Can AI help us solve the hardest problems in Mathematics? Terry Tao - #535

Answer my survey to get a chance to win a $100 Amazon Gift Card! 👉 https://forms.gle/uWgVRCf3BC2xxR2Y7 Please join my mailing list to get FREE notes & resources from this show! Click 👉 http://briankeating.com/yt Will AI solve future math proofs? Every time you type a password, buy something online, or send an encrypted message, you’re trusting an assumption about prime numbers: that they don’t hide an exploitable pattern. Modern cryptography depends on primes behaving “randomly enough,” yet many fundamental questions about primes remain unproven. In this episode, I’m joined by Fields Medalist Terence Tao to explore what mathematicians can prove, what they strongly suspect, and what could change if unexpected structure appeared in the primes. We discuss pseudorandomness and why it matters for encryption, the twin prime conjecture, and how quantum computing reshapes what is feasible in computation and security. We also get into AI and mathematics: why large language models can sound convincing even when unreliable, how AI can help with idea generation and literature recall, and why verification and proof assistants will matter if AI is to contribute to real mathematical progress. Along the way, Tao explains proof techniques like proof by contradiction, why complex numbers and the square root of minus one are so central, and how high-dimensional geometry breaks low-dimensional intuition. Finally, Tao shares a real-world example of how math breakthroughs translate into technology: compressed sensing, which has enabled much faster MRI scans by reconstructing images from far less data. Timestamps: 00:00 Discrepancy Theory Explained 09:35 Induction: Science and Mathematics 14:38 "Proof Concept Through Play" 18:43 "Complex Numbers and Completeness" 25:36 Prime Numbers and Cryptography 29:51 "Computability and Complexity in Mathematics" 35:04 AI Discovers New Knot Theory Insights 42:02 "Elegance in Nature's Laws" 47:12 "AI as Complementary Research Tools" 51:27 "Humility in Pursuit of Proofs" 54:46 "Multiple Approaches to Mathematics" 01:03:04 Rethinking Reality and Physics 01:07:14 "Origins of Compressed Sensing" 01:10:17 "Shannon Bound and Information Limits" Follow Terry's Blog: https://terrytao.wordpress.com/ Get My NEW Book: Focus Like a Nobel Prize Winner: https://a.co/d/hi50U9U - Join this channel to get access to perks like monthly Office Hours: https://www.youtube.com/channel/UCmXH_moPhfkqCk6S3b9RWuw/join My tell-all cosmic memoir Losing the Nobel Prize: http://amzn.to/2sa5UpA Follow me to ask questions of my guests: 🏄‍♂️ Twitter: https://twitter.com/DrBrianKeating 🔔 Subscribe https://www.youtube.com/DrBrianKeating?sub_confirmation=1 📝 Join my mailing list; just click here http://briankeating.com/list ✍️ Detailed Blog posts here: https://briankeating.com/blog 🎙️ Listen on audio-only platforms: https://briankeating.com/podcast #universe #podcast #briankeating #intotheimpossible #science #astronomy #cosmology #cosmicmicrowavebackground #intotheimpossible #briankeating #terrencetao Learn more about your ad choices. Visit megaphone.fm/adchoices

Speaker names marked ? are inferred from speaking time and the show’s known hosts — they are not voice-verified.

• 0:00Speaker 00

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• 0:31Speaker 01

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• 0:56Brian Keating?

  Every time you enter a password, or buy something online, or send any kind of encrypted WhatsApp message, you're betting your security on a pattern in prime numbers. A pattern no mathematician on earth has ever been able to prove. They're the atoms of multiplication. They're supposed to be random, unpredictable, and our entire digital security infrastructure assumes that they are. But here's the thing: we don't actually know that. We've tested it with supercomputers on trillions of cases, but mathematical proof, it's still elusive. Today, the man sitting across from me has solved more legendary math problems than almost any human alive. Terrence Tao won the Fields Medal, the Nobel Prize in Math, and he's tackled questions that have stumped the greatest minds for centuries. And he just told me there could be an undiscovered pattern hiding in prime numbers. A pattern that, if it exists, could break the encryption protecting every financial transaction you'll ever do.

• 1:41Brian Keating?

  We're gonna talk about the beauty of numbers, why AI keeps getting the math wrong, what it was like to meet the legendary Paul Erdosch as a 10-year-old, and whether or not mathematics is invented or discovered. Let's go deep into the impossible with the Mozart of math. First question I always ask a mathematician is, how do you like your coffee?

