"The following is a conversation with Terrence Tao. Widely considered to be one of the greatest mathematicians in history. Often referred to as the Mozart of math, he won the Fields Medal and the Break..."

"What was the first really difficult research level math problem that you encountered? One that gave you pause maybe. Well, I mean in your undergraduate um education, you learn about the really hard im..."

"from a little puzzle by the Japanese mathematician Soji Kaya uh in like 1918 or so. Um, so the puzzle is that you you you have um a needle um in on the plane. Um think like like a like driving like on..."

"doing it would pass through every intermediate direction. Is this in the two dimensional plane? This is in the two dimensional plane. Yeah. So we understand everything in two dimensions. So the next q..."

"So this seems like a puzzle. Why is it interesting? So it turns out to be surprisingly connected to a lot of problems in partial differential equations, in number theory, in geometry, comics. For exam..."

"Um and so it's possible to do that. Uh and geometrically what's going on is that there's always s of light rays. Um so like if if if this wave represents light for example um you can imagine this wave..."

"There's a famous unsolved problem called the Navia Stokes regularity problem. So the Navia Stokes equations equations that govern the fluid flow for incompressible fluids like water. The question asks..."

"you speak to the neighbors? So the existence and smoothness like you said millennial prize problem right you've made a lot of progress on this one in 2016 you published a paper finite time blow up for..."

"clay price problem concerns what's called the incompressible navio stokes which governs things like water. There's something called the compressible navio stokes which governs things like air. And tha..."

"shows up a lot in mathematics. Um a basic example is the digits of pi 3.14159 and so forth. The digits look like they have no pattern and we believe they have no pattern. On the long term, you should ..."

"maybe half as as long as as the previous one and then you you could you could actually uh converge to all the energy concentrating in one point in a finite amount of time. Um and that that's uh that s..."

"interacts and only keep the ones that I want. Um, so in particular, um, if, um, if there's a fluid and it could transfer energy from a large Eddie into this small Eddie or this other small Eddie, I wo..."

"technique that is going to work and applying it but you you need to not take the techniques that don't work. Um and for the problems that are really hard, often there are dozens of ways that you might..."

"mathematician ladish skaya she in the 60s shows in two dimensions there is no blow up and in two dimensions the nav equations is what's called critical the effect of transport and the effect of viscos..."

"way. Can you describe this this Yeah. So this came out of of this work of constructing this this this average equation that that blew up. Um so one um as as part of how I had to do this. So there this..."

"and then you you push that in as well. So um by doing that it kind of the energy inches forward scale by scale in such a way that it's always um localized at one scale at a time. Um and then it can re..."

"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 um like if you can imagine kind of a steampu..."

"machine had the ability to mine the planet, create some more materials to smelt them and build more copies of the same machine. Um, then you could colonize a whole planet um over time. Um, so uh if yo..."

"there's always errors um you you have to 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 that it doesn't interfere with the the run..."

"that's a big leap. So there's precedent. I mean um so the the thing about mathematics is that it's really good at um spotting connections between what you think of what you might think of as completel..."

"to create um and gates and or gates for gliders. Like there's this massive ridiculous structure which if you if a if you have a stream of gliders um coming in here and a stream of gliders coming in he..."

"directly take the constructions in the game of life and plunk them in. But again it just it shows it's possible. You know, there's a kind of emergence that happens with these cellular automa. Local ru..."

"and they contain arithmetic progressions of any length um so in that case it's obvious because the the odd numbers are really really structured I can just take 11 13 15 17 I just I can I can easily fi..."

"that there's arithmetic progressions of arbitrary length within a random? Yes. Um have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorists, but occasionally..."

"extremely long random sequence for this to happen. I suppose that's intuitive. It's just infinity. Yeah. Infinity absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity? Well, y..."

"work with there. I wonder how often using infinity uh forces us to deviate from um the physics of reality. Yeah. So there's a lot of pitfalls. Um so you know we we spend a lot of time in undergraduate..."

"Okay. Okay, so it's such a if I have don't have an infinite number of monkeys but but a large finite number of monkeys, how long do I have to wait for H to come out? Um and that's a more quantitative ..."

"like decades earlier and then later on people finize them. So since we mentioned a lot of math and a lot of physics uh what is the difference between mathematics and physics as disciplines as ways of ..."

"um so there's definitely a symbiosis um it's ma I guess mathematics is is unusual among other disciplines is that we start from hypothesis like the axims of a model and ask what conclusions come up fr..."

