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Terence Tao

lines on the record: 252
first: 2025-06-15
most recent: 2025-06-15
sittings or episodes: 1
sources: podcast

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“Well, the problem is that I used arguments from probability theory. And there's always this exceptional event. So in probability, we have these low, large numbers, which tells you things like if you play a casino over a game at a casino with a losing expectation, over time you are guaranteed, or almost surely with probability as close to 100% as you wish, you're guaranteed to lose money. But there's always this exceptional outlier. It is mathematically possible that even when the game is the odds are not in your favor, you could just keep winning slightly more often than you lose. Very much like how in Nabia Stokes, there could be most of the time your waves can disperse. There could be just one outlier choice of initial conditions that would lead you to blow up. And there could be one outlier choice of a special number that they stick in, that shoes off infinity, while all other numbers crash to Earth.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes. So the result that I proved roughly speaking that statistically, like 99% of all inputs would drift down to maybe not all the way to one, but to be much, much smaller than what you started. So it's like if I told you that if you go to a casino, most of the time you will end up, if you keep playing for long enough, you end up with a smaller amount in your wallet than when you started. That's kind of like those out that I proved.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. Yeah, so it's Yeah, if you plot it these sequences, they look like Brownian motion. They look like the stock market. They just go up and down in a seemingly random pattern. And in fact, usually that's what happens, that if you plug in a random number, you can actually prove at least initially that it would look like random walk. And actually a random walk with a downward drift. It's like if you're always gambling on a roulette at the casino with odds slightly weighted against you. So sometimes you win, sometimes you lose. But over in the long run, you lose a bit more than you win. And so normally your wallet will go to zero if you just keep playing over and over again.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“13, 40, 20, 10, so forth. These are also called hailstone sequences because there's an oversimplified model of hailstone formation, which is not actually quite correct, but some are thought to high school students as a first approximation, is that a little nugget of ice gets a nice crystal forms on cloud, and it goes up and down because of the wind. And sometimes it's cold, it acquires 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 eventually causes hellstone. And eventually it falls out the earth. So the conjecture is that no matter how high you start up, like you take a number which is in the millions or billions, this process that goes up if you're odd and down if you're even eventually goes down to Earth all the time.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, yeah. So it's a problem that you can explain. It helps with some visual aids. But yeah, so you take any natural number, like say 13, and you apply the following procedure to it. So if it's even, you divide it by 2. And if it's odd, you multiply it by 3 and add 1. So even numbers get smaller, odd numbers get bigger. So 13 will become 40 because 13 times 3 is 39, add one, you get 40. So it's a simple process for odd numbers and even numbers, they're both very easy operations. And then you put them together, it's still reasonably simple. 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 becomes 40. 40 is now even divided by 2 is 20. 20 is still even. Divided by 2, 10, 5, and then 5 times 3 plus 1 is 16, and then 8, 4, 2, 1. And then from one, it goes 1, 4, 2, 1, 4, 2, 1. It cycles forever. So the sequence I just described.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Have any kind of secret pattern, but what is mysterious is what is the mechanism that really forces the randomness to happen. And this is just absent.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So that's a good question. So, like conjecturally, we have a good model of them. I mean, as I said, I mean, they have certain patterns, like the primes are usually odd, for instance. But apart from this of obvious patterns, they behave very randomly. And just assuming that they behave, so there's something called the Kramer random model of the primes, that after a certain point, primes just behave like a random set. And there's various flight modifications to this model, but this has been a very good model. It matches the numerics. It tells us what to predict. I can tell you with complete certainty the true prime connector is true. The random model gives overwhelming odds to this true. I just can't prove it. Most of my mathematics is optimized for solving things with patterns in them. And the primes have this anti-pattern, as Doom, almost everything, really. But we can't prove that. I guess it's not mysterious at the primes we ran. It's kind of random because there's no reason for them to be...”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“No one has any serious proposal. And there's various ways of saying that you can modify the primes a little bit and you can destroy the Ruman hypothesis. So it has to be very delicate. You can't apply something that has huge margins of error. It has to just barely work. There's like all these pitfalls that you dodge very adeptly.