• 1:58Speaker 03

  I actually don't drink coffee much, except on social occasions, actually. Black, no sugar.

• 2:02Brian Keating?

  Okay. So the reason I ask that, maybe you'll recognize, is Erdo, I believe, said, what did he say about mathematicians and coffee?

• 2:09Speaker 03

  He said that mathematicians are a means for turning coffee into theorems. There's a very nerdy follow up joke to that, which is that a co-mathematician is a way of turning co-theorems into feet.

• 2:20Brian Keating?

  Into feet?

• 2:21Speaker 03

  Yeah. That's a very, it's a very inside joke.

• 2:24Brian Keating?

  That's right. That's a dad joke plus a mathematician joke. That's really, really, but really good at the same time. Well, the reason I bring up Erdo, of course, you actually met him when you were a kid, didn't you?

• 2:33Speaker 03

  Yes, I think I was 10 at the time. So he had a collaborator in Adelaide, which is the city where I grew up, George Segares. So he would visit every now and then. At the time, I think one of the math professors at the local university introduced me to him. And Erdo was always very good at, he was known for meeting bright young kids. And so we had a nice conversation. I wish I'd remembered more of it, actually. I was just, I was too young at the time to realize just sort of how, how much of an honor it was really. The one thing I remember was that he really treated me like an equal, like, you know, he didn't condense him as a kid. And he later sent me a postcard that it just had, oh, thank you for your nice hospitality. Here's a math problem, which I didn't solve, actually, but it did get solved later by someone else.

• 3:17Brian Keating?

  That's amazing. Yeah, he was one of the most prolific mathematicians in at least modern history, maybe in all time history. And famously, there's a relationship between the number of authors you have to go through before you're related to him, right? What is your Erdo number?

• 3:30Speaker 03

  Yeah, so this is concept called the Erdo number. So Erdo was a lot in graph theory. And so this, this concept is inspired by graph theory. So Erdo himself has an Erdo number of zero. If you've written a paper with Erdo, you get an Erdo number of one. If you've written a paper, if someone's written a paper with Erdo, you have an Erdo number of two. So I have an Erdo number of two, for instance. And I think nowadays people, it's common of Erdo numbers of four or five. People have made similar numbers in other fields. There's a Bacon number. So if you've starred in a film with Kevin Bacon, you have a Kevin Bacon number of one and so forth. And then there's something called the Erdo Bacon number, which is the sum of your Erdo number and Bacon number, which is usually infinite. Cause you either don't have a chain of papers going to Erdo. Sure, you don't have a chain of movies going through to Bacon, but there are a half dozen people who have like the combined of like seven or eight.

• 4:15Brian Keating?

  Yes, yes, I've heard of random things like that, but yeah, he was known in many ways. I remember hearing from Jim Simons, who was my late great mentor and obviously you knew him well. That he had, he was incredibly productive, but part of his productivity relied on the use of amphetamines. He used to take some, is that true?

• 4:34Speaker 03

  Uh, that's what I've heard. Apparently, one of his friends convinced him to give up amphetamines for a month. For a bet or something and Erdos grudgingly did it. And then he's, at the end, he just went back and just said, you just set mathematics back by one month. So he was, I guess, hardcore. You don't see that so often nowadays. I think, I think back then, maybe there was, there was less of a work life, less of a stress and work life balance than you are today.

• 4:58Brian Keating?

  Now, you had a work related to Erdos, right? Erdos discrepancy theorem or something. What is that? Can you explain that for my audience?

• 5:05Speaker 03

  Okay, so Erdos was famous for posing many, many problems and I solved a few of them in over my career. Discrepancy theory is a theory about how irregular sequences can be. So like if you have a sequence of plus ones and minus ones. And if they're random, you expect in, if you, if you pick, say, 1,000 numbers, plus or minus ones at random. You'd expect 500 of them to be plus ones, 500 minus ones. So the discrepancy is defined as the difference between the number of plus ones and minus ones. A sequence can have low discrepancy if you view it over the whole sequence, but if you look at sub sequences, the, this sequence can have a higher discrepancy. So for example, if you take an out, out next sequence of plus one, minus one, plus one, minus one, a thousand times.