"this is um this is always the case. You know they're always far apart to begin with. Um but you need one to figure out where to push the other you know. So um if your model is predicting anomalies um ..."

"have powerful computers, only some mathematical things can be um explored numerically. There's something called the comatorial explosion. If you want us to study, for example, Zodius the you want to s..."

"do hope that uh that mathematics will will have a larger experimental component in the future perhaps powered by AI. We'll of course talk about that but in the case of chess and there's a similar thin..."

"always be distinct. Um, but they can get closer um over time. Um, you know, so um and the process of getting closer often means that you you have to discard your initial intuitions. Um so um like astr..."

"fudge factors you know with with enough fudge factors you can explain anything. Um but uh the mathematical point of the model is that um you want to have fewer parameters in your model than data point..."

"at the macro scale and normally because of the common form of explosion you would think that uh the macros scale equations must be like infinitely exponentially more complicated than than the uh the m..."

"global financial crisis was a a famous example of this. Uh people thought that uh um mortgage defaults um had this sort of um Gaussian type behavior that that if you if you ask if a population of of o..."

"topic of universality. Mhm. You're known and celebrated for working across an incredible breadth of mathematics reminiscent of Hilbert a century ago. In fact, the great Fields Medal winning mathematic..."

"they were not really considered related. Um I mean a little bit like you know you could say that that this length was five times this length because you could take five copies of this length and so fo..."

"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. Um and in mathematic..."

"these extra tools. I mean you said that you can be both the hedgehog and and the fox depending on the context depending on the collaboration. So what can you if it's at all possible speak to the diffe..."

"mean I can do that too but uh there are other people who are extremely good at that. Let's step back and uh uh maybe look at the the a bit of a romanticized version of mathematics. Mhm. So, uh I think..."

"gave some examples of well-known theorems and then he would give what he thought was was the extreme proof um in these different aspects. Um and I I just found that really eye opening um that that um ..."

"mean it may seem like a frivolous exercise but it can generate all these insights which if you didn't have this artificial um objective to to to pursue you might not see. What to you is the most beaut..."

"connects together all these tools mathematics. Yeah. Yeah. dynamic structure and complex and complex and um the complex numbers they all considered almost yeah they were all next door neighbors in mat..."

"mathematical um um concepts the analog Hamiltonian that 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 th..."

"analogies are so important, you know. I mean, so yeah, the round earth is not intuitive because we're stuck on it, but you know, but you know, but round objects in general, we have pretty good intuiti..."

"a lot of evidence for this kind of thing. But you know, we're on a round rock. Yeah. Flying through space. Yeah. Yeah. And it's a big leap and you have to take a chain of those leaps the more and more..."

"Yeah, that's uh I highly recommend that. I believe you give 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 t..."

"if you propose axioms then the mathematics lets you follow those a to their conclusions and sometimes you can get quite a quite a long way from you know initial hypothesis. If we can stay in the land ..."

"relatively low in the hierarchy was this thing called the wave maps equation. So it's a wave which at any given point uh is fixed to be like on a sphere. Um so uh I can think of a bunch of arrows in s..."

"techniques to um solve that problem. So part of it is it was um this problem is really nonlinear uh because of the curvature of the sphere. Um this there was a certain nonlinear effect which was a non..."

"fine. You know, you're a young man. I don't ask questions. I I I have to ask about the you know um how do you approach solving difficult problems? What if it's possible to go inside your mind when you..."

"if there are 10 things that are making your life difficult. Find a version of the problem that turns off nine of the difficulties but only keeps one of them. Um and so that um and then that just so yo..."

"sometimes I just have to write everything I know about the problem on the four blackboards and then sit my couch and just sort of see the whole thing. Is it all symbols like notation or is there some ..."

"Let's talk about AI a little bit if we could. So um maybe a good entry point is just talking about computer assisted proofs in general. Can you describe the lean formal proof programming language and ..."

"they made the compiler really small and you can there are several different compilers available for the same for um can you give people some intuition about the the difference between writing on pen a..."

"Oh, I see. Um yeah, so it's it's designed for reliability. So uh modern AIs are not used in it's a disjoint technology. People are beginning to use AIS on top of lean. So when a mathematician tries to..."

"10 15% of the time it doesn't quite work but it it's close enough that I can say oh if I just change it here and here it it will work and then like half the time it gives me complete rubbish um so but..."

"wrote a paper through this theorem with this number 12 and then a few weeks later someone said oh we can actually improve this 12 to an 11 by reworking some of these steps and when this happens with p..."

"Um, and if you program things correctly, um, with sort of 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 if you don't hard..."