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And there's a very precise way to quantify that. And the Riemann hypothesis is a very elegant way that captures this. But as with many other ways in mathematics, we have very few tools to show that something really genuinely behaves really random. And this is actually not just a little bit random, but it's asking that it behaves as random as actually random set. This square root cancellation. And we know because of things related to the parity problem, actually, that most of us usual techniques cannot hope to settle this question. The proof has to come out of left field. Yeah, but while that is”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, yeah, it states that sort of viewed multiplicatively. For questions only involving multiplication, no addition. The primes really do behave as randomly as you could hope. So there's a phenomenon in probability called square root cancellation that if you want to poll say America on some issue and you ask one or two voters and you may have sampled a bad sample and you get a really imprecise measurement of the full average. But if you sample more and more people, the accuracy gets better and better and the accuracy improves like the square root of the number of people you sample. So yeah, if you sample a thousand people, you can get like a 2% margin of error. So in the same sense, if you measure the primes in a certain multiplicative sense, there's a certain type of statistic you can measure. It's called the Riemann state of function. And it fluctuates up and down. But in some sense, as you keep averaging more and more, if you sample more and more and more, the fluctuation should go down as if the”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“I think we'll Keep getting more possible results. It does need at least one. This parity barrier is the biggest remaining obstacle. There are simpler versions of the conjecture where we are getting really close. So I think we will, in 10 years, we will have many more, much closer results. We may not have the whole thing. So trend primes is somewhat close. Riemann hypothesis, I have no clue. It has to happen by accident.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“When I have nothing better to do, which is less and less, which is busy with so many things these days. But yeah, when I have free time and I'm not and I'm too frustrated to work on my sort of view or research projects and I also don't want to do my ministry. I don't want to do some errand for my family. I can play with these things for fun. And usually you get nowhere. You have to learn to just say, okay, fine. Once again, nothing happened. I will move on. Very occasionally one of these problems I actually solved, or sometimes, as you say, you think you solved it and then you forward for maybe 15 minutes and then you think I should check this because this is too easy too good to be true and it usually is.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, yeah. Yeah, sometimes you try something and it works super well. Again, the sense of methodical smell we talked about earlier, you learn from experience when things are going too well. Because there are certain difficulties that you sort of have to encounter. I think the way a colleague might put it is that if you are on the streets of New York and you put in a blindfold and you put in a car and after some hours the blindfold is off and you're in Beijing you know I mean that was too easy somehow like like there was no ocean being crossed even if you don't know exactly what how What was done? You're suspecting that something wasn't right”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Suitable set of almost primes. And whereas the primes are very sparse. Overall, relative to the almost prime, so actually much less sparse. You can set up a set of almost primes where the primes of density like say 1%. And that gives you a shot at proving by applying some sort of original principle that there's PESR primes that are distantly about. But in order to prove the Trinity conjecture, you need to get the density of primitive side that also has up to a threshold of 50%. Once you get up to 50%, you will get trin drives. But unfortunately, there are barriers. We know that no matter what kind of goods that are almost primaries you pick, the density of price can never get above 50%. It's called the parity barrier. And I would love to find one of my long-term dreams is to find a way to breach that barrier because it would open up not only to trend power conjecture, but the go back conjecture and many other problems in number theory are currently blocked because our current techniques would require going beyond this theoretical parity barrier. It's like going fast in the speed of light.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Because you can divide up the norms from 1 to 100 into 100 pigeon holes. Let's say you have 101 numbers. If you have 101 numbers, then two of them have to be distanced less than 10, because two of them had to belong to the same pigeon. So it's a basic feature of a basic principle in mathematics. So it doesn't quite work with the primes directive because if the primes get sparser and sparser as you go out, that fewer and fewer numbers are prime. But it turns out that there's a way to assign weights to numbers. So there are numbers that are kind of almost prime, but they don't have no factors at all other than themselves and one. They have very few factors. And it turns out that we understand almost primes a lot better than understand primes. And so, for example, it was known for a long time that they would trin almost primes. This has been worked out. So almost primes are something we kind of understand. So you can actually restrict attention to”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So it's ultimately based on what's called the pigeonhole principle. So the pigeonhole principle, it's the statement that if you have a number of pigeons and they all have to go into pigeonholes and you have more pigeons than pigeonholes, then one of the pigeon holes has to have at least two pigeons there. So there has to be two pigeons that are close together. So for instance, if you have 100 numbers and they all range from one to a thousand, two of them have to be at most 10 apart.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Sexy primes that differ by six. The name is much less exciting than the name suggested. So you can make a conspiracy, rule out one of these, but once you have like 50 of them, it turns out that you can't rule out all of them at once. It requires too much energy somehow in this conspiracy space.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, so like this twin primes, think of cousin primes that differ by four, this thing called sexy primes that differ by six.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So, one unreasonable thing is how to disprove it. But more than one, there are tools. So yeah, so for example, we know there's infinitely many primes that are Node 2, so the infinite pair, which differ by most 246 actually is the code.