• 5:47Speaker 03

  The discrepancy over the whole interval is zero because you have 500 plus ones, 500 minus ones, it sums to zero. But if you look only over the even numbers, you just see plus one, plus one, plus one, plus one, or minus one, minus one, minus one. Then you have a very large discrepancy, I think 500. Some sequences, they can be very well balanced overall, but when you restrict to subs. Sub sequences, they can have a higher discrepancy. So Erdos was interested in whether you could design a sequence which had what's called bounded discrepancy over all.

• 6:18Speaker 03

  Called homogeneous arithmetic progressions. So could you create a very, very long sequence of plus ones and minus ones? Where if you look at any finite segments, say from one to a hundred, one to a thousand, the number of plus ones and minus ones only differ by at most two, say. But also, if you look over the even numbers, same thing happens. You look over multiples of three, same thing happens. So I wonder if it's possible to make an extremely uniformly distributed sequence that was always balanced, no matter how you looked. And you can do it for quite a while. I think someone constructed a sequence of like one thousand, one hundred and sixty four elements also, where the discrepancy was was never bigger than plus or minus two. So extremely well balanced. So there are some extremely uniform sequences, but using a really huge supercomputer and someone called a SAT Silver, they could show that past that point, you had to have a discrepancy of three.

• 7:04Speaker 03

  The sequence becomes more and more unbalanced over time. But until several years ago, that was the record. So three was the best lower bound for how much discrepancy these sequences had to have. So Firdish asked, do these sequences, if you continue these sequences on forever and ever, do you must you eventually must the discrepancy eventually go to infinity? And this is what I was able to show.

• 7:27Brian Keating?

  Oh, wow. So it does go to it diverges?

• 7:29Speaker 03

  Yes, yes, extremely slowly, as far as we know, like logarithmic or double logarithmic, but it does it does go to infinity. And I had to use tools from information theory and number theory.

• 7:40Brian Keating?

  I've heard that there are people that use some applications of of your work to detect cheating in the following sense, that when a student cheats, of course, our students never cheat, but you know, say they're doing a true, false exam or they're, you know, want to mirror something and they want to simulate that they actually got the answers, maybe they'll put down randomly, true, false, true, false, true. And it'll be too close, the sum would be a plus minus would go to zero. Is that is that any relevant?

• 8:02Speaker 03

  It is connected. Yeah, so there are other statistical patterns that random sequences have and artificial sequences don't. I don't think my low discrepancy work directly bears to that, but there are other patterns. The most famous is called Benford's law, which is a very unintuitive law that roughly speaking, thirty percent of all numbers in the world start with one, which sounds very weird because, you know, numbers can start with one, two, three or up to nine. But you can take, for example, take all the countries in the world. Population and there's about about a hundred odd countries and about about a third of them, the the population is out of one, um, China, for instance, um, but, um, or you can take, um, the the the net wealth of of, um, of several millionaires and billionaires, whatever. And you also find that Muslims start with one or birthdays.

• 8:46Speaker 03

  Yeah, it it the pattern i- is quite universal. Um, but whereas if you if you pick numbers randomly, if you like fudge your accounting books, you you, um, uh, when you pick numbers ran artificially, they don't necessarily obey that.

• 8:58Brian Keating?

  They're uniform. So you.

• 8:59Speaker 03

  Yeah.

• 8:59Brian Keating?

  Or you try to make them, you think that uniform.

• 9:01Speaker 03

  Yeah. So so humans are actually really quite bad at at creating truly random patterns. Um, and so, yeah, you can distinguish natural patterns from from human generated ones.

• 9:09Brian Keating?

  Interesting. So one thing that I've, you know, been dying to talk to you about for a long time are kind of the the limits of mathematical induction. So you mentioned that you start with a small number and then you kind of add on to it. And I do want to hearken to, uh, the work of of Jim Simons. He's most famously known for being a multi billionaire, uh, establishing philanthropies that support my research and and and hopefully, uh, other very, uh, highly competent scientists. But, um, but one of his lesser known things that was actually very important and at least in my understanding of how mathematical induction works or violates is his work on minimal surfaces where he showed something really fascinating. So I I should I should have you to explain what a minimal surface is. But as I understand it, if you can sort of think of it physically, if you had some some shape,

• 9:54Brian Keating?

  say a coat hanger, and you made it into a loop, and then you wanted to attach it to another loop using a soap bubble, the shape that would would obtain would be called a minimal surface. Is that correct? Okay. And then he showed that there are such minimal surfaces and zero or one dimension, or it was known that that was true. And then he showed it in dimension two, it exists in dimension three and dimension four and dimension six, seven, and then he got to eight.