"through a math paper. So, one thing that lean really enables is actually collaborating on proofs at a really atomic scale that you really couldn't do in the past. So traditionally with pen and paper u..."

"of thing or are you brains are you focusing on a particular part and you're brainstorming? There's always a brainstorming process first. Yeah. So math research projects sort of by their nature when yo..."

"of Ben Green which called the green tower theorem. Um it's a statement that the primes contain arithmetic progressions of any length. So it was a modification of this theoret and the way we collaborat..."

"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 s..."

"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 distributed. So I told you before about the spli..."

"and tell you what what the project is. Okay. So abstract algebra studies operations like multiplication and addition and the abstract properties. Okay. So multiplication for example is commutive. X * ..."

"became possible with with lean. Um we were hoping to use a lot of AI as well. Um so the project is almost complete. Um so of these 22 million all but two had been settled. Um wow and uh well actually ..."

"Yeah. I know. It's rational. So what we've done for this project is is self-report. So um there are actually standard categories um from the sciences of what types of contributions people give. So the..."

"actually turned out to be not so great for a couple of reasons. So, so one is that if you actually wanted to be considered for tenure or whatever, you could not use this paper in your uh uh as your su..."

"of mathematics. Um so I have to ask you here about the integration of AI into this whole process. So deep mind's alpha proof was trained using reinforcement learning on both failed and successful form..."

"is is an unsolved basically unsolved problem. That is fascinating. Okay. So uh but once you have an informal language they're using um their RL train model. So some something akin to alpha zero that t..."

"time period and and uh um but there are smaller competitions um there are competitions where the the answer is a is a number rather than a long form proof um and that's that's um AI are actually a lot..."

"the date of Easter uh and there was really complicated uh calculations you know but it's all automated been automated for centuries we don't need that anymore you know they used to navigate to do sphe..."

"can you can tell you can tell immediately like, okay, there's signs. But with with a generate code of and then you're right eventually you find an obvious dumb thing that just looks like good code. Ye..."

"foresaw like in in 2000 he was envisioning what mathematics would look like in in actually two and a half decades and that's funny yeah He he wrote in his in in his article like a a a hypothetical con..."

"happened? Yeah. There are there are problems that were solved uh by a complicated process conversing with AI to propose things and the human goes and tries it and the contract doesn't work but it migh..."

"success rate right now, but uh it's there's so much garbage. Uh so much the signal to noise ratio is so poor that it's it's um it's most helpful when you already somewhat know the literature. Um and y..."

"um and in fact even worse than good some ways. I mean another way of asking the Fields metal question is what year do you think you'll wake up and be like real surprised? you read the headline, the ne..."

"mean so um a version of this is um I mean the physicists have a dream of getting the AI to discover new new laws of physics. Um you know the the dream is you just feed it all this data. Okay. and and ..."

"sometimes joke that basically AI has to go through um grad school and actually you know go to grad courses, do the assignments, go to office hours, make mistakes, um get advice on how to correct the m..."

"and and there are also 3D spaces that can't even fit into four dimensions. you need five or six or or higher. But anyway, uh mathematically you can still pose this question that if you have a bounded ..."

"uh this process would give you a sphere or it would create a singularity. Um actually very much like how PDS either they have global regularity or finite blow like basically it's almost exactly the sa..."

"quantities kind of like energy that look the same at every single scale and turned the problem into a critical one where the nonlinearities actually suddenly looked a lot less scary than they did befo..."

"showing up that that that for which your tool doesn't work, you can just assume by fiat this this bad case doesn't occur. So you you do some magical thinking um for the but but but strategically okay ..."

"solve the problem. So we we solve a problem after 2 years but if we hadn't had that initial false dawn of nearly solving a problem we would have given up by month two or something and and worked on an..."

"proof of all these things. you know, or um you promise to make some progress or discover some interesting phenomena. Uh and maybe you don't solve the problem, but you find some related problem that yo..."

"someone to recognize that it that would be a useful thing to transport into this problem. So we we should maybe step back for a little bit and just talk about prime numbers. Okay. So they're often ref..."

"view. Um and separately they're not so bad. Um so like any question about that only was addition is relatively easy to solve and any question that only was multiplication is easy to solve. Um but what..."

"prime conjecture is just that it posits that there are infinitely many pairs of prime numbers that differ by do. Yes. Now the interesting thing is that you have been very successful at pushing forward..."

"there's not I mean 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 you know if if someone was given access to the d..."

