and maybe on 10 pages earlier, this was defined.

1:37:09And this, um, the proof is scattered all over the place and you basically are forced to read fairly sequentially. Um, it's, it's not like say a novel where like, you know, in a theory, you could, you know, open up a novel halfway through and start reading. There's a lot of context, but when I've driven lean, if you put your cursor on a line code, every single object there, you can hover over it and it would say what it is, what it came from, what, where stuff is justified, you can trace things back much easier than sort of flipping through a math paper. So one thing a lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past.

1:37:41So traditionally, pen and paper, when you want to collaborate with another mathematician, um, either you do it at a blackboard where you, you can really interact, but if you're doing it sort of by email or something, um, basically, yeah, yeah, yeah, you have to segment it. Yeah, I'm gonna, I'm gonna finish section three, you do section four, but you can't really sort of work on the same thing, collaborate at the same time. But with lean, you can be trying to formalize some portion of the proof and say, oh, I got stuck at line 67 here, I need to prove this thing, but it doesn't quite work. Here's the three lines of code I can trouble with, um, but because all the context is there, someone else can say, oh, okay, I recognize what you need to do, you need to apply this trick or this tool and you can do extremely atomic level conversations.

1:38:22So because of lean, I can collaborate, you know, with dozens of people across the world, most of them I don't have never met in person, um, and I may not know actually even whether they're, um, how reliable they are in, in, in their, um, in, in the proof take a break, but lean gives me a certificate of trust. Um, so I can do, I can do trust the mathematics. So there's so many interesting questions, there's one, you're, you're known for being a great collaborator. So what is the right way to approach solving a difficult problem mathematics when you're collaborating? Are you doing a divide and conquer type of thing or are you brain, are you focused in a particular part and your brain storming?

1:39:05There's always a brain storming process first. Yeah, so math research projects sort of by their nature, when you start, you don't really know how to do the problem. Um, it's not like an engineering project where some other theory has been established for decades and it's, it's implementation is the main difficulty. You have to figure out even what is the right path. And so this is what I said about cheating first, you know, um, it's like, um, you go back to the bridge building analogy and I said, first assume you have an infinite budget and, and like unlimited amounts of work force and so forth.

1:39:34Now can you, can you build this bridge? Okay. Okay. Now have infinite budget, but only finite workforce. Now can you do that and so forth? Um, so, uh, I mean, of course, no, no, no engineering can actually do this, like, is there a fixed requirements? Yes, there's this sort of jam sessions, always at the beginning where you try all kinds of crazy things and you, you make all these assumptions that aren't realistic, but you plan to fix later. Um, and you try to see if there's even some skeleton, I'm gonna push them might work. Um, and then hopefully that breaks off the problem into smaller subproblems, which you don't know how to do, but then you, uh, you focus on on the subones and sometimes different collaborators are better at working on certain things.

1:40:14Um, so one of my themes I'm known for is the theme of Ben Green, which is not called the Green Talfeerum. Um, it's a statement that the primes contain earth make precautions of any event. So it was a modification of this theme was already. And the way we collaborated was that Ben had already proven a similar result for progressions of then three. Um, he showed that sets like the primes contain lots of other subquestions of plan three, um, even, and even, um, subsets of the primes, certain subsets do. Um, but his techniques only worked for, um, for them three precautions, they didn't work for longer progressions.

1:40:45Um, but I had these techniques coming from a gothic theory, which is something that I had been playing with and, and, uh, I knew better than Ben at the time. Um, and so if I could justify certain randomness properties of some set relating to primes, like there's a certain technical condition, which if I could have it, if Ben could supply me the respect, I could, I could conclude the theorem. But I, what I asked was a really difficult question in number theory, which, um, he said, no, there's no way we can prove this. Can you, so he said, can you prove your part of the theorem using a weak hypothesis that I have a chance to prove it?

1:41:18And he proposed something which he could prove, but it was too weak for me. Uh, I can't use this. Um, so there's this, there was this conversation going back and forth. Um, it's still a different cheats to, yeah, yeah, yeah. I want to cheat more. He wants to cheat less. Yeah. But eventually we found a, a, a, a, a, a, a property which A he could prove, and B I could use. Um, and then we could prove out here. And, um, yeah, so there's, there's a, there's a, there's a, all kinds of dynamics, you know, I mean, it's, it's, every, every, um, collaboration has, has, has, has some story. It's no two of the same.

1:41:50And then on, on the flip side of that, like you mentioned, with lean programming, now that's almost like a different story because you can do, you can create, I think you've mentioned a kind of a blueprint, right, for a problem. And then you can really do a divide and conquer with lean where you're working on separate parts. Right. And they're using the computer assistant proof checker essentially to make sure that everything is correct along the, yeah. So it makes everything compatible and, uh, yeah, and, and, and trustable. Um, yeah. So currently, you know, only a few mathematical projects can be cut up in this way.

1:42:25At the current state of the art, most of the lean activity is on formalizing proofs that have already been proven by humans. Math paper basically is a boop, a blueprint in a sense. It is taking a difficult statement, like big theorem and breaking up into me are a hundred little lambs. Um, but often not all written with enough detail that each one can be sort of directly formalized. A blueprint is like a really pedantically written version of a paper where every step is explained as much detail as, as, as possible. And to try to make each step kind of self -contained, um, and or depending on only a very specific number of previous statements, I mean, proven so that each node of this blueprint graph that gets generated can be tackled independently of all the others.

1:43:09And you don't even need to know how the whole thing works. Um, so it's like a modern supply chain, you know, and if you want to create an iPhone or some other complicated object, um, no one person can, can build a, um, a single object, but you can have specialists to, to just, if they're given some widgets from a similar company, they can combine them together to form a slightly bigger widget. I think there's a really exciting possibility because you can have, if you can find problems that could be broken down this way, then you can have, you know, thousands of contributors, right? Yes, yes, yes.

1:43:38So I told you before about the split between theoretical and experimental mathematics, and right now, most mathematics is theoretical and when you're tired, it's experimental. I think the platform that lean and other software tools, so, um, get help and things like that, um, allow, they will allow experimental mathematics to be to scale up, um, to a much greater degree than we can do now. So right now, if you want to, um, um, do any mathematical exploration, uh, of some mathematical patterns, some of you need some code to write out the pattern and, I mean, sometimes there's some computer algebra packages that can help, but often it says one mathematician coding lots and lots of Python, whatever.

1:44:14And because coding is such an error for an activity, it's not practical to allow other people to collaborate with you on writing modules for your code because if one of the modules has a bug in it, the whole thing is unreliable. Um, so it's these up, uh, so you got these bespoke, uh, spaghetti code written by non -professional programmers with my mathematicians, you know, and they're clunky and, and, and slow and, um, and so because of that, it's, it's hard to, to really mass -produce experimental results. Um, but, um, yeah, but I think with lean, I mean, so I'm already starting some projects where we are not just experimenting with data, but experimenting with proofs.

1:44:54So I have this project called the Equational Theory's project. Basically, we generated about 22 million little problems at abstract algebra. We should back up and tell you what the project is. Okay, so abstract algebra studies operations like multiplication and addition and the abstract properties. Okay, so multiplication, for example, is commutative. X times Y is always Y times X is for numbers. Um, and, uh, it's also associative. X times Y times Z is the same as X times Y times Z. Um, so, um, these operations obey some laws that don't obey others. For example, X times X is not always equal to X.

1:45:25So that laws are not always true. So given any, any, any operation, it obeys some laws and not others. Um, and so we generated about 4 ,000 of these possible laws of algebra that's certain operations. It's satisfying. And our question is which laws imply which other ones? Um, so for example, does commutativity imply associativity? And the answer is no because it turns that you can describe an operation which obeys the cognitive law, but it doesn't obey the association of law. So by producing an example, you can, you can show that commutatively does not imply associativity. But some of the laws do imply other laws by substitution and so forth.

1:45:57And you can write some algebraic proof. So we look at all the pairs between these 4 ,000 laws and there's about 22 million of these pairs. And for each pair, we ask, does this law imply this law? Well, if so, give a give, uh, give a proof, if not, give a count example. Um, so 22 million problems, each one of which you could give to like an undergraduate algebra student and they had a decent chance of solving the problem. Although there are a few, at least 22 million, there are like a hundred or so that are really quite hard. Okay, but a lot are easy. And the project was just to work out to determine the entire graph, like, like which ones imply which other ones?

1:46:31That's an incredible project, by the way. Such a good idea, such a good test that the very thing we've been talking about at a scale that's remarkable. Yeah, so it would not have been feasible. You know, I mean, the state of the art in the literature was like, you know, 15 equations and sort of highly implied, that sort of as a limit of what a human repentant paper can do. So you need to scale that up. So you need to crowdsource, but you also need to trust all the, um, yeah, yeah, yeah. I mean, no one person can check 22 million of these proofs. You need to be computerized. And so it only became possible with, with Lean.

1:47:02Um, we were hoping to use a lot of AI as well. Um, so that the project is almost complete. So, uh, these 20 million, all the two have been settled. Um, wow. And, uh, well, actually, and all those two, we have a pen and paper proof of the two. And we were formalizing. In fact, I was, this morning, I was working on it. So we're almost done on this. Um, it's incredible. It's, yeah, a, a, a, how many people were able to get 50, um, which in mathematics is considered a huge number. It's a huge number. Yeah. Crazy. Yeah. So we're going to have a paper 50 all this, and a big appendix of food contributor.

1:47:37What? Here's an interesting question. Not to maybe speak even more generally about it. When you have this pool of people, is there a way to, uh, organize the contributions by level of expertise to the people of the contributors? No, okay. Uh, I'm asking you a lot of pot head questions here, but I'm imagining a bunch of humans and maybe in the future some AI's. Yeah. Can there be like an elo rating type of situation where, like, a gamification of this, the beauty of all these lean projects is automatically you get all this data. Yeah. So like, everything is week uploaded for this guitar and could have tracks who contributed what?

1:48:15So you could generate statistics from, yeah, at any, at any later point in time, you can say, oh, this person contributed this made this many lines of code or whatever. I mean, these are very crude metrics. I would, I would definitely not want this to become like, you know, part of your tenure review or something. But, um, I mean, I think already in, in enterprise computing, right, people do use some of these metrics as part of the assessment of performance of an employee. Again, this is the direction, which is a bit scary for academics to go down. We don't like metrics so much. And yet academics use metrics.