• 10:16Speaker 03

  Right.

• 10:16Brian Keating?

  Actually, it didn't work and I would have stopped at two, right? So most mathematical induction, you know, seems to continue to infinity. But you already told me one thing that doesn't continue to infinity, as you might naively expect. What are the limits of mathematical induction? Maybe define what it is first. What is mathematical?

• 10:31Speaker 03

  Right. Yeah. So induction means different things. I think the philosophers and in philosophy of science, induction refers to something slightly different where you. You take facts that you observe from small examples and you you induce from that what what things a prediction for what will happen for larger cases. And it's it's it's a very basic procedure, a scientific method, because you know, you do experiments and then you extrapolate from the experiments. Mathematical deduction is a more precise form of reasoning where so there's a precise principle of mathematical induction, but if you have a statement that you want to be true for all natural numbers, one, two, three, four, and so forth, and you know it's true for one, and whenever you know it's true for some number n, you it you can you you know

• 11:15Speaker 03

  for sure that it implies the same thing for n plus one, then it implies it's true for all for all numbers. The analogy often given is just a row of dominos. So if each domino represents one case of what you're going to prove, and you can prove the first case, you knock the first domino over, and you know that each domino, whenever you can prove it, it tips over the next domino, then no matter how long the the string of dominos is, you you can you can knock over every single domino chain. But it's really important that your argument is a one hundred percent watertight. You know, if it's really dominoes and like the ninety seventh domino doesn't tip over the ninety eighth, you know, it then it it stops there. So it's a principle that only works in the water mathematics, which is one of the few places where you really can have one hundred percent guarantees. So Simons, yeah, so he discovered what's called the Simons Cone.

• 12:00Speaker 03

  Yeah, you're pushing a little bit because this is geometry is not completely my my area of mathematics, but yeah, so minimal surfaces are, yeah, the most famously are two dimensional surfaces like soap forms. But in mathematics, there's nothing stopping you from considering the same notion in other dimensions. In one dimension, it's just like rubber bands, or at one dimension, minimal surfaces are very boring, it's just straight lines. But you can consider them three dimensional surfaces in four dimensional space, which already is hard to visualize, but mathematically you can consider it and five and six. Weirdly, sometimes problems become easier in higher dimensions. So even if you care about the physical world and you only care about two and three dimensions, sometimes it makes sense as a method addition to first study higher dimensions. It gets you some intuition, which can help guide you with with the problems that you

• 12:45Speaker 03

  do care about. Yeah, so it turns out, yeah, that in up below eight dimensions, I think, yeah, all minimal surfaces are smooth. You you can't tie a soap bubble and create any kind of knot. Yeah, because there's always some way to pull it apart and reduce the surface tension. Yeah, starting in eight dimensions, he discovered a very surprising fact that singularities do can actually form. Yeah, that there is, there is this, yeah, it looks like a cone, except in much higher dimensions. And there is no way to to modify the cone to make it to reduce the surface area. If you made a cone, if you try to arrange soap into a cone in three dimensions, you could just remove the, do what sort of surgery? You remove the vertex of the cone and replace it by two rounded knobs. Okay, and that would reduce the surface tension.

• 13:30Brian Keating?

  Interesting.

• 13:30Speaker 03

  You can't do that in higher dimensions. So nowadays, because of data science, actually, we, we, we need to understand high dimensional geometries much, much better than we used to. And a lot of our old intuition is actually, which you get from low dimensional geometry, is actually completely false in, in high dimensions. So just to give you one example, like if you inscribe a, a, a circle inside a square, it occupies a fair, you know, a pretty large chunk of, of, of, of the, of, of the square, maybe like seventy five percent or something. And you inscribe a ball inside a cube, it's still pretty big, I think about half the volume of a cube. But like, if you take a thousand dimensional cube and you inscribe a thousand dimensional ball inside it, it's like incredibly tiny. It's like point zero zero zero one percent, like balls become extremely.

• 14:14Brian Keating?

  Poor space filling.