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, yeah. So the funny thing about conspiracies is that any one conspiracy theory is really hard to disprove. That if you believe the word is one by lizards, you say, here's some evidence that it's not by lizards, but that evidence was planted by lizards. You may have encountered this kind of phenomenon. There's almost no way to... Definitively without, and the same is true in mathematics, that a constructor says solely devoted to limiting twin primes. You have to also infiltrate other areas of mathematics. But it could be made consistent at least as far as we know. But there's a weird phenomenon that you can make one conspiracy rule out other conspiracies. So if the world is run by this, they can't also be won by aliens. Right.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And then you put the two together. But in twin primes, if the primes are random, then you're happy, you win. But if your primes are structured, they can be structured in a specific way that eliminates the twins. And we can't rule out that one conspiracy.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“We don't know, yeah. We believe the prize behaves like a random set. And so the reason why we care about the Premier conjecture is a test case for whether we can genuinely, confidently say with 0% chance of error that the priors behave like a random set. Random versions of the primes we know contain twins, at least with 100% probability, or probably tending to 100% as you go out further and further. Yeah, so the primers, we believe that they're random. The reason why asthma progressions are indestructible is that regardless of whether it looks random or looks structured, like periodic, in both cases, athmic progressions appear, but for different reasons. And this is basically all the ways in which there are many proofs of these sort of mathematical progression epithems, 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.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. If the primes were really genuinely random, if the primes were generated by monkeys, then yes, in fact, the infinite monkey theorem would.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah. It's again an infinite monkey type phenomenon for any fixed length of your set, you don't get arbitrary length of progressions. You only get quite short progressions.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah. On the other hand, ethnic progressionist has turned out to be much more robust. You can take the primes and you can eliminate 99% of the primes, actually. And you can take any 90% you want. And it turns out, and another thing we prove is that you still get azmic regressions. Athmic progressions are much, you know, they're like cockroaches.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“To do this. And so you could present a sensitive base of the prize which passes all Doesn't contain any shrimp farms anymore. And this is a real obstacle for the trim farm conjecture. It means that any proof strategy to actually twin tribes in the actual primes must fail when applied to these slightly edited tribes. And so it must use some very subtle, delicate feature of the primes that you can't just get from aggregate statistical analysis.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right. So, what we've realized because of this type of research is that different patterns have different levels of instructibility. So what makes the Trincum problem hard is that if you take all the primes in the world, three, five, seven, eleven, so forth, there are some twins in there, 11 and 13 is a twin prime, a pair of twin primes and so forth. But you could easily, if you wanted to redact the primes to get rid of these twins. The twins, they show up and there are infinitely many of them, but they're actually reasonably sparse. Initially there's quite a few, but once you got to the millions, the trillions, they become rarer and rarer. And you could actually just, if someone was given access to the database of primes, you just edit out a few primes here and there, they could make the TrinPack conjecture false by just removing like 0.01% of the primes or something. Just well chosen to”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“There are certain polynomials in some number of variables. Is there a solution in the natural numbers? And the asset depends on an undecidable statement, like whether the axioms of mathematics are consistent or not. But even the simplest problems I combine, sound more applicative, such as the primes with something additive, such as shifting by two, separately we understand both well. But if you ask to prime by two, can you get up? How often can you get another prime? It's been amazingly hard to relate the two.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So the natural numbers have two basic operations attached to some addition and multiplication. So if you want to generate the natural numbers, you can do one of two things. You can just start with one and add one to itself over and over again. And that generates you the national numbers. So additively, they're very easy to generate. One, two, three, or five. Or you can take the prime number. If you want to generate multiplicatively, you can take all the prime numbers 2, 3, 5, 7, and multiply them all together. And together, that gives you all the national numbers, except maybe for one. So there are these two separate ways of thinking about the natural numbers added to point of view and a multiplicative point of view. And separately, they're not so bad. So any question about that international only was addition is relatively easy to solve. And any question that only was multiplication. But what has been frustrating is that you combine the two together. And suddenly you get this extremely rich. I mean, we know that there are stables in number theory that are actually as undecidable.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, there's no even viable strategy. Like even if I Activate all my other cheats that I know of This is still no way to get me to beat. 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.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“For people who fought 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 grant proposals, we sort of say we will study this set of problems. But even there, we don't promise definitely by five years we will supply a proof of all these things. You promised 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.