1:48:51They just use old ones. Number of papers. Yeah. Yeah. It's true. It's true that, I mean, it feels like this is a metric while flawed is going in the more in the right direction, right? Yeah. It's interesting. At least it's a very interesting metric. Yeah. I think it's interesting to study. I mean, I think you can study the off -whether these are to incentivize performance, it becomes gained. And then it is no longer a useful measure. Oh, humans always give me these rational. So what we've done for this project is self -report. So there are actually standard categories from the sciences of what types of contributions people give.

1:49:34So there's this concept and validation and resources and coding and so forth. So we, we, there's a standard list of pro or so categories. And we just ask each contributor to this big matrix of all the all the authors and all the categories just to tick the boxes where they think they're contributed. And just give a rough idea, you know, like also you did some coding and and you provided some compute, but you didn't do an A for pen and paper verification or whatever. And I think that that works out traditionally mathematicians just order alphabetically by surname. So we don't have this tradition as in their sciences of lead author and second author and so forth.

1:50:11Which we're proud of, you know, we make all the authors equal status, but it doesn't quite scale to this size. So a decade ago I was involved in these things called polymath projects. It was the crowdsourcing methodics, but without the lean component. So it was limited by you needed a human moderator to actually check that all the contributions coming in were actually valid. And this was a huge bottleneck actually. But still we had projects that were, you know, 10 authors or so, but we had decided at the time not to try to decide who did what, but to have a single pseudonym. So we created this fictional category called DHJ Polymath in the Spirul Bobecky, Milwaukee.

1:50:51This is the pseudonym for a famous group of mathematicians in the 20th century. But and so the paper was altered on the pseudonym. So none of us got the author credit. This actually turned out to be not so grateful a couple of reasons. So one is that if you actually wanted to be considered for 10 year old or whatever, you could not use this paper in your, as you submitted it as a money publication, because it didn't have the form of author credit. But the other thing that we've recognized a much later is that when people refer to these projects, they naturally refer to the most famous person who was involved in the project.

1:51:28Oh, so this was Tim Gowas' playoff project. This was Tim's playoff project and not mentioned the other 19 or whatever people that were involved. So we're trying something different this time around, where we have everyone's an author, but we will have an appendix with those matrix. I mean, so both projects are incredible, just the fact that you're involved in such huge collaborations. But I think I saw a talk from Kevin Buzzer about the lean programming language a few years ago and you're saying that this might be the future of mathematics. And so it's also exciting that you're embracing one of the greatest mathematicians in the world embracing this which seems like the paving of the future of mathematics.

1:52:11So I have to ask you here about the integration of AI into this whole process. So deep minds, alpha proof was trained using reinforcement learning on both failed and successful formal lean proofs of IMO problems. So this is sort of high level high school or very high level. Yes, very high level high school level mathematics problems. What do you think about the system? And maybe what is the gap between this system that is able to prove the high school level problems versus gradual level problems? Yeah, the difficulty increases exponentially with the number of steps involved in the proof. It's a commentorial explosion.

1:52:54So I think of large language models is that they make mistakes. And so if your proof has got 20 steps and your high school level has a 10 % failure rate at each step going in the wrong direction, it's extremely unlikely to actually reach the end. Actually, just to take a small tangent here, how hard is the problem of mapping from natural language to the formal program? Oh, yeah, it's extremely hard actually. Natural language, you know, it's very fault -tolerant. Like you can make a few minor grammatical errors and speak in the second language, you can get some idea of what you're saying. But formal language, if you get one little thing wrong, I do not know the whole thing, it's nonsense.

1:53:37Even formal to formal, it's very hard. There are different incompatible proofs of some languages. There's lean, but also cock and Isabelle and so forth. And actually, even converting from a formal language is an unsolved problem. That is fascinating. Okay, so but once you have an informal language, they're using their RL -trained model. So something akin to alpha zero that they used this go. To then try to come up with proofs. They also have a model, I believe, as a separate model for geometric problems. So what impresses you about the system and what do you think is the gap? We talked earlier about things that are amazing over time, become kind of normalized.

1:54:23So now somehow, of course, geometry is still vulnerable from the problem. Right, that's true. That's true. I mean, it's still beautiful. Yeah, yeah, I know. It's a great work. The chose was possible, having it. The approach doesn't scale currently. Three days of Google's service, server time to sort of one high school math formula. This is not a scalable prospect, especially with the exponential increase in complexity increases. Which mentioned that they got a silver medal performance. The equivalent of, I mean, they took way more time than was allotted. And they had this assistance with where the humans started helped by formalizing.

1:55:02But also, they're giving us those formats for the solution, which I guess is formally verified. So I guess that's fair. There are efforts. There will be a proposal at some point to actually have an AI method limpiate where at the same time as the human contestants get the actual little bit problems, AI will also be given the same problems, the same time period. And the outputs will have to be created by the same judges. Which means that will have to be written in natural language, rather than formal language. I hope that happens. I hope this is what happens. I hope I hope it won't happen. This IMO, the performance is not good enough in the time period.

1:55:47But there are smaller competitions. The competitions where the answer is a number rather than a long form proof. And that's, that's, AI is actually a lot better at problems where there's a specific numerical answer. Because it's easy to to reinforce to reinforcement learning on it. Yeah, you've got the right answer, you've got the wrong answer. It's a very clear signal. But a long form proof either has to be formal and then the lean can give it thumbs up, thumbs down, or it's informal. But then you need a human to create it, to tell. And if you try to do billions of reinforcement learning, once you're not, you can't hire enough humans to create those.

1:56:30It's already hard enough for the last time I was to do reinforcement learning on just the regular text that people get. But now if you hire people, not just give thumbs up thumbs down, but actually check the output mathematically. Yeah, that's too expensive. So if we explore this possible future, what is the thing that humans do that's most special in mathematics? So that you could see AI not cracking for a while. So inventing new theories. So coming up with new conjectures versus proving the conjectures, building new abstractions, new representations, maybe an AI -terrestrial with seeing new connections between disparate fields.

1:57:17That's a good question. I think the nature of what mathematicians do of a time has changed a lot. So a thousand years ago, mathematicians had to compute the date of Easter. And there's really complicated calculations. But it's all made in the whole of the centuries. We don't need that anymore. They used to navigate to do spherical navigation, spherical trigonometry to navigate how to get from the old part of the new. I'm sorry, a very complicated calculation. Again, we'd been automated. Even a lot of undergraduate mathematics, even before AI, like Booth and Alpha, for example, is not a language, but it can solve a lot of undergraduate level math tasks.

1:57:55So on the computational side, verifying routine things, like having a problem and say, here's a problem in partial differential equations. Could you still be using any of the 20 standard techniques? And there's a try -doll 20, I hear that 100 different permutations and disease, my results. And that type of thing, I think, it worked very well. Type of scaling to, once you solve one problem to make the AI attack 100 adjacent problems. The things that humans do still, yes or so, where the AI really struggles right now is knowing when it's made a wrong turn. And you can say, oh, I'm going to solve this problem.

1:58:33I'm going to split up this problem into these two cases. I'm going to try this technique. And sometimes, if you're lucky, it's a simple problem. It's the right technique and you solve the problem. Sometimes it will have a problem. It would propose an approach which is just complete nonsense. But it looks like a proof. So this is one annoying thing about LLM generated mathematics. So we've had human -generative mathematics as a very low quality. Like, you know, some missions people who don't have the formal training and so forth. But if a human proof is bad, you can tell it's bad pretty quickly.

1:59:10It makes really basic mistakes. But the AI -generative proofs, they can look superficially flawless. And that's partly because that's what the reinforcement learning has, like, you train them to do, to make things to produce text that looks like what is correct. For many applications, it's good enough. So it was often really subtle. And then when you spot them, they're really stupid. Like, no human would have vacuumed that mistake. Yeah, it's actually really frustrating in the programming context because I program a lot. Yeah, when a human makes low quality code, there's something called code smell.

1:59:45Right? You can tell immediately. They're signs. But with the AI -generative code, they're old of us. And then you're right. Eventually, you find an obvious dumb thing that just looks like good code. Yeah, so it's very tricky to... And frustrating for some reason to... Yeah, so the sense of smell. This is one thing that humans have. And there's a metaphorical mathematical smell that is not clear how to get there. Eventually, I mean, so the way alpha zero and so forth make progress and go and chest and so forth is... In some sense, they have developed a sense of smell for go and chest positions.

2:00:29That this position is good for white. That's good for black. They can't initiate why. But just having that sense of smell lets them strategize. So if AI's gained that ability to sort of sense the viability of certain proof strategies. So you can say, I'm going to try to break up this problem into two small subtasks and they can say, oh, this looks good. The two tasks look like they're simpler tasks than your main task. And they still got a good chance of being true. So this is good to try. Or you made the problem worse because each of the two sub -problems is actually harder than your original problem, which is actually what normally happens if you try a random thing to try.

2:01:08Normally, it's very easy to transform a problem into an even harder problem. Very rarely do you transform a simpler problem. So if they can pick up a sense of smell, then they could maybe start competing with a human level method. So this is a hard question, but not competing, but collaborating. If, okay, hypothetical, if I gave you an oracle that was able to do some aspect of what you do, and you could just collaborate with it. Yeah, yeah. What would that oracle... What would you like that oracle to be able to do? Would you like it to maybe be a verifier, like check to the code smut? Like your...

2:01:48Yes, a professor tell, this is the correct... This is a promising fruitful direction. Yeah, yeah, yeah. Or would you like it to generate possible proofs and then you see which one is the right one? Or would you like it to maybe generate different representation, totally different ways of seeing this problem? Yeah, I think all of the above. We don't know how to use these tools, because it's a paradigm that we have not had in the past assistance that are competent enough to understand complex instructions that can work at massive scale, but also unreliable. It's an interesting, unreliable in subtle ways, whereas we were sufficiently good output.

2:02:38It's an interesting combination. You have graduate students who work with who kind of like this, but not at scale. And we had previous software tools that can work at scale, but very narrow. So we have to figure out how to use... So Tim Kowd, you imagine Yaku's foresaw, like in 2000, he was envisioning what mathematics would look like in like two and a half decades. And I'm surprised. Yeah, he wrote in his article, like a hypothetical conversation between a mathematical assistant of the future and himself, you know, a financial problem, and they would have to have a conversation that sometimes the human would propose an idea and the AI would evaluate it and sometimes the AI would propose an idea and sometimes a competition was quite an able to scope and say, okay, I've checked the 100 cases needed here, or the first you said this is for all end objective, but end up to 100 and it looks good so far, or hang on, there's a problem at 8 equals 46.