• 14:15Speaker 03

  Yeah, they're nowhere near space filling in high dimensions. And this is important when you look at clouds of data and, you know, like if you have some, you're taking a thousand measurements and that's like a thousand data points, but, but there's some errors in them. You know, do you measure the, the root mean square error, which is like trying to place this, your, your measurement inside some ball? Or do you measure the worst error, worst of the one thousand errors, which is like placing? The question is, do you want your error bars in high dimensions to be like a ball or a cube? And it starts making a difference.

• 14:42Brian Keating?

  Right, they're, they're significant differences between that approximation. So you mentioned that technique of going to higher dimensions to solve problems in lower dimensions. That's one of the many tools that mathematicians use. Others include, you know, proof by, uh, ridiculous absurdium. Can you talk about what is your favorite type of mathematical proof? And when you're on to it, you just get so excited to finish the film.

• 15:04Speaker 03

  Proof by contradiction, I think Hardy had a. Great quote that in chess, a chess player may offer, you know, a pawn or a bishop, but a mathematician offers the entire game, you know, so.

• 15:17Brian Keating?

  Sacrifice, yeah.

• 15:18Speaker 03

  Yeah, so it's okay, we want to prove this is a conclusion. I will give you that the conclusion is false. Okay, I will just let you run with it. Okay, and but you do that and I will show that it gives you a contradiction. It actually is a technique. So it on the one hand, it is very unintuitive. The undergraduate students that we teach, they struggle a lot with the notion of proof of contradiction. On the other hand, it is a concept that I have seen primary school students teach each other. So in recess, you might see kids play the game of who can name the largest number. So they say, okay, one thousand and then a million, a billion, a billion, billion. And they'll go on like this. But at some point, someone will realize, one of the kids will realize that no matter what number the other kid says, they can just say that number plus one.

• 16:03Speaker 03

  They have proven that there was no largest number in natural numbers. And this is a proof of contradiction, because if anyone ever did claim that natural numbers is largest natural numbers, they add one and you have contradicted them. So it is actually a very intuitive proof technique, but you have to teach it the right way and sometimes case condition themselves. Type of mathematics that I that the type of proof arguments that I like the best are ones that make unexpected connections between different areas of mathematics. Like say between discrete mathematics and continuous mathematics, we talk about low dimensions and high dimensions. You can you can have a problem which has to do with combinatorics and nothing to do with the real world, but you find that there is there is some physical model of it and you can use ideas from physics. Of course, as physicists do this all the time, they also have correspondences which which are really quite quite amazing.

• 16:48Speaker 03

  So, yeah, those feel like magic to me.

• 16:51Brian Keating?

  Yeah, I mean, the most famous one, at least to my physicist's mind, is the proof that by contradiction, that square root of two is irrational. So that's the Euclid's original proof or what does it trace back to before him?

• 17:02Speaker 03

  Euclid or Pythagorean, I think, the Pythagoreans.

• 17:05Brian Keating?

  Maybe Pythagoreans, yeah. Euclid proved the prime basically by the same sort of ideas, infinity plus one.

• 17:11Speaker 03

  Yeah, yeah, yeah. No, we have a lot to thank Euclid for, actually. I mean, he wasn't the first to write down. Many of the theorems like the Pythagorean theorem, for example, I think the Babylonians had a version, the Chinese had a version. But really, he was the one who introduced this this notion of proof that complex facts about mathematics, you could you could deduce from from simpler axioms. And it was extremely influential way of thinking, which you hadn't seen before.

• 17:37Brian Keating?

  So the square root operation, just as a notion, has always fascinated me. You know, for for one thing, it seems to occur in physics, you know, quite regularly. And I'll get into some examples that pique my curiosity. And eventually I do want to tie this to Wigner's famous statement about the unreasonable effectiveness of mathematics to the physical world, and we'll talk about that in just a bit, which also tangentially involves the square root. But the square root, in physics at least, for example, in classical mechanics, you can construct things, operators that involve the position of momentum called Poisson brackets.

• 18:12Speaker 00

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• 18:42Brian Keating?

  And as soon as you take them from the classical world to the quantum world, instead of commuting being equal to zero, they become equal not to zero, but times a fundamental constant, times the square root of negative one. And it's just so baffling to me that once you introduce the concept of a square root and imaginary number, then so much mathematics is open to physicists. And I wonder, you know, is there like, could we, an intelligent alien, you know, who knew all of mathematics, could they have taught us this? Or is there something special about the square root operation? In L L M's, they use L U decomposition. And we have spinners that would have a spinner representation in our square roots. Is there something special about the square root? Or like, in other words, why don't we say it with cube roots or fifth roots or hundred?