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, I think different mathematicians have different levels of emotional investment in what they do. I mean, I think for some people it's a job. You have a problem and if it doesn't work out, you go on the next one. Yeah, so the fact that you can always move on to another problem, it reduces the emotional connection. There are cases, you know, so there are certain problems that are what I call diseases where just latch on to one problem and they spend years and years thinking about nothing but that one problem. And maybe the career suffers and so forth. But this big win, once I finish this problem, I will make up for all the years of lost opportunity. I mean, occasionally, occasionally it works. I really don't recommend it.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“The problem, we would have given up by month two or something and worked on an easier problem. If we had known it would take two years, not sure we would have started the project. Sometimes actually having the incorrect, you know, it's like Columbus traveling in the New World, they had an incorrect version of the measurement of the size of the Earth. He thought he was going to find a new trade route to India. Or at least that was how he sold it in his prospectus. I mean, it could be that he actually secretly knew.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“We have to estimate these 13 terms that show up in this expansion. And we estimate 12 of them, but in our notes, I can't find the estimation of the 13th, can you? Can someone supply that? And I said, sure, I'll look at this. And yeah, we didn't, we completely omitted this term. And this term turned out to be worse than 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 this was very frustrating. But because we had already invested months and months of effort investment ready, we stuck at this, which we tried increasingly desperate things and crazy things. And after two years, we found an approach which was somewhat different, but quite a bit from our initial strategy, which did actually generate these problematic terms and actually solve the problem. So we solve the problem after two years. But if we hadn't had that initial full storm of nearly some of”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Sometimes actually it's even productive to make mistakes. So one of the projects which actually we won some prizes for four other people. We worked on the SPDE problem. Again, actually this blow of regularity type problem. And it was considered very hard. Jean Bougain, who was another fields methodist, who worked on a special case of this, but he could not solve the general case. And we worked on this problem for two months. And we thought we solved it. We had this cute argument that if anything fit and we were excited, we were planning celebrationary to all get together and have champagne or something.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“You can modify the problem too. I mean, you can ask them if there's a specific thing that's blocking you, that there's some bad case keeps showing up for which your tool doesn't work. You can just assume by fiat this bad case doesn't occur. So you do some magical thinking, 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. But if this is the only problem, then everything else checks out, then it's still worth fighting. So yeah, you have to do some sort of forward reconnaissance sometimes.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Of that. So he managed to classify all the singularities of this problem and show how to apply surgery to each of these and who that was able to resolve the Point College conjecture. So a lot of really ambitious steps and nothing that a large language model today, for example, could, I mean, at best I could imagine a model proposing this idea as one of hundreds of different things to try. But the other 99 would be complete dead ends, but you'd 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.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Singularities showed up that you couldn't see how to resolve in any way, that you couldn't do any surgery to. So you need to classify all the singularities. Like what are all the possible ways that things can go wrong? So what Perlman did was, first of all, he made the problem, he turned the problem a supercritical problem to a critical problem. I said before about how the invention of energy, the Hamiltonian, really clarified Newtonian mechanics. So he introduced something which is now called permanent reduced volume and permanence entropy. He introduced new quantities kind of like energy that looked the same at every single scale and turned the pump into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did before. And then he had to solve, he still had to analyze the singularities of this critical problem. And that itself was a problem similar to this Wayfaff thing I worked on, actually. So on the level of difficulty,”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, these are quite sophisticated equations on par with the Einstein equations that are slightly simpler, 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 actually supercritical. It had the same promise as Nabier Stokes. As you blow up, maybe the curvature could get concentrated in finer and smaller and smaller regions. And it looked more and more nonlinear and things just looked worse and worse. And there could be all kinds of singularities that showed up. Some singularities, like there's these things called neck pinches where the surface sort of behaves like a barbell and it pinches at a point. Some singularities are simple enough that you can sort of see what to do next. You just make a snip and then you can turn one surface into two and evolve them separately. But the prospect that there's some really nasty, like non-”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Smooth out, and it will turn into a nice round sphere unless, of course, it was a toy or something, in which case it would get stuck at some point. Like if you instead of us, there'll be a point in the middle. When the inner ring shrinks to zero, you get a singularity and you can't blow up any further. You can't flow any further. So he created this float, which is now called Ritchie 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. He wanted to show that either this process would give you a sphere or it would create a singularity very much like how PDEs either they have global regularity or Finite-Hein block. Basically, it's almost exactly the same thing. It's all connected. And he showed that for two dimensions, two dimensional surfaces, if you started something connected, no singularities ever formed. You never ran into trouble and you could flow and it will give you a sphere. So he got a new proof of the two dimensional.