2:03:42And so just a free form conversation, where you don't know in advance where things are going to go, but just based on, I think ideas that could propose in both sides, calculations could propose in both sides, I've had no solution to it. So I tried to prompt it, okay, so here's the problem, I suggest using this talk, and it'll find this lovely argument using a completely different tool, which eventually goes into the weeds and say, no, no, no, try using this, okay, and I start using this and then you'll go back to the tool that I wanted to do before. And you have to keep railroading it onto the path you want, and I could eventually force it to give the proof I wanted, but it was like hurting cats, and the amount of personal effort I had to take to, not just to prompt it, but I also check it output, because a lot of what it looked like is going to work, I know there's a problem on 917, and basically arguing with it, it was more exhausting than doing it on assisted, so like it, but that's the currency to be hard.

2:04:43I wonder if there's a phase shift that happens to words no longer, feels like hurting cats, and maybe you'll surprise us how quickly that comes. I believe so. So in formalization, I mentioned before that it takes 10 times longer to formalize a proof that I had. With these modern AR tools, it's also just better tooling, the lean developers are doing a great job adding more and more features and making it user -friendly. It's going from 9 to 8 to 7, okay, no big deal, but one day you'll drop a little one. And that's the phase shift, because suddenly it makes sense when you write a paper to write it in lean first, or through a conversation with AI, which is generating lean on the fire with you, and it becomes natural for journals to accept, and maybe they'll offer expedite refereeing.

2:05:36If a paper has already been formalized in lean, they'll just have to referee to comment on the significance of the results and how it connects to the literature, not worry so much about the correctness, because that's been certified. Papers are getting longer and longer in mathematics, and it's harder and harder to get good refereeing for the really long ones, unless they're really important. It is actually an issue, which in formalization is coming in just the right time for this to be. And the easier and easier it gets because of the tooling and all the other factors, then you're going to see much more like math, label, grow, potentially exponentially.

2:06:12It's a virtuous cycle. One phase shift of this type that happened in the past was the adoption of late tech. So late tech is the type of language that all mathematicians use now. So in the past, people used all kinds of word processes and typewriters and whatever. But at some point, late tech became easier to use than all other competitors, and people switched within a few years. It was just a dramatic phase shift. It's a wild out there question, but what year, how far away are we from AI system being a collaborator on a proof that wins the field's model, of that level? Well, it depends on the level of collaboration.

2:06:57No, it deserves to get the field's model. So I can imagine if it was a metal -witting paper having some AI systems in writing it, like they all complete alone, I use it like it speeds up my own writing. You can have a theorem and you have a proof of three cases. And I write down the proof of first case and the autocomplete just suggests that now these other proof of second case could work. And it was exactly correct. That was great. Save me like five to ten minutes of typing. But in that case, the AI system doesn't get the field's model. No.

2:07:37I was talking 20 years, 50 years, 100 years. What do you think? Okay. Sorry, I gave a prediction in print about 2026, which is now next year. There will be mathematical operations with AI. So it's not a fuel's model winning, but like actual research level, like published ideas that are generally by AI. Maybe not the ideas, but at least some of the computations. The verifications. I mean, is that already happened? There are problems that were solved by a complicated process, conversing with AI to propose things and the human goes and tries it. And the contact doesn't work. If I pose a different idea, it's hard to disentangle exactly there are certainly math results which could only have been accomplished because there was a human mathematician and an AI involved.

2:08:33But it's hard to sort of disentangle credit. I mean, these tools, they do not replicate all the skills needed to mathematics, but they can replicate sort of some non -trivial percentage of them, you know, 30, 40 percent. So they can fill in gaps. You know, so coding is a good example. You know, so it's annoying for me to code and Python. I'm not a native professional programmer, but with AI, that the fiction cost of doing it is much reduced. So it fills in that gap for me. AI is getting quite good at literature review. I mean, there's still a problem with hallucinating references that don't exist.

2:09:20But this, I think, is a civil problem. So if you train in the right way and so forth, you can verify using the internet. In a few years, get the point where you have a lemma that you need. And say, anyone proving this lemma before, and we'll do basically a fancy web search AI system. So yeah, there are these six papers where something similar has happened. I mean, you can ask you right now and it'll give you six papers, of which maybe one is legitimate and relevant. One exists, but is not relevant for our hallucinate. It has a non -zero success rate right now, but there's so much garbage, so much the signal noise ratio is so poor that it's most helpful when you already somewhat know the literature.

2:10:08And you just need to be prompted to be reminded of a paper that was really subconsciously in your memory. Or just helping you discover new you were not even aware of, but is the correct citation. Yeah, that's, yeah, that it can sometimes do, but when it does, it's buried in a list of options to which the other are bad. Yeah, I mean, being able to automatically generate a related work section that is correct. Yeah. That's actually a beautiful thing that might be another phase shift because it assigns quite a correctly. Yeah, it does, it breaks you out of the silos of... Yeah, yeah, yeah, yeah. Yeah, now there's a big hump to overcome right now.

2:10:45I mean, it's like self -driving cars. The safety margin has to be really high to be feasible. So yeah, so there's a last mile problem with a lot of AI applications that they can do at all that work 20%, 80 % of the time, but it's still not good enough, and in fact, even worse than good, in some ways. I mean, another way of asking the field's amount of question is, what year do you think you'll wake up and be like real surprised? You read the headline, the news, or something happened that AI did, you know, real breakthrough, something. It doesn't, you know, it feels metal, it even hypotheses. It could be like really just...

2:11:29This Alpha Zero moment would go that way. Right, right. Yeah, this decade, I can see it like making a conjecture between two unrelated two things that people thought was unrelated. Oh, interesting. Generating a conjecture, that's a beautiful conjecture. Yeah, and actually has a real chance of being correct and then meaningful. Because that's actually kind of doable, I suppose, but the word of the data is... No, that would be truly amazing. The current model struggle a lot. I mean, so a version of this is the physicists have a dream of getting the AI to discover new rules of physics. You know, the dreams you just feed it all this data, okay?

2:12:11And it says that he was a new patent that we didn't see before. But it actually even struggled, the current state of the art, even struggles to discover all laws of physics from the data. I mean, or if it does, there's a big concern with contamination that it did only because, like, it's somewhere in a training data, it did some new, you know, boils law or whatever, if you're trying to reconstruct. Part of it is that we don't have the right type of training data for this. Yeah, so for laws of physics, we don't have like a million different universes, we have a million different laws of nature.

2:12:46And a lot of what we're missing in math is actually the negative space of... So we have published things of things that people have been able to prove and conjectures that end up being verified, or we can't examples produced. But we don't have data on things that we've proposed, and they're kind of a good thing to try. But then people quickly realized that it was the wrong conjecture, and then they... They said, oh, but we should actually change our claim to modify it, and in this way to actually make it more plausible. There's a trial and error process, which is a real integral part of human mathematical discovery, which we don't record, because it's embarrassing.

2:13:23We make mistakes, and we only like to publish our wins. And yeah, it has no access to the data to train on. I sometimes joke that basically, you know, I just get an AI, I have to go through a grad school, and actually, you know, go to grad courses, do the assignments, go to office hours, make mistakes, get advice on how to correct the mistakes, and learn from that. Let me ask you, if I may, about Gregory Perlman. You mentioned that you try to be careful in your work, and not let a problem completely consume you. Just you've really fallen love with the problem, and it really cannot rest until you solve it.

2:14:03But you also hasted to add that sometimes this approach actually can be very successful. And the example you gave is Gregory Perlman, who proved the Poincare conjecture, and did so by working alone for seven years with basically little contact with the outside world. Can you explain this one millennial prize problem that's been solved Poincare conjecture, and maybe speak to the journey that Gregory Perlman's been on? All right, so it's a question about curved spaces. That was a good example. So I think it was a 2D surface. In just being round, you could maybe hit the uterus with a hole in it, or kind of many holes.

2:14:41And there are many different topologies, upreory, that the surface could have. Even if you assume that it's bounded and smooth and so forth. So we have figured out how to classify surfaces. As a first approximation, everything is determined. I've got the genus. How many holes it has. So the sphere has a genus zero, a donut has a genus one, and so forth. And one way you can tell the surface is apart, probably the sphere has, which is constantly connected. If you take any closed loop on the sphere, like a big closed row of rope, you can contract it to a point and while staying on the surface. And the sphere has this property.

2:15:13But a torus doesn't, if you're on the torus, and you take a rope that goes around, say, the outer diameter, of course, there's no way it can't get through the hole. There's no way to contract a total point. So it turns out that the sphere is the only surface with this property of contractability. Up to like continuous deformations of the sphere. So some things that I want to call topologically, equivalence to the sphere. So point where you ask the same question, higher dimensions. So this becomes hard to visualize, because surface you can think of as embedded in three dimensions. But as curved free space, we don't have good intuition of four years face to to to to live it.

2:15:49And then there are also three these ways that can't even fit into four dimensions. You need five or six or higher. But anyway, mathematically, you can still pause this question. That if you have a bounded three dimensional space now, which is also has this simply connected property that every loop can be contracted, can you turn it into a three dimensional version of a sphere. And so this is the point where you can actually weirdly in higher dimensions, four and five, it was actually easier. So it was solved first in higher dimensions. There's somehow more room to do the deformation. It's easier to to to move things around to your sphere.

2:16:20But three was really hard. So people try many approaches. There's sort of commentary approaches where you chop up the the surface into little triangles or tetrahedral. And you just try to argue based on how the faces interact each other. There were algebraic approaches. There's various algebraic objects like the fundamental group that you can attach to these homology and homology and and and and all these very fancy tools. They also didn't quite work. But Richard Hamilton proposed a partial differential equations approach. So you take you take so the problem is that you're so you have this object which is of secret is a sphere.

2:16:58But it's given to you in a really in a in a weird way. So I think of a ball that's been kind of crumpled up and twisted and it's not obvious that it's the ball. But like if you have some sort of surface which is which is a deformed sphere, you could you could for example think of it as a surface of a balloon. You could try to inflate it. You pull it up and naturally as you fill the air, the wrinkles were sort of smooth out. And it will turn into a nice round sphere. Unless of course it was a toy or something which is it would get stuck at some point. Like if you inflate a toy or it would there would be a point in the middle when the entering strings to zero you get a singularity and you can't pull up any further.