• 19:25Speaker 03

  Why is the square root something that's that's so prevalent in physics, for example? Yeah, so so we experience the world as in a continuum Euclidean space. And the notion of numbers that are most natural to us from our spatial intuition are the real numbers. Right, so real numbers have lots of wonderful properties. The algebra of real numbers works really well, you know, things like addition is commutative. X plus Y is Y plus X and X times Y is Y times X, multiplication is commutative and so forth. But they have one flaw, which is that not every polynomial equation has roots. So if you take the equation X squared plus one equals zero. And in the real numbers, X squared plus one is never zero, because X squared is always positive.

• 20:09Speaker 03

  So it's what's called, it's not what's called algebraically complete. But it's very close to being complete. So if you take a polynomial which is of odd degree, like a cubic, X cubed plus three, X plus one. It must have a root because a cubic polynomial or odd degree polynomial, when you make X very, very big, it becomes very large and positive. And when you make X very, very large and negative, it becomes negative. And because the reals are continuous, the polynomials are continuous. Therefore, at some point in between, you must hit zero to get from negative to positive. Sort of a half of all the polynomials in the world, the you can solve them in the reals and half you can't. So in the, with the benefit of hindsight, this really suggests that you should make the reals so twice as big

• 20:53Speaker 03

  in order to get this really useful property of algebra completeness. And so as it turns out, there are these numbers called the complex numbers, which are twice as big as the so the real numbers are one dimensional and the complex numbers are two dimensional. And they have wonderful, wonderful properties, lots of very nice geometric structure, nice algebraic structure, ultimately coming from this algebra completeness.

• 21:14Speaker 03

  And so we also know from algebra that the way to make something twice as big, a number system twice as big is to add a square root that you didn't have before. If you want to make a system three times as big, you should add a cube root that you didn't have previously. So once you know that you're looking for a new number system that's twice as big as what you started with, it's very natural to look for, to throw in a square root of a number that doesn't currently have a square root, such as minus one. So that's kind of the in actual spec.

• 21:42Brian Keating?

  How you might have predicted.

• 21:43Speaker 03

  Yeah, I mean, this is not historically how complex numbers were discovered, but this could be sort of one explanation.

• 21:47Brian Keating?

  Hey, everybody. I'm usually the one that asks my guests to judge their books by their covers. But today I'm asking myself to judge my own book by its cover. My newest book, Focus Like a Nobel Prize Winner, is chock full of advice, life tips, and focus and productivity tips from nine of the world's greatest minds, Nobel laureates ranging from economics to peace to physics, of course.

• 22:10Brian Keating?

  I'll go check it out, and my publisher's got Amazon to run a special. So go to Amazon and get the Kindle copy today. Right. And then there's a whole other class of numbers, transcendental numbers, where you have, they don't solve polynomial equations, correct?

• 22:25Speaker 03

  Yeah, yeah.

• 22:26Brian Keating?

  When they have roots of polynomial.

• 22:27Speaker 03

  We've learned that the notion of number is very flexible. I mean, people get upset when they learn that what, you know, it feels simpler to have, you know, one notion of everything taught in school and then not have that change. You know, like people were very upset when the notion of a planet got changed like 10, 15 years ago. And we occasionally do that with math, too. Like, the number one used to be prime about 100 years ago.

• 22:51Brian Keating?

  Oh, really? I mean, I still consider it prime, but I still consider it a planet.

• 22:56Speaker 03

  Yeah, yeah, but it's because in all mathematicians inside any, in any field of study, when you first study a subject, you don't really know what concepts are the most fundamental and important and which ones are not. So you make a guess based on your experience. So maybe you think that numbers that don't have any smaller factor are important. So you call them prime numbers, or maybe the stars that move in the sky are important. So you call them planets. But over time, you realize that actually there are slightly better definitions that have better properties. Okay, so I can't speak to the astronomers why

• 23:30Speaker 03

  the new notion of a planet is better. But, but for example, one of the, what we've learned is that one of the really important properties of primes is what's called the fundamental theorem of Earthnatic, that any number can be broken up into primes in exactly one way, other than rearranging the factors. So 12 is 2 times 3 times 2, or 2 times 2 times 3. Other than interchanging the the order, that's the only way to break up a number of primes. Just like there's only one way to break up a chemical compound into atoms. So primes are like the atoms of multiplication. But if you made one prime, then you wouldn't, you would lose, you have to give up the fundamental theorem of arithmetic. Because now twelve is also one times two times two times three. And you just add too many exceptions to this really important factor in in number theory.