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Argue based on how the faces interact with each other. There were algebraic approaches, there's various algebraic objects, I think it's called the fundamental group that you can attach to these homology and cohomology and all these very fancy tools. They also didn't quite work. But Richard Hamilton's proposed partial differential equations approach. So you take the problem is that you have this object which is so secretly the sphere, but it's given to you in a really weird way. So I think a ball has been kind of crumpled up and twisted. And it's not obvious that it's a ball. But if you have some sort of surface which is a deformed sphere, you could, for example, think of it as a surface of a balloon. You could try to inflate it, you blow it up. And naturally, as you fill the air, the wrinkles were...”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So it turns out that the sphere is the only surface with this property of contractibility. Equivalent of the sphere. So Poincoi asked the same question high dimensions. So it becomes hard to visualize because the surface you can think of as embedded in three dimensions, but a curved free space, we don't have good intuition of 4D space to live in. And then there are also 3D spaces that can't even fit into four dimensions. You need five or six or higher. But anyway, mathematically, you can still pose this question that if you have a bounded three-dimensional space now, which 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 query conjecture. 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 move things around into a sphere. But three was really hard. So people tried many approaches. There's sort of commentory approaches where you chop up the surface into little triangles or tetrahedral and you just try to”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“All right, so it's a question about curbspaces. I think you think it was a 2D surface interesting round you could maybe be a torus with a hole in it or could have many holes. And there are many different topologies a priori that a 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's determined by some called the genus, how many holes it has. So a sphere has genus zero, a donut has genus one and so forth. And one way you can tell these surfaces apart, probably the sphere has, which is called simply connected. If you take any closed loop on the sphere, like a big closed rope, you can contract it to a point and while staying on the surface. And the sphere has this property, but a torus doesn't. If you're on a torus and you take a rope that goes around, say the outer diameter, there's no way it can't get through the hole. There's no way to contract it to a point.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“And 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 ended up being verified or maybe counter examples produced. But we don't have data on things that were 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 said, oh, but we should actually change our claim to modify it 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. We make mistakes and we only like to publish our wins. And the AI has no access to this data to train on. I sometimes joke that basically AI has to go through a grad school and actually go to grad courses, do the assignment.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, no, that would be truly amazing. The current models struggle a lot. I mean, so a version of this is, I mean, the physicists have a dream of getting the AIs to discover new laws of physics. The dream is you just feed it all this data. And here's a new patent that we didn't see before. But it actually even struggled, the current state of the art even struggles to discover old laws of physics from the data. Or if it does, there's a big concern of contamination. It did it only because it's somewhere in his training data iterativity, somehow new Boyle's law or whatever you're trying to reconstruct. Of it is we don't have the right type of training data for this. So for laws of physics, we don't have a million different universes or a million different laws of nature.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, right. Yeah, this decade, I can see it like making a conjecture between two unrelated two things that people thought was unrelated.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, yeah, yeah. Yeah, no, there's a big hump to overcome right now. I 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 develop tools that work 20%, 80% of the time, but it's still not good enough. And in fact, even worse than good in some ways.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Has happened. You can ask it right now and it'll give you six papers, of which maybe one is legitimate and relevant. One exists but is not relevant and four are hallucinated. It has a non-zero success rate right now, but there's so much garbage, so much, the signal-to-noise ratio is so poor that it's most helpful when you're already somewhat knowledge literature and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“So coding is a good example. So it's annoying for me to code in Python. I'm not a native, I'm not a professional programmer. But with AI, the friction 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, it's still a problem with hallucinating the references that don't exist. But this, I think, is a civil war promise. If you train in the right way and so forth, and verify using the internet, you should in a few years get to the point where you have a lemma that you need and has anyone proven this lemma before and it will do basically a fancy web search AI assistant say yeah yeah there are these six papers where something”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source
“Has it already happened? Yeah, 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. But it might pose a different idea. It's hard to disentangle exactly. There are certainly math results which could only have been accomplished because a human mathematician and an AI involved. It's hard to sort of disentangle credit. I mean, these tools do not replicate all the skills needed to do mathematics, but they can replicate sort of some non-trivial percentage of them, 30-40%. So they can fill in gaps.”
2025-06-15 · Lex Fridman Podcast · #472 – Terence Tao: Hardest Problems in Mathematics, Physics & the Future of AI · IDENTIFIED FROM THE TRANSCRIPT · source