2:17:40You can't pull any further. So he created this flow which is called witty flow which is a way of taking an arbitrary surface or space and smoothing it out to make it rounder and rounder to make it look like a sphere. And he wanted to show that either this process would give you a sphere or it would create a singularity. Like you very much like how PDs either have global regulatory or finite and blow up. Basically it's almost exactly the same thing. It's all connected. And so and he showed that for two dimensions two dimensional surfaces, if you start with some clear neck no singularities ever formed.

2:18:13You never manage a trouble and you could flow and it will give you a sphere. And so he got a new proof of the two dimensional results. But whether that's a beautiful explanation or reach your flow and its application and its contacts. How difficult is the mathematics here for the 2D case? Yeah these are quite sophisticated equations. On par with the Einstein equations. Yeah. It's slightly simpler but yeah but they were considered hard nonlinear equations to solve. And there's lots of special tricks in 2D that helped. But in 3D the problem was that this equation was like super critical. This has the same problem as Navier's talks.

2:18:48As you blow up maybe the curvature could get concentrated in finite smaller smaller regions. And it looked more and more nonlinear and things just look worse and worse. And we all kinds of singularities that showed up. Some singularities, there's these things called neck pinches where the surface sort of creates the bits like a barbell and it pinches at a point. Some singularities are simpler than you can sort of see what you do next. You just make a snip and then you can turn one surface into two and then ebkel them separately. But there was the prospect that this from really nasty like notive singularities showed up that you couldn't see how to resolve it in any way.

2:19:28But you couldn't do any surgery too. So you need to classify all the singularities. Like what are all the possible ways that things can go wrong? So what Promenade did. First of all he made the problem. Did you turn the problem a supercritical problem to a critical problem? I said before about how the invention of energy, the Hamiltonian, that really clarified Newtonian mechanics. So he introduced something which is now called Promenade's reduced volume and Promenade's entropy. He introduced new quantities kind of like energy that looked the same at every single scale and turned the problem into a critical one where the nonlinearities suddenly looked a lot less scary than they did before.

2:20:07And then he had to solve, you still had to analyze the singularities of this critical problem. And that itself was a problem similar to the way it's happening at work on actually. So on the level of difficulty of that, so he managed to classify all the singularities of this problem and show how to apply surgery to each of these and through that was able to resolve the point where he can jack you. So quite a lot of really ambitious steps and like nothing that a large language model today, for example, could, I mean, at best, I could imagine a more proposing this idea as one of hundreds of different things to try.

2:20:42But the other night would be complete dead ends, but you don't only find out after months of work. He must have had some sense that this was the right track to pursue because it takes years to get from A to B. So you've done like you said actually, you even strictly mathematically, but more broadly in terms of the process, he's done similarly difficult things. What can you infer from the process he was going through because he was doing it alone? What are some low points in a process like that? When you start to like, you've mentioned heart share, like, AI doesn't know when it's failing. What happens to you, you're sitting in your office when you realize the thing you did the last few days, maybe weeks is a failure.

2:21:27Well, for me, I switch to a different problem. So I'm a fox, I'm not a hair truck, but you will generally, that is a break that you can take is to step away and look at it and yeah, I'm a problem. You can modify the problem too. I mean, you can ask them to, if there's a specific thing that's blocking you, that just some bad case keeps showing up for which your tool doesn't work, you can just assume by fear, this bad case doesn't occur. So you do some magical thinking for the, but strategically, okay, for the point to see if the rest of the argument goes through. If it's multiple problems with your approach, then maybe you just give up.

2:22:05But if this is the only problem that we know, but everything else checks out, then it's still worth biting. So yeah, you have to do some some so forward reconnaissance sometimes. That is sometimes productive, just assume like, okay, we'll figure it out. Oh yeah, eventually. Sometimes actually it's even productive to make mistakes. So one of the, I mean, there's a project which actually we wanted some prizes for, we worked on this PD problem, again, actually this blow -off regulatory type problem. And it was considered very hard. Sean McCain, who was another few as my first to work on a special case of this, but he could not solve the general case.

2:22:48And we worked on this problem for two months, and we found, we thought we solved it. We had this, this cute argument that if anything fit, and we were excited, we were planning celebration, we will get together and have champagne or something. And we started writing it up, and one of us, not me, I could, but another call of a, said, oh, in this, in this lemma here, we, we have to estimate these 13 terms that, that true of this expansion, and we estimate 12 of them, but in our notes, I can't find the estimation of 13, can you, can someone supply that? And I said, sure, look at this and I like you, yeah, we didn't cover that we completely omitted this term.

2:23:22And this term, we were in the other 12 terms, put together. In fact, we could not estimate this term. And we tried for a few more months, and all different permutations, and there was always this one thing, one term, that we could not control. And so like, this was very frustrating. But because we had already invested months and months of effort and was already, we stuck at this, which we tried increasingly desperate things and crazy things. And after two years, we found that the picture is somewhat different, but quite a bit from our initial strategy, which didn't actually, didn't generate these problem edit terms and actually solve the problem.

2:23:58So we solved the problem after two years. But if we hadn't had that initial full -storm of nearly solving a problem, we would have given up by month two or something and worked on an easier problem. Every year, no, no, take two years, not sure we would have started the project. Yeah, sometimes actually having the incorrect, you know, it's like Columbus, trying to knew what they had an incorrect version of the measurement of the size of the earth, and he thought he was going to find a new trade route in India. Or at least that was how he sold it in his prospectus. I mean, it could be that he actually secretly knew it, but just on the psychological element, do you have like emotional or like self -doubt, the just overwhelming moments like that?

2:24:41You know, this stuff, it feels like math, it's so engrossing that like it can break you. When you like invest so much yourself in the problem and then it turns out wrong, you could start to be similar way chess has broken some people. Yeah, I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people is as a job, you know, you have a problem and if it doesn't work out, you will call the next one. Yeah, so the fact that you can always move on to another problem, it reduces the emotional connection. I mean, there are cases, you know, so there are certain problems that are what are called macro diseases, where we have just latch on to the one problem and they spend years and years thinking about nothing but that one problem and you know, maybe the career suffers and so forth.

2:25:34I say, oh, but how could this big win this world? You know, once I once I finish this problem out, I will make up for all the years of off lost opportunity. And that's, that's, I mean, occasionally occasionally it works, but I really don't recommend it for people who have got the right fortitude. Yeah, so I've never been super invested in any one problem. One thing that helps is that we don't need to call our problems in advance. Well, when we do crowd proposals, we sound to say we will study this set of problems, but even that we don't promise, definitely by five years, I will supply a proof of all these things.

2:26:13You know, you promise to make some progress or discover some interesting phenomena and maybe you don't solve the problem, but you find some related problem that you can say something new about. And that's a much more feasible task. But I'm sure for you, there's problems like this. You have, you have made so much progress towards the hardest problems in the history of mathematics. So is there, is there a problem that just haunts you? It sits there in the dark corners, you know, twin prime conjecture, Raymond hypothesis, go up conjecture, twin prime, that's, again, so I mean, the problem is like the Raymond hypothesis, those are so far out of reach.

2:26:54I think so. Yeah, there's no even viable stretch, like even if I activate all the cheats that I know of in this book, like there's still no way to get made to be. Like it's, I think it needs a breakthrough in another area of mathematics to happen first, and for someone to recognize that it would be a useful thing to transport into this problem. So we should maybe step back for a little bit and just talk about prime numbers. Okay. So they're often referred to as the atoms of mathematics. Can you just speak to the structure that these atoms have? So the natural numbers have two basic operations, attach them in addition and multiplication.

2:27:35So if you want to generate the natural numbers, you can do one or two things. You can just start with one and add one to itself over and over again. And that generates you the natural numbers. So additively, they're very easy to generate, one, two, three, five, or you can take the prime number, if you want to generate multiple, get it. You can take all the prime numbers, two, three, five, seven, and more, five, more together. And together, they're, because you all the natural numbers except maybe four, one. So there are these two separate ways of thinking about the natural numbers. We added to point of view and a more significant point of view.

2:28:03And separately, they're not so bad. So like any question about that natural was only it was addition, it's rather easy to solve. And any question that only was multiplication is rather easy to solve. But what has been frustrating is that you combine the two together. And suddenly you get extremely rich, I mean, we know that there are statements in numbers that are actually as understandable. There are certain polynomials in some number variables. You know, it's the solution in the natural numbers. And the answer depends on an undecidable statement, like whether the axioms of mathematics are consistent or not.

2:28:38But yeah, but even the simplest problems that combine something more applicative such as the primes with some additives such as shifting by two. Separately, we understand both from well, but if you ask when you shift the prime by two, do you, can you get up, how often can you get another prime? It's been amazingly hard to relate the two. And we should say that the twin prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by two. Now, the interesting thing is that you have been very successful at pushing forward the field and answering these complicated questions of this variety.

2:29:18Like you mentioned, the green tile theorem, it proves that prime numbers contain arithmetic progressions of any length. She's mind blown. You could boost something like that. Right. Yeah. So what if realized because of this type of research, is that there's different patterns have different levels of indestructibility. So what makes the twin prime problem hard is that you could take all the primes in the world, you know, three, five, seven, eleven, so forth. There are some twins in there, eleven and thirteen is a twin prime, perfect in prime and so forth. But you could easily, if you wanted to, um, redact the primes to get rid of, to get rid of the, um, these twins, like, the twins, they chop and they infinitely many of them, but that you recently sparse.

2:30:01Not there's not, I mean, initially it's quite a few, but once you got the millions, the trillions, they become rarer and rarer. And you could actually just, you know, if someone was given access to the database of prime, you just edit it out a few, a few primes here and there, they could make the trim package at your false by just removing like point zero, one percent of the prime, so something, just well chosen to, to, um, to do this. And so you could present a censored database of the primes, which passes all of the statistical tests of the primes, you know, that it obeys things like the problem with theorem and other sex of the primes, but doesn't contain any twin primes anymore.

2:30:37Um, and this is a real obstacle to the twin prime conjecture. It means that any proof strategy to actually find twin primes in the actual primes must fail when applied to these slightly edited primes. And so it must be some very subtle, delicate feature of the primes that you can't just get from like, like, I could get statistical analysis. Okay, so that's all. Yeah. On the other hand, I think progression has turned out to be much more robust. Like, you can take the primes and you can eliminate 99 % of the primes, actually, you know, and you can take any 90 % of anyone and it turns out, and another thing we prove is that you still get has -make progressions.