• 24:13Speaker 03

  So we made a decision to therefore redefine.

• 24:17Brian Keating?

  Ah, what it was, okay. But just like with Pluto, there's no, there's no consequence to life on Earth. It's it's more sort of.

• 24:24Speaker 03

  Yeah, it's it's a human convention, but it's it's, but we we update our human conventions to match reality better over time.

• 24:31Brian Keating?

  You, I've done work in prime pairs, is that right?

• 24:34Speaker 03

  Yes, yeah, so primes are one of the oldest subjects in mathematics. We Euclid had the first theorem almost ever. And it was about prime numbers almost more than two thousand years ago. And so it's very frustrating and annoying that even the most basic questions about primes, we still cannot answer definitively. We can, we have good guesses, like, so for almost all questions of primes, we can predict the answer. But we cannot get the one hundred percent mathematical standard of proof for for many of them. And one of those basic questions, which is at least three hundred years old, is called the Trinpime Conjecture, that there should be. So Euclid showed there's infinitely many primes, the primes never end. You can always find primes bigger than any number you wish.

• 25:17Speaker 03

  But we cannot find, we cannot say the same yet for prime twins. So these are pairs of primes that are differed by the closest they can, which is two. For example, eleven and thirteen. Well, okay, two and three are closer, but but after two, all primes are odd. So the the closest you can get is is two. And so we can observe that every so often, you know, the primes, they they don't seem to obey a pattern. Sometimes the prime gaps are large, sometimes they're small. But every so often, they they come close to each other and you get a twin. And they seem to occur infinitely often, you know, as we we can find trillions and trillions at least by computer. But we have never been able to prove that they go on forever. We have this prediction that the primes behave like, basically like a random sequence of numbers.

• 25:59Speaker 03

  Um, random sequences with this, if you, if you have a random sequence of, of the same density as the primes, they will hit, um, um, form twins infinitely often, but the primes are not random. We believe they're what's called pseudo-random, um, that they have no obvious pattern besides the ones that we can, we can obviously see such as them being, um, uh, being odd. Yeah. So I mean, it's, it's a very likely hypothesis, but we can't prove it.

• 26:22Brian Keating?

  Pseudorandomness, meaning that it could be derived from some algorithm, but not in all cases or something. What is pseudorandom versus random distinction?

• 26:29Speaker 03

  Um, random means not deterministic, that, that there's no single, like, so the primes, if you've forgotten what the primes were and you had to regenerate them by a computer program, you would generate exactly the same set. Uh, whereas if you were generating a set by rolling dice or flipping coins, you would get a different set. Pseudorandom are sets which are either random or deterministic, but statistically they are indistinguishable from random noise. So, um, for example, um, a random number should just, if you have a random sequence of numbers, there should be just as many numbers that are end in five as end or end in seven as end in six, like the, the digits should be equally distributed. Now, the primes, they're not perfectly pseudorandom because they, they do avoid certain, uh, they do have certain patterns, like they, they tend to be odd, um, for instance, but there's

• 27:10Speaker 03

  ways of excluding those sort of obvious biases. And once you exclude them, um, um, there should, it's expected that there's, there's no, no tests that can distinguish them from random, um, um, numbers. This is important for cryptography, actually, because there are many crypto systems, like the ones we use to encrypt web traffic, uh, cryptocurrency, um, you know, uh, financial transactions, where data like sensitive data like a passwords or, or, um, or credit card numbers are encrypted using mathematical, um, routines that rely implicitly on primes having no pattern. And so they, they use primes in various mathematical ways to mix up these numbers. And we believe by doing so, the data that we actually send looks indistinguishable from random noise and, and conveys no information about, um,

• 27:51Speaker 03

  your, your personal data. And we really hope that that's, that that's true. I mean, um, so one reason why it's important for mathematicians to actually study prime numbers is that We occasionally get a shock that, I mean, it hasn't really happened in number theory in centuries, in decades at least, but