"all yeah on the other hand progressions has turned out to be much more robust. um like you can take the primes and you can eliminate 99% of the primes actually you know and you can take take any 99% y..."

"behave like a random set. Okay. Random. Yeah. Random versions of the primes we know contain twins. Um at least with with 100% probability or probably tending to 100% as you go out further and further...."

"the one funny thing about conspiracies is that any one conspiracy theory is really hard to disprove that you know if if you believe the water is won by lizards you say here's some evidence that that i..."

"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 difference between the prize that okay so um that there's..."

"one but they have very few factors. Um and it turns out that we understand almost primes a lot better than primes. Um and so for example it was known for a long time that there were twin almost primes..."

"something and it it works super well. Um you you again again the sense of methac smell uh we talked about earlier uh you learn from experience when things are going too well because there are certain ..."

"primes in a certain multiplicative sense there's a certain type of statistic you can measure and it's called the reman's data function and it fluctuates up and down but in some sense um as you keep av..."

"question. Um the proof has to come out of left field. Um yeah but uh what that is yeah no one has any serious proposal. Um yeah and and there's there's various ways to sort of as I said you can modify..."

"Another incredibly surprisingly difficult problem is the colots's conjecture. Oh yes. simple to state, beautiful to visualize in its simplicity and yet extremely uh difficult to solve and yet you have..."

"bigger. So, 13 will become 40 because 13 * 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 it together. It..."

"random pattern and in Usually that's what happens that that if you plug in a random number, you can actually prove at least initially that it would look like um random walk. Um and that's actually a r..."

"money. But there's always this exceptional outlier. Like it is mathematically possible that even in when the game is is the odds are not in your favor, you could just keep winning slightly more often ..."

"undecidable and and do things like this. In fact, he invented a programming language for uh these kind of fractional linear transformations. He called a factrat as a play on forrat. Uh and he showed t..."

"would have follow on effects for um cryptography um because a lot of cryptography uses number theory um it uses number theory constructions involving primes and so forth and um it relies very much on ..."

"But then how do you then say stuff about the the primes? Yeah. That you're going towards the collect conjecture again. Um because if I I you you want it to be random, right? You want it to be randomly..."

"equal to NP. I mean it seems like it's one of those cases similar similar to reman hypothesis that I think the evidence is le leaning pretty heavily on the no. Certainly more on the no than on on the ..."

"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 um it just shows your humility. But anyway, uh maybe you could speak to the Fields Medal,..."

"shorthand like a very like pi. Yeah. Steve Jobs. Yeah. Yeah. As as a starting point, you know, as a first approximation that's how you and then read some biographies and then look into much deeper. Fi..."

"understand sort of high level details um fact there's an ongoing project to formalize it in lean right Kevin puzzly yeah can can we take that small tangent is it is it how difficult does that cuz as a..."

"things that he needs to rely on as black boxes are things that were known by 1980 to um to number theorist at the time. Um and then some other person some other work would have to done to to to get fr..."

"point where you can see you see something you recognize. What uh inspires you about his journey that we similar as we talked about seven years mostly working in secret? Yeah. Uh that is a romantic uh ..."

"And we should say for people who 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 of very high quality. So ..."

"Um but our other centers are sophisticated enough that different people we we we can repurpose other areas of our brain to do mathematics. So some people have figured out how to use the visual center ..."

"same goal. Um, that's beautiful. And yeah, but I mean the way we educate unless you have like a personalized tutor or something. I mean education sort of just by natural scale has to be mass-produced ..."

"I said, you know, there are communities of non- mathematicians where they're deploying math for some very specific purpose, you know, like like optimizing their poker game and and for them then math b..."

"directly of everything that's happened in the 20th century yeah Hilbert spaces we have lots of things that are named after him of course just the arrangement of mathematics and just the introduction o..."

"you know like the next step then is to try anything like no matter how stupid um and in fact almost as stupid of the better um which you know and one a technique which is almost guaranteed to fail but..."

"you're motivated to do it. Is there something our human mind will never be able to comprehend? Well I sort of as a mathematician I mean you there must be some suffer that you can't understand. That wa..."

"mathematical community plural is is is incredibly super intelligent uh entity um that uh no single human mathematician can can come closer to to replicating. You see it a little bit on these like ques..."

"be really difficult can become extremely you know can become like trivial to solve. you know, I mean, like it was like navigation, you know, just just knowing where you were on the planet was this hor..."

"know that cuz in the next 100 years, 200 years, just imagine showing showing up in 200 years. Yeah. Well, already plenty has happened, you know, like if if you could go back in time and and talk to yo..."