2:31:17Um, has -make progressions are much, you know, they're like cockroaches of arbitrary length. Yes. Yes. That's crazy. Yeah. So, so this is for, for people who don't know arithmetic progressions, there's a sequence of numbers that differ by some fixed amount. Yeah. But it's again, like, it's an infinite monkey type phenomenon. For any fixed length of your set, you don't get arbitrary, that's the progressions. You only get quite short progressions. But you're saying twin primes not an infinite monkey of the nautilus. I mean, it's a very subtle mon - it's still infinite monkey phenomena. Right. Yeah.

2:31:48If the primes were really genuinely random, if the primes were generated by monkeys, um, then yes, in fact, the infinite monkey theorem would all, but you're saying that twin prime is - it doesn't - you can't use the same tools like the - it doesn't appear random almost. Well, we don't know. Yeah. We - we - we - we believe the primes behave like a random set. And so, the reason why we care about the trim -how -conjecture is - is - is - is - is - a test case for whether we can genuinely - completely say we're - we're 0 % chance of error that the primes behave like a random set. Okay. Random - yeah.

2:32:21Random versions of the primes we know contain twins. At least we're - we're 100 % probably - or probably 10 - to 100 % as you go out for them further. Um, yeah. So the primes we believe that they're random. The reason why primes and progressions are indestructible is that regardless of whether you're saying - it looks random or looks structured, like periodic, in both cases, um, the arithmetic progressions appear, but for different reasons. Um, and this is basically all the ways in which the thing - there are many proofs of - of these sort of arithmetic progression algorithms, and they're all proven by some sort of dichotomy where your set is either structured or random, and in both cases you can say something, and then you pull the two together.

2:33:01Um, but in twin primes, if - if the primes are random, then you're - you're happy, you win, if you're primes are structured, they can be structured in a specific way that eliminates the twins. Uh, and we can't rule out that one conspiracy, and yet you're able to make a azange term progress on the K -2 -po version. Right. Yeah. So, um, the - the one thing about conspiracies is that any one conspiracy theory is really how to disprove. Uh -huh. That, you know, if you believe the word is what - by lizards is that here's some evidence that - that - that - that - that not what I mean, this is what - that - that evidence was taught about the lizards.

2:33:34Yeah. Hi. Hi. You may have encountered this kind of phenomena, yeah. So, it's like, um, a pure - like there's there's almost no way to it, um, definitely without experts, and the same is true in mathematics, but a conspiracy is taught solely devoted to learning twin primes. You know, like you would - you have to also infiltrate other erasian mathematics, sort of, but - but like it could be made consistent, at least as far as we know. But there's a weird phenomena that you can make one - one can do, um, one conspiracy, rule out other conspiracies. So, you know, if the - if the word is one by this is the kind of one by the - it's right.

2:34:08Right. So one unreasonable thing is - is - is - is hard to disprove, but more than one there are - there are tools. Um, so yeah. So for example, we - we know there's infinitely many primes that are, um, uh, no two, which are - so they're infinitely pairs of pride which differ by at most, um, 246 actually is - is - is the code - so there's like a bound yes on the - right. So like, there's twin primes, there's - there's a thing called cousin primes that differ by - by four, um, there's a thing called sexy primes that differ by six. Uh, what are sexy primes? Primes that differ by six. Right. The - the - the name - the name is much less - of course there's much less exciting than the name suggests.

2:34:43Yeah. Um, so you can make a conspiracy rule out one of these, but like once you have like 50 of them, it turns out that you can't rule out all of them at once. It just - it requires too much energy somehow in this conspiracy space. How do you do the bound part? How do you - how do you develop a bound for the different teen primes that - okay, so, uh, there's an infinite number of - so it's ultimately based on what's called the pigeon principle. Um, so the pigeon principle, uh, it's a statement that if you have a number of pigeons and they all have to go into pigeon holes and you have more pigeons than pigeon holes, then one of the pigeon holes has to have at least two pigeons in.

2:35:16So there has to be two pigeons that are close together. So for instance, if you have a hundred numbers and they all range from one to a thousand, um, two of them have to be at most 10 apart. Because you can divide up the numbers from one to a hundred into one hundred pigeon holes. Let's - let's say they have a hundred - if you have 101 numbers, 101 numbers, then two of them have to be a distance less than 10 apart because two of them have to belong to the same pigeon hole. So it - it's a basic, um, basic feature of a basic principle in mathematics. Um, so it doesn't quite work with the primes directly because if the primes get sparser and sparser as you go out, that - that fewer and fewer numbers are pried.

2:35:52But it turns out that there's a way to assign weights to the - to - to numbers. Like, um, so there are numbers that are kind of almost pried, but they're not - they - they don't have no factors at all other than themselves in one. They have very few factors. Um, and it turns out that we understand almost primes a lot better than those end primes. Um, and so, for example, it was known for a long time that they were trying almost primes. Uh, this has been worked out. So almost primes are some we can't understand. So you can actually restrict the attention to a - a suitable set of almost primes. And, uh, whereas the primes are very sparse overall relative to the almost primes, actually, are much less sparse, they make - you can set up a set of almost primes where the primes of density likes a 1%.

2:36:35Um, and that gives you a shot at proving by applying some sort of pigeonhole principle that - that there's pairs of primes that are just nearly 100 - 100 apart. But in order to - with the dendritious primes, you need to get the density of primes, this is something almost as up to - up to a threshold of 50%. Um, once you get up to 50%, you would get dendritious primes. But, uh, unfortunately, there are barriers. Um, we know that - that no matter what kind of goods that are almost primes you pick, the dendritious primes can never get up off 50%. It's called the parody barrier. Um, and I would love to find - yes, so one of my long -term dreams is to find where to breach that barrier.

2:37:08Because it would open up not only to dendritious primes, but the go -back ejecture and many other problems in number theory are commonly blocked because our current techniques would require improvement going beyond this theoretical parody barriers. It's like - it's like pulling past the speed of light. Yeah, so we should say a 20 -prime ejecture. One of the biggest problems in the history of mathematics, go -back and texture also. Um, they feel like - next door neighbors. A -Zert, when days when you felt you saw the path? Oh, yeah. Um, um, yeah. Uh, sometimes you try something and it works super well.

2:37:42Um, you - you - again, again, the sense of mathematical smell, uh, we talked about earlier. Uh, you learn from experience when things are going too well. Because there are certain difficulties that you sort of have to encounter. Um, um, I think the way of - of calling my put it is that, um, you know, like if you are on the streets of New York and you put in a blindfold and you put in a car and - and, uh, after some hours, um, you - the blindfolds often in you and Beijing. Um, you know, I mean, that was too easy somehow. Like, there was no ocean being crossed. Even if you don't know exactly what how - what - what was done, uh, you're suspecting that - that something wasn't right.

2:38:21But is that still in the back of your head to - do you return to these - to the prime - do you return to the prime numbers every once in a while to see? Yeah, when I have nothing better to do, which is less than that time time, which is like it busy with so many things these days. But yeah, when I free time and I'm not - and I'm too frustrated to - to work on my, sort of, view all these such projects and - I also don't want to do my ministerial sub -order. I don't want to do some errands for my family. Um, I can play with these things, um, for fun, uh, and usually you get nowhere. Yeah, you have to - you have to just say, okay, fine.

2:38:52Once again, and nothing happened, I - I will move on. Um, yeah, very occasionally one of these problems I actually solved, uh, well sometimes as you say, you think you solved it and then you're - you're forward for uh, maybe 15 minutes and then you think I should check this because this is too easy to be true and usually is. What's your gut say about when these problems would be, uh, solved? When prime and go back to prime, I think we'll keep getting - keep getting more partial results. Um, it doesn't need at least one - this parity barrier is - is the biggest remaining obstacle. Um, there are simpler versions of the conjecture where we are getting really close.

2:39:31Um, so I think we will - in 10 years we will have it many more much closer results. We may not have the whole thing. Um, yeah, so trend times is somewhat close. Uh, the remanopause is - I have no - I mean, it has happened by accident, I think. So the remanopause is kind of more general conjecture about the distribution of prime numbers, right? Yeah, it's - it's - it's states that sort of viewed modulatively, like for questions only involving replication, no addition. The primes really do behave - well, I had as randomly as - as you could hope. So there's a phenomenon in - probably called sclerobic cancellation that, um, you know, like if you want to pull, say, America, uh, on - on some issue, um, and you - you ask one or two voters and you may have sampled a bad sample and then you get - you get a really imprecise, um, measurement of the full average.

2:40:19But if you sample more and more people, the accuracy gets better and better and at the actual - improves like the square root of the number of - of people you - you sample. So, yeah, if you sample, um, a thousand people, you can get like a two, three percent margin of error. So in the same sense, if you measure the primes in certain multiplicative sense, there's a certain type of statistic you can measure, it's called the remanisator function and it fluctuates up and down. But in some sense, um, as you keep averaging more and more, if you sample more and more, the fluctuation should go down as if they were random.

2:40:48And there's a very precise way to quantify that and the women hypothesis is a very elegant way to capture this. But, um, as with many other asian mathematics, we have very few tools to show that something really genuine behaves like Bidi -Vandam. And this is actually not just a little bit random, but it's - it's asking that behaves as random as a dachyly random set. This - this - this square root cancellation. And we know here because of things related to the parody property, that most of us - usual techniques cannot hope to set all this question. The proof has to come out of left fuel. Yeah, but what that is, no one has any serious proposal.

2:41:30Yeah, and there's various ways to sort of - I said you can modify the primes a little bit and you can just avoid the remanapartis. So like it has to be very delicate. You kind to apply something that has huge margins of error. It has to just barely work. And like, there's like all these pit - pitfalls, they like dodge very adeptly. The prime numbers are just fascinating. Yeah, yeah, yeah. What - what do you, um, most mysterious about the prime numbers? I said that's a good question. So like, conjectually we have a good model of them. I mean, like, as I said, I mean, they have certain patterns like the primes are usually odd, for instance.

2:42:07But, apart from these obvious patterns, they behave very randomly and just assuming that they behave - so there's something called the cream of random model for the primes. But, that - that after a certain point, primes just behave like a random set. And there's various flight modifications as a model. But this has been a very good model. It matches the numerics. It tells us what to predict. Like, I can tell you with complete certainty the trim back and it's true. The random model gives overwhelming odds to this trim. I just can't prove it. Most of our mathematics is optimized for solving things with patterns in them.

