"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..."

"So the next question is what happens in three dimensions. So suppose like the Hubble space telescope is tube in space and you want to observe every single star in the universe. So you want to rotate t..."

"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..."

"the Navia Stokes regularity problem. So the Navia Stokes equations equations that govern the fluid flow for incompressible fluids like water. The question asks if you start with a smooth velocity fiel..."

"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..."

"statistically it's extremely unlikely but mathematically it's possible that this can happen and we can't rule it out. Um and this is a situation that shows up a lot in mathematics. Um a basic example ..."

"and each time it does this uh it takes 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 amoun..."

"the past there have been many attempts to try to obtain what's called global regularity for Navio Stokes which is the opposite of final time blow up that velocity say smooth and it all failed there wa..."

"had you must use some feature of the true equation which which my artificial equation um does not satisfy. So it it rules out certain um certain approaches. So um the thing about math is is it's not j..."

"So the viscosity are the things that calm things down. Um and so this is um um this is why the problem is hard in two dimensions. So the Soviet mathematician ladish skaya she in the 60s shows in two d..."

"Lots of really strange things are going on at very fine scales. So, whenever there is some huge source of nonlinearity, yeah, that can create a huge problem for predicting what's going to happen. Yeah..."

"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..."

"locks. So um 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 from ..."

"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..."

"that this thing is possible. Um there are other groups who are now pursuing ways to make navis blow up which are nowhere near as ridiculously complicated as this. Um um they they actually are pursuing..."

"game of life. Um so for example they discovered this thing called a 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 a..."

"It's so incredible. A lot of this was like community crowdsourced by like amateur mathematicians actually. Um so I knew about that that that work and so that is part of what inspired me to propose the..."

"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..."

"Um have you heard of the infinite monkey theorem? Usually mathematicians give boring names to theorists, but occasionally they they give colorful names. Yes. The popular version of the infinite monkey..."

"It's just infinity. Yeah. Infinity absorbs a lot of sins. Yeah. How are we humans supposed to deal with infinity? Well, you can think of infinity as as as just an abstraction of um a finite number for..."

"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..."

"started taking results that are true in infinite limits and what's called finetizing them. Um so you know that something's true eventually but um you don't know when. Now give me a rate. Okay. Okay, s..."

"infinite ones are found first usually 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 ..."

"don't yet have the observation we'd like to a prediction. Um and then we have these simplified models sometimes making unrealistic assumptions you know spherical cow type things. Those are the mathema..."

"the path to get there. Um a lot there there's a lot less sort of speculation about suppose I did this, what would happen? Um you know, planning and and and modeling um uh speculative fiction maybe is ..."

"where to push the other you know. So um if your model is predicting anomalies um that are not picked up by experiment that tells experimenters where to look you know um to to to to find more data to r..."

"yeah I mean theoretical mathematics was just much more successful I mean because doing complicated mathematical computations is uh was just not not feasible until very recently. Uh and even nowadays, ..."

"computation. Uh chess is another um famous example. The number of chess positions uh we can't get a computer to fully explore. But now we have AI um um we have tools to explore this space not with 100..."

"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 thing in mathematics that I don't believe it's providing a kind of formal ..."

"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..."

"itself is this evidence that this non-constant and uh the explanation behind why that is it's catching up. Um it's catching up. I mean it's still you know the dark matter or dark energy this this kind..."

"have. Um you can think of of a theory like one way to think about um physical math theory theory is it's a compression of of the universe um and data compression. So you know you have these pabytes of..."

"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..."

"simpler toy models where we do um have a good understanding of why univers universality occurs. Um um most basic one is is the central limit theorem that explains why the bell curve shows up everywher..."

"that this is systemic risk is actually a much bigger issue and uh just because the model is pretty uh and nice uh it may not match reality. Right. So, so the mathematics of working out what models do ..."

"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..."

"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..."

"that um so the if ei= minus one um so yeah people oh uses all the fundamental constants okay that that's I mean that's cute um but but to me so the exponential function was interested by oil to measur..."

"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..."

"know they saw the things they could they could measure they could measure mass and acceleration and force and so forth and so Newtonian mechanics for example F= ma was the famous Newton second law of ..."

"secretly behind the scenes in classical mechanics also is the key uh object in um um in quantum mechanics that there's there's also an object called Hamiltonian. It's a different type of object. It's ..."

"don't know how to what to replace it with um We don't actually have the mathematical um um concepts the analog Hamiltonian that sort of organized everything. Does your gut say that there is a theory o..."

"made progress before. There's no reason why we should stop. Do you think it will be a mathematician that develops uh theory of everything? What often happens is that when the physicists need uh um som..."