2:42:37Um, and the primes have this anti -pattern, um, as doom, also everything really. But we can't prove that. Yeah, I guess it's not mysterious at the price of event. It's kind of random because there's sort of no reason for them to be, um, to have any kind of secret pattern. But, but what is mysterious is - what is the mechanism that really forces the randomness to happen? This is - yes, absent. Another incredibly surprisingly difficult problem is the collots conjecture. Oh, yes. Simple to state. Beautiful to visualize. Yeah. In the simplicity. And yet extremely difficult to solve. And yet you have been able to make progress.

2:43:20Uh, Paul Radar said about the collots conjecture that mathematics may not be ready for such problems. I mean, others have stated that it is an extraordinarily difficult problem completely out of reach. This is in 2010. Out of reach of present -day mathematics and yet, yeah, made some progress. Why is it so difficult to make? Can you actually even explain what it is? Oh, yeah. So it's, it's, it's, it's, it's a problem that you can explain, um, yeah, um, it, um, it helps with some, um, visual aids. But yeah, so you take any natural number like, it's a 13. And you apply the following procedure to it.

2:43:53So if it's even you divide it by two, and it was odd, you multiply it by three and add one. So even numbers get smaller, odd numbers get bigger. So 13 would become 40 because 13 times three is 39 and add one to your 40. So it's a simple process for odd numbers and even numbers. They're both very easy operations. And then you put together that's still reasonably simple. Um, but then you ask what happens when you iterate it. You take the output that you just got and feed it back in. So 13 comes 40. 40 is now even divided by two is 20 20 is still even divided to 10, five and then five times C plus one is 16 and then eight four two one.

2:44:30So, uh, and then for one, it goes one four two one four two one is cycles forever. So this sequence I just described, um, yeah, 13 40 20 tens of both, uh, these are also called hailstorm sequences because there's an over simplified model of, of hailstorm formation, yeah, which is not actually quite correct, but it's so some of thought to high school students as a boxmation is that, um, like a, a little nugget of ice gets gets a nice crystal forms in, and clouded it goes up and down because of the wind and, and sometimes it's cold, it gets, it's a bit more mass and maybe it melts a little bit and this process is going up and down, creates this sort of partially melted ice, which I mentioned because it's hearthstone and eventually it falls down to the earth.

2:45:09So the conjecture is that no matter how high you start up, like you take the number which is in the millions of billions, equal, this process that it goes up if you're hard and down if you even eventually goes down to earth all the time. No matter where you start, it was very simple algorithm you end up at one, right? And you might climb for a while. Right. Yeah, so it's, yeah, if you plot it, um, these sequences, they look like brownie and motion, um, they look like the stock market, you know, they just go up and down in a, in a seemingly random pattern. And in fact, usually that's what happens, that, that if you plug in a random number, you can actually prove that, at least initially that it would look like, um, random walk.

2:45:46And that's actually a random walk with a downward drift. Um, it's like if you're always gambling on a roulette at the casino with odds slightly weighted against you. So sometimes you, you win, sometimes you lose, but over in the long run, you lose a bit more than you win. Um, and so normally your wallet will hit, will go to zero, um, if you just keep playing over and over again. So statistically, it makes sense. Yes. So, so the result that I proved roughly speaking asserts that, that statistically, like nine, 10 % of all inputs would, would drift down to, I'm not, maybe not all the way to one, but to be much, much smaller than what you started.

2:46:22So it's, it's like if I told you that if you go to a casino, most of the time you end up, if you keep playing off long enough, you end up with a smaller amount of any wallet, then when you start, um, that's kind of like the result that I proved. So why is that result like can you continue down that thread to prove the full conjecture? Well, the problem is that, um, my, I used arguments from probability theory. Um, and there's always this exceptional event. So, you know, so in probability, we have this, this low large numbers, um, which tells you things like if you play a casino with a, um, a game at a casino with a losing, um, expectation over time, you are guaranteed, well, almost surely, you know, probably, probability as close to 100 % as you wish, you're guaranteed to lose money.

2:47:06But there's always this exceptional outlier. Like it is mathematically possible that even in the game is, it's also not in your favor, you could just keep winning slightly more than you lose. Very much like how in Navier Stokes, it could be, you know, most of the time, um, your waves can disperse, there could be just one outlier choice of initial conditions that would lead you to and there could be one outlier choice of, um, um, special number that you stick in, that shoe soft infinity, or all other numbers crashed to earth, uh, crash to one. Um, in fact, um, there's some mathematicians, um, who've, Alex Contorovich, for instance, who've proposed that, um, that actually, um, these collapse, uh, iterations are like these similar automata.

2:47:51Um, that, yeah, yeah, actually, if you look at what they happen on, on, in binary, they do, if you look a little bit like, like these game of life type pattenessing, um, and in an analogy to how the game of life can create these massive, like self -opticating objects and so forth, possibly you could create some sort of heavier than air flying machine, uh, a number which is actually encoding this machine, which is, it's just, whose job it is is to encode is to create a version of a server, which is larger. heavier than air machine encoded in a number. Yeah. That flies forever. Yeah. So Confer, in fact, worked on worked on this problem as well.

2:48:25Oh, so Confer, um, so similar, in fact, I, I was more on inspirations for the Navi Stokes project, that Confer studied generalizations of the collapse problem. We're instead of, more than we're three and adding one or divided by two, you have more complicated branching rules, but but instead of having two cases, maybe you have 17 cases and then you go up and down and he showed that once your iteration gets complicated enough, you can actually encode two ring machines and you can actually make these problems undecidable and do things like this. In fact, he met a programming language for, uh, these kind of fractional linear transformations, he could affect track as a play on, on full track.

2:49:01Uh, and he showed that, that he, you couldn't, um, you can program is, it was too incomplete. You could, you could, you could, uh, you could make a program that if you, if your number you inserted in was encoded as a prime, it would sink to zero, it would go down otherwise it would go up, uh, and things like that. Um, so the general cluster problems is, is really, uh, as complicated as all the mathematics. Some of the mystery of the cellular terminal that we talked about, uh, having a mathematical framework to say anything about cellular terminal, maybe the same kind of framework is required. Yeah, yeah, clocks and gesture.

2:49:35Yeah. If you want to do it not statistically, but you really want 100 % of all inputs to, to photo earth. Yeah. So one might be feasible is, is, yeah, assisting 99 %, you know, but you go to one, but like everything, yeah, well, that looks hard. What would you say is out of these within reach famous problems is the hardest problem we have today? Is there any mental hypothesis? We want to, it's up there. Um, Pico's MP is a good one, because like, uh, that's, that's a meta problem. Like if you solve that in the, um, in the positive sense that you can find a Pico's MP algorithm, then potentially this solves a lot of other problems as well.

2:50:13And we should mention some of the conjectures we've been talking about, you know, a lot of stuff is built on top of them now. There's ripple effects. Pico's MP has more ripple effects than basically any other. Right. If the readman hypothesis is disproven, um, that would be a big mental shock to the number of theorists, but it would have follow on effects for, um, cryptography. Um, because a lot of cryptography uses number theory, um, uses number theory, contraptions, evolving primes and so forth. And, um, it relies very much on the intuition that number of theorists have built over many, many years of what operations evolving primes behave randomly and what runs don't.

2:50:52And in particular, um, encryption methods are designed to turn text information on it into text, which is indistinguishable from, um, from random noise. So, um, and hence, we believe to be almost impossible to crack, um, at least mathematically. Um, but, uh, if something has caught our police at the Rebunner hypothesis is wrong, it means that they've had, there are actual patches of the primes that we're not aware of. And if there's one, there's probably going more, um, and suddenly a lot of our crypto systems are in doubt. Yeah. But then, how do you then say stuff about the primes? Yeah. That you're going towards the coaxing conjecture again, um, because if I, do you want it to be random, right?

2:51:41Yes. Yeah. So more broadly, I'm just looking for more tools, more ways to show that, that, yeah, that things are right now. How do you prove it conspiracy doesn't happen? Right. Is there any chance to you that P equals NP? Is there some, can you imagine a possibly universe? It is possible. I mean, this is various scenarios. I mean, there's one way it is technically possible, but in fact, it's never actually implementable. The evidence is sort of slightly pushing in favor of no, that we probably, P is not equit NP. I mean, it seems like it's one of those cases, you know, similar to Rebunner hypothesis, I think the evidence is leaning pretty heavily on the no.

2:52:19Certainly more on the no than on the, yes, the funny thing about P equals NP is that we have also a lot more obstructions than we do for almost any other problem. So while there's evidence, we also have a lot of results ruling out many, many types of approaches to the problem. This is the one thing that the computer science is actually very good at. It's actually saying that certain approaches cannot work, no go theorems. It could be understandable. We don't, yeah, we don't know. There's a funny story I read that when you won the field's medal, somebody from the internet wrote you and asked, you know, what are you going to do now, the one this procedures award.

2:52:55And then you just quickly very humbly said that, you know, this shiny metal is not going to solve any of the problems I'll currently working on. I'm going to keep working on them. It's just, first of all, it's funny to me that you would answer an email in that context. And second of all, it just shows your humility. But anyway, maybe you could speak to the fields metal, but it's another way for me to ask about Gregor at Pearlman, what do you think about him famously declining the fields metal and the millennial prize, which came with a $1 million of prize money? He stated that I'm not interested in money or fame.

2:53:33The prize is completely irrelevant for me if the proof is correct, then no other recognition is needed. Yeah. Well, he's somewhat of an outlier, even among mathematicians who tend to have some idealistic views. I've never met him. I think I'd be interested in him one day, but I'd never have a chance. I know people who met him. He's always had strong views about certain things. It's not like he was completely isolated from the math community. I mean, he would give talks and papers and so forth. But somewhere he just decided not to engage with the rest of the community. He was dissolution or something.

2:54:08I don't know. And he decided to piece out and collect mushrooms in St. Petersburg or something. That's fine. You can do that. That's another flip side. We are not a lot of problems that we solve. Some of them do have practical application. That's great. If you stop thinking about a problem, he hasn't published since in this field, but that's fine. There's many many other people who have done this as well. I guess one thing I didn't realize initially with the field's metal is that it sort of makes you part of the establishment. Most mathematicians, there's just career mathematicians. You just focus on publishing your next paper, maybe getting one, promote one rank and starting a few projects, maybe having to take some students or something.