"Dwick's unreasonable effectiveness of mathematics. I think the theories that work well to explain the universe tend to also involve the same mathematical objects that work well to solve mathematical p..."

"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..."

"equation. I found this transformation while visiting my aunt in Australia and I was trying to understand the dynamics of all these fields and I I couldn't do it with pen and paper. Um and I had not en..."

"sort of uh I call it cheating strategically um so u the the beauty of mathematics is that is that you get to change the rule change the problem change the rules as you wish this you don't get to do th..."

"a lot of these Hong Kong action movies. Um it's from a culture. Um and uh one thing is that every time there was a fight scene, you know, so maybe the the hero will get swarmed by a hundred bad guy go..."

"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 ..."

"correct if you trust the compiler of but 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 th..."

"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..."

"quick process. You um, 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 mo..."

"and paper um when you want to collaborate with another mathematician um either you do it as a blackboard where you um you can really interact but if you're doing it sort of by email or something um ba..."

"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..."

"three progressions. They didn't work for longer progressions. Um but I had these techniques coming from agotic theory which is something that I had been playing with and and uh I knew better than Ben ..."

"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 system proof checker essentially to make sure ..."

"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 of some mathematical pattern or something you need some code to write out the patte..."

"* Y * Z. Um so um these operations obey some laws that don't obey others. For example, x * x is not always equal to x. So that law is not always true. So given any any operation, it obeys some laws an..."

"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..."

"know using the internet um you know um you should in a few years get to the point where you you have a a lema that you need and uh we say has anyone proven this lema before and it will do basically a ..."

"it's it's um it's most helpful when you already somewhat know the literature. Um and you just need to be prompted to be reminded of a paper that was already subconsciously in your memory versus helpin..."

"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..."

"All right. So it's it's a question about curb spaces. Earth is a good example. So you can think of a 2D surface in being round could maybe be a Taurus with a hole in it or it can have many holes and t..."

"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 ..."

"you so you have this object which is so secretly is a sphere but it's given to you in a in a really um in in a weird way. So like like think of a ball that's been kind of crumpled up and twisted and i..."

"of the two dimensional result but by the way that's a beautiful explanation of reach flow and its application in this context how difficult is the mathematics here like for the 2D case is it yeah thes..."

"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..."

"in terms of the process he's done similarly difficult things what 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..."

"still worth fighting um so yeah you have to do some some sort of forward reconnaissance sometimes to uh you know and that is sometimes productive to assume like okay we'll figure it out oh yeah yeah e..."

"at this we we tried increasingly desperate things and and crazy things um and after two is we found an approach which was actually somewhat different by quite a bit from our initial um strategy which ..."

"connection. I mean there are cases you know so there are certain problems that are what I call back diseases where where where just latch on to that one problem and they spend years and years thinking..."

"need to call our problems in advance. Uh um well uh when we do grant proposals we s say we we will study this set of problems. But even then we don't promise definitely by 5 years I will supply a proo..."

"Yeah. there's no even viable strate like even if I activate all my all the cheats that I know of in this problem like it there's just still no way to get me to be um like it's um I think it needs a br..."

"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...."

"you know like it would you have to also infiltrate other areas of mathematics to sort of but but like it could be made consistent at least as far as we know but there's a weird phenomenon that you can..."

"between the prize that okay so um that there's an infinite number of so it's ultimately based on what's called the pigeon hole principle um so the pigeon hole principle uh it's a statement that if you..."

"sparse. They make um you can set up a set of almost primes where the primes have density like say 1%. Um and that gives you a shot at proving by applying some sort of original principle that that thos..."

"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..."

"together. It'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 becomes 40. 40 is now even divide by 2 ..."

"to prove the full conjecture? Well, the problem is that um my I I used arguments from probability theory um and there's always this exceptional event. So you know, so in probability we have this this ..."

"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..."

"Um POS 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 PMP algorithm that potentially this solves a lot of othe..."

"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 ..."

"keep 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 um it just shows your humility. But anyway, uh ma..."

"see these people try to only solve like a really big math problems and not work on on on things that are less uh sexy if you wish but but but actually still interesting and instructive as you say like..."

"to humanize a subject you know if you identify a small number of people and say you know these representative people of the subject role models for example um that has some role um but it can also be ..."

"people of a different style. Um and you know even if and sometimes too much focus is given on the on the person who does the last step to complete um a project in mathematics or elsewhere that's that'..."

"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 ..."

"to just work on one atomic thing. There's something about the formalization here that also at as a very first step opens it up to the programming community too. The people who are already comfortable ..."

"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..."

"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..."

"healthy, you know, the community of humans can be so much more intelligent and mature and and and rational than the individuals within it. Well, one place I can always count on rationality is the comm..."