2:55:00But then suddenly people want your opinion on things and you have to think a little about things that you might just so foolishly say because you know no one's going to listen to you. It's more important now. Is it constraining to you? Are you able to still have fun and be a rebel and try crazy stuff and play with the idea? I have a lot less free time than I had previously. Mostly by choice. I mean, obviously I have the option to sort of decline. I could decline even more. Or I could acquire a repetition of things so unreliable that you would always been asked anymore. I love the different algorithms here.

2:55:40It's always an option. There are things that I don't spend as much time as I do as a postdoc. Just working in the one -pop middle time or fooling around. I still do that a little bit. As you advance in your career, the more soft skills, the more front nodes, all the technical skills to the early stages of a career. As a postdoc, it's a publisher -apparish. You incentivise to basically focus on proving very technical terms, to prove yourself as well as prove the theorems. But then as you get more senior, you have to start mentoring and giving interviews and trying to shape direction of field both research -wise and sometimes you have to do certain things.

2:56:33It's the right social contract because you need to work in the trenches to see what can help mathematicians. The other side of the establishment, the really positive thing, is that you get to be a light that's an inspiration to a lot of young mathematicians or young people that are just interested in mathematics. It's like, it's just how the human mind works. This is where I would probably say that I like to feel it's metal that it does inspire a lot of young people somehow. This is just how human brains work. At the same time, I also want to give respect to somebody like Gregorio Proman who is critical of awards in his mind.

2:57:14Those are his principles and any human that's able for their principles to do the thing that most humans would not be able to do is beautiful to see. Some recognition is necessary and important, but yeah, it's also important to not be able to take over your life and only be concerned about getting the next big award or whatever. Again, you see these people try to only solve like a really big math problems and not work on things that are less sexy, be wish but actually it's still interesting and instructive. As you say, the way the human mind works, it's we understand things better when they're attached to humans.

2:57:57Also, if they're attached to a small number of humans, the way our humans mind is wide, we can comprehend the relationship between the 10 or 20 people. But once you get beyond that, like a 100 people, there's a limit. I pick up the other. We have to simplify the whole matter. The 9 .9 % of humanity becomes the other. Another of these models are incorrect in this course. This is all kinds of problems. So to humanize a subject, if you identify a small number of people, these are representative people of the subject. I say role models, for example. That has some role, but it can also be too much of it can be harmful because it's, I'll be the first to say that my own career trough, rather, is not that of a typical mathematician.

2:58:53I, the very accelerated education. I skipped a lot of classes. I think I was very fortunate mentoring opportunities. I think I was at the right place at the right time. Just because someone doesn't have my trajectory, it doesn't mean that they can't be good mathematicians. I mean, they're very different style. I'm we need people with different style.

2:59:17Sometimes too much focus is given on the person who did the last step to complete a project in mathematics or elsewhere that's really taken centuries or decades with lots and lots of building a lesson previous work. But that's a story that's difficult to tell if you're not an expert. It's easy to just say one person did this one thing. It makes it much simpler history. I think on the whole, it is a hugely positive thing to talk about Steve Jobs as a representative of Apple. When I personally know, and of course, everybody knows the incredible design, the incredible engineering teams, just the individual humans on those teams.

2:59:58They're not a team. They're individual humans on a team, and there's a lot of brilliance but it's just a nice shorthand, like a very like pie. Steve Jobs. Yeah, as a starting point, you see that. As a first approximation, that's how you can read some biographies and then look into which deeper first approximation. That's right. So you were a person to Andrew Wiles at that time. Oh yeah. A professor there. It's a funny moment how history is just all interconnected. At that time, he announced that he proved the Fermat last year. What did you think maybe looking back now with more context about that moment in math history?

3:00:37Yeah, so I was a graduate student at the time. I mean, I vaguely remember, you know, there was press attention and we all had the same we had pigeonholes in the same mail room, you know, so we all particularly on mail and like suddenly Andrew Wiles' mailbox exploded to be all beflowing. That's a good metric. Yeah. You know, so yeah, we all talked about it at T and so forth. I mean, we didn't understand most of us and we understand sort of high level details. Like is it an ongoing project to formalize it in lean? Kevin Bonsid is actually. Yeah. Can we take that small tangent? Is it how difficult does that?

3:01:13Because as I understand the Fermat last, the proof for Fermat last term has like super complicated objects. Yeah. Yeah. It's really difficult to formalize now. Yeah, I guess, you know, the objects that they use, you can define them. So they've been defined in lean. Okay, so just defining what they are can be done. That's really not trivial, but it's been done there. But there's a lot of really basic facts about these objects that have taken decades to prove in that they're in all these different math papers. And so a lot of lots of these have reformed as well. Kevin's, Kevin Bonsid's goal actually has five year grant to formalize Fermat last year.

3:01:52And his aim is that he doesn't think we'll be able to get all the way down to the basic axioms. But you want to formalize it for the point where the only things that he needs to rely on as black boxes are things that were known by 1980 to number zeroes at the time. And then some other person or some other work we're happy done to get from there. So it's a different area mathematics than the type of mathematics I'm used to in analysis, which is kind of my area. The objects we study are kind of much closer to the ground. We study things like prime numbers and functions and things that are within scope of a high school math education to at least define.

3:02:35But then this is very advanced algebraic side of number three, where people have been building structures of one structures for quite a while. And it's a very sturdy structure. It's been very, at the base, it releases extremely hard development with text books and so forth. But it does get to the point where if you haven't taken these years of study and you want to ask about what is going on at level six of this tower. You have to spend quite a bit of time before they can even get to the point where you can see when you recognize. What inspires you about his journey that was similar as we talked about seven years, most working in secret?

3:03:15Yeah, that is a romantic, so it kind of fits with the romantic image that people have of mathematicians to think of, they think of things that are at all as these kind of eccentric wizards or something. So that sort of needs kind of an accentuated that perspective. I mean, it is a great achievement. A hairstyle of solving problems is so different from my own. But which is great. I mean, we need people like that. It's bigger. In terms of like you like the collaborative. I like moving on from a problem if it's giving too much difficulty. But you need the people who have the tenacity and the fearlessness.

3:03:58I've collaborated with people like that where I want to give up because the first approach that we tried to work in the second one didn't approach, they convinced and they have a third, fourth and the fifth approach works. And I have to eat my words. Okay, I didn't think this was going to work, but yes, you were right all along. And we should say for people that don't know, not only are you known for the brilliance of your work, but the incredible productivity, just the number of papers, which are all very high quality. So there's something to be said about being able to jump from top of your topic.

3:04:30Yeah, it works for me. Yeah, I mean, also people who have very productive and they take focus very deeply on, yeah. I think everyone has to find their own workflow. Like one thing which is a shame in mathematics is that we have mathematics, there's sort of a one -size -fits -all approach to teaching mathematics. And so we have a certain curriculum and so forth. I mean, you know, maybe like if you do math competitions or something, you get a slightly different experience. But I think many people, they don't find their native math language until very late, or usually too late, so they stop doing mathematics.

3:05:06And they have a bad experience with a teacher who's trying to teach them one way to do mathematics, they don't like it. My theory is that humans don't come, evolution has not given us a math center or a brain directly. We have a vision center and a language center and some other centers which have evolution as home. But it doesn't, we don't have innate we can repurpose other areas of our brain to do mathematics. So some people have figured out how to use the visual center to do mathematics. And so they think very visually when they do mathematics, some people have repurposed their language center and they think very symbolically.

3:05:49Some people like it's if they are very competitive and they like gaming, there's a type, there's part of your brain, it's very good at solving puzzles and games and that can be repurposed. But like when I talk to my mathematicians, they don't quite think, I can tell that they're using some of different styles of thinking. I mean, not, not, just joint, but they, they may prefer visual. I don't like to prefer visual so much, I need also visual aids myself. Mathematics provides the common language, so we can still talk to each other even if we are thinking in different ways. But you can tell there's a different set of subsystems being used in the thinking process.

3:06:32They take it from past, they're very quick at things that I struggle with and vice versa. And yet they still get to the same goal. And it's beautiful. And, yeah, but I mean, the way we educate, unless you have like a person like Tudor or something, I mean, education, sort of just financial skill has to be mass -produced, you know, you have to teach the 30 kids. You know, they have 30 different styles, you can't, you can't teach 30 different ways. On that topic, what advice would you give to students, young students who are struggling with math and better interested in it, and would like to get better?

3:07:05Is there something in this, yeah, in this complicated educational context, what would you... Yeah, it's a tricky problem. One nice thing is that there are now lots of sources for my faculty in Richmond outside the classroom. So in my day, there are already math competitions. And, you know, they're also like popular math books in the library. But now you have YouTube, there are forums just devoted to solving math puzzles. And math shows up in other places. Like, for example, there are hobbyists who play poker for fun. And they're for very specific reasons, interested in very specific probability questions.

3:07:44And they actually, in this community of amateur probabilists in poker, in chess and baseball. I mean, there's this, there's, yeah. There's math all over the place. And I'm hoping actually with these new sort of tools for lean and so forth, that actually we can incorporate the broader public into math research projects. Like, this is almost, this doesn't happen at all currently. So in the sciences, there's some scope for citizen science, like astronomers, they're, the amateurs who would discover comets and there's biologists, there are people who could identify butterflies and so forth. And in math, there are smaller activities where amateur mathematicians can like discover new primes and so forth.

3:08:30But, but previously, because we have to verify every single contribution, like most mathematical research projects, it would not help to have input from the general public. And I can tell you what it would just be, be time consuming because just error checking and everything. But, you know, one thing about these formalization projects is that they are bringing together more, bringing in more people. So I'm sure that high school students have already contributed to some of these formalizing projects, contributed to math with them. You know, you don't need to be a PhD holder to just work on one atomic thing.

3:09:03There's something about the formalization here that also, as a very first step, opens it up to the programming community too. Yes. The people who are already comfortable with program, it seems like programming is somehow maybe just the feeling, but it feels more accessible to folks than math. Math is seen as this like extreme, especially modern mathematics, seen as this extremely difficult to enter area and programming is not, so that could be just an entry point. You can actually code and it can get results, you know, you can you know, if programming was taught as an almost entirely theoretical subject where you just talk with the computer science, the theory of functions and routines and so forth, and outside of some very specialized homework assignments, you would not like your program like on the weekend for fun.

3:09:54Or yeah, there would be as consider as hard as math. So as I said, there are communities of non -matheticians where they're deploying math for some very specific purpose, you know, like optimizing the poker game. And for them, then math becomes fun for them. What are the advantages you give in general to young people? How to pick a career? How to find themselves? Like, that's a tough tough tough question. Yeah, so there's a lot of certainty now in the world, you know, I mean, there was this period after the war where at least in the West, you know, if you came from a good demographic, you know, like you there was a very stable path through it to a good creator, you go to college, you get an education, you pick one profession and you stick to it, it's becoming much more think of the past.

3:10:43So I think you just have to be adaptable and flexible. I think people will have to get to the goals that are transferable, you know, like learning one specific program in language or one specific subject mathematics or something, it's it's that itself is not a super transferable skill, but sort of knowing how to reason with abstract concepts or how to problem solve and things go wrong and so on. These are things which I think we will still need, even as our tools get better and you know, you would be working with AI and so forth. But actually, you're an interesting case study. I mean, you're like one of the great living mathematicians, right?

3:11:22And then you had a way of doing things, and then all of a sudden you start learning, first of all, you kept learning new fields, but you've learned lean. That's not that's a non trivial thing to learn. Like that's a yeah, that's a for a lot of people. That's extremely uncomfortable leap to take, right? Yeah, a lot of mathematicians. Plus, I've always been interested in new ways to do mathematics. I feel like a lot of the ways we do things right now are inefficient. Like I spend many of my colleagues, you spend a lot of doing very routine computations or doing things that other mathematicians would instantly know how to do and we don't know how to do and why can't we search and get a quick response and so on.

3:12:03So, that's why I've always been interested in exploring new workflows. About four or five years ago, I was on a committee where we had to ask for ideas for interesting workshops to run at a math institute. And at the time Peter Schultz had just formalized one of his new theorems and there's some other developments in computer assisted proof that look quite interesting. And I said, oh, we should we should we should want to workshop on this. This is a pretty good idea. And then I was a bit too enthusiastic about this idea. So I got violent told to actually write it. So I did with a bunch of other people, Kevin Poise and Jordan Ellenberg and and I'm part of other people.

3:12:43And it was it wasn't a nice success. We put together a bunch of mathematicians and computer scientists and other people and we got up the speed and let's say we are. And it was really interesting developments that most mathematicians didn't know what was going on. That lots of nice proofs of concept. You know, it's just so hints of what was going to happen. This was just before chat GBD, but there was even then there was one talk about language models and the potential capability of those in the future. So that got me excited about subject. So I started giving talks about this is some of which more of us just start looking at.

3:13:18Now that I mentioned the run as conference and then chat GBD came out and suddenly AI was everywhere. And so I got interviewed a lot about this topic and in particular the interaction between AI and former professors and I said, yeah, they should be combined. This is this is this is this perfect synergy to happen here. And at some point I realized that I have to actually do not just talk the talk but walk the walk. You know, like, you know, I don't work in machine learning and I don't work in proof formization. And there's a limit to how much I can just rely on authority and saying, I'm a wonder mathematician.

3:13:51Just trust me, you know, when I say that this is going to change my fatigue, but I'm not doing it any when I don't do any of it myself. So I thought I had to actually justify it. A lot of what I get into actually, I don't quite see an advice as how much time I'm going to spend on it. And it's only after I'm sort of waist deep in a project that I realized, but that point I'm committed. Well, that's deeply admirable that you're willing to go into the fray be in some small way beginner, right? Or have some of the sort of challenges that a beginner would, right? And yeah, new concepts, new ways of thinking, also, you know, sucking at a thing that others, I think, I think in that talk, you know, you could be a feels, no matter winning mathematician and undergrad knows something better.

3:14:41Yeah. I think mathematics inherently, I mean, mathematics is so hugely safe. Nobody knows all of modern mathematics. And inevitably we make mistakes. And, you know, you can't cover up your mistakes with just sort of provato and I mean, because people will ask for your proofs and if you don't have the proofs, you know, the proofs. I don't love math. Yeah. So it does keep us honest. I mean, you can still, it's not a perfect panacea, but I think we do have more of a culture of admitting error than because we're forced to all the time. Big ridiculous question. I'm sorry for it once again. Who is the greatest mathematician of all time?

3:15:24Maybe one who's no longer with us. Who are the candidates? Doiler, Gauss, Newton, Ramonnogen, Hilbert? So first of all, I say, I mentioned before, like there's some time dependence. But on the day, yeah, like if you, if you, if you poke cumulatively over time, for example, you clip like, like, sort of like, this is one of the good tenders. And then maybe some unnamed anonymous math, which is before that, you know, whoever came up with the concept of odd numbers, you know, do mathematicians today still feel the impact of Hilbert just directly of everything that's happened in the 20th century. Yeah, Hilbert spaces.

3:16:01We have lots of things that are named up in of course, just the arrangement of mathematics and just the introduction of certain concepts. I mean, 23 problems have been extremely influential. There's some strange power to the declaring which problems. Yeah, hard to solve the statement of the open problems. Yeah, I mean, you know, this is bystander effect everywhere. Like if no one says you should do X, and I just sort of mills around, waiting for somebody else to do something and like nothing gets done. So and like it's the one thing that actually you have to teach undergraduates in mathematics is that you should always try something.

3:16:39So you see a lot of paralysis in an undergraduate trying a math problem. If they recognize that there's a certain technique that can be applied, they will try it. But there are problems for which they see none of their standard techniques obviously applies. And the common reaction is just paralysis. I don't know what to do. I think there's a quote from the Simpsons. I've tried nothing and I'm all that of ideas. So, you know, like the next step then is to try anything like no matter how stupid. In fact, almost the stupid of the better. Which, you know, I think we just almost guaranteed to fail.

3:17:16But the way it fails is going to be instructive. Like it fails because you do not at all take into account this hypothesis. Oh, this hypothesis must be useful. That's a clue. I think you also suggested somewhere this fascinating approach which really stuck with me as they're using it. It really works. I think you said it's called structured procrastination. No, yes. It's when you really don't want to do a thing. The imagine a thing you don't want to do more. Yes. Because that's worse than that. And then in that way you procrastinate by not doing the thing that's worse. Yeah. It's a nice hack. It actually works.

3:17:51Yeah. Yeah. With anything like, psychology is really important. You talk to athletes like now, fun, runners and so forth. They talk about what's the most important thing? Is it the training of edgerman or the diet and so so much of it is psychology. Just tricking yourself to think that the form is feasible. So you're more of a to do it. Is there something our human mind will never be able to comprehend? Well, I sort of, I guess in mathematician, I mean, yeah, it's a bit of a reduction. There must be some of it. It's a bit of a large number that you can't understand. That's the first thing I came to mind.

3:18:30So that, but even broadly, is there, something about our mind that's we're going to be limited even with the help of mathematics? Well, okay. I mean, it's like, how much augmentation are you willing? Like, for example, if I didn't even have pen and paper, I gave, I had no technology whatsoever. Okay. So I'm not allowed blackboard pen and paper. You're already much more limited than you would be. Incredibly limited. Even language, the English language is a technology. It's one that's been very internalized. So you're right. They're really, the formulation of the problem is incorrect because there really is no longer a just a solo human already augmented in extremely complicated, intricate ways, right?

3:19:17Yeah. Yeah. So we're already like a collective intelligence. Yes. Yes. So humanity plural has much more intelligence in principle on his good days. Then the individual humans put together. It can all have less. Okay. But, yeah. So yeah, math, math, math, the mathematical community plural is incredibly super intelligent entity that no single human mathematician can can come close to to replicating. You see it a little bit on these questions analysis sites. So this mathematical flow, which is the math version of stackable flow. And like, sometimes you get like this very quick responses to very difficult questions from the community.

3:19:56And it is a pleasure to watch actually, as an expert, I'm a fan spectator of that of that site, just seeing the brilliance of the different people there. The deaf analogies that people have. And the willingness to engage in the in the rigor and the nuance of the particular questions, pretty cool to watch. It's honestly just fun to watch. What gives you hope about this whole thing we have going on human civilization? I think, yeah, the younger generation is always like really creative and enthusiastic and and inventive. It's a pleasure working with the young students.

3:20:40Progress of science tells us that the problems that used to be really difficult can become extremely you know, can become like trivial to solve. I mean, it was navigation. Just knowing where you work on the planet, was this horrendous problem people would be, would died. And you know, I lost fortunes because they couldn't navigate. And we have devices in our pockets that do this automatically for us. I guess a completely solved problem. So things that I've seen unfeasible for us now could be maybe just homework exercises for me. Yeah, one of the things I find really sad about the finiteness of life is that I won't get to see all the cool things would create as a civilization, you know, because in the next 100 years, 200 years just imagine showing up in 200 years.

3:21:26Yeah, well, I really think this happened. You know, like if you could go back in time and talk to you if you're teenage self or something, you know, I mean, yeah, I'm just the internet and AI. I mean, I get into, they've been internalized and say, yeah, of course, and I can understand a voice and give reason why, you know, slightly incorrect answers to any question, but yeah, this was mind blowing even two years ago. And in the moment, it's hilarious to watch on the internet and so on. The drama, people take everything for granted very quickly. And then they, we humans seem to entertain ourselves with drama.

3:22:01Well, out of anything that's created, somebody needs to take one opinion, another person needs to take an opposite opinion, argue with each other about it. But when you look at the arc of things, I mean, just even in progress of robotics, just to take a step back and be like, wow, this is beautiful. The way humans are able to create this. Yeah, when they infrastructure and the culture is healthy, the community of humans can be so much more intelligent and mature and rational than individuals within it. Well, one place I can always count on rationality is the comment section of your blog, which I'm a fan of.

3:22:36There's a lot of really smart people there. And thank you, of course, for putting those ideas out on the blog. And I can't tell you how honored I am that you would spend your time with me today. I was looking forward to this for a long time. Terri, I'm a huge fan. You inspire me. You inspire millions of people. Thank you so much for talking. Thank you. It was a pleasure. Thanks for listening to this conversation with Terrence Tao to support this podcast. Please check out our sponsors in the description or at lexfreedman .com subsponsors. And now, let me leave you some words from Galileo Galilei.

3:23:14Mathematics is a language with which God has written the universe. Thank you for listening and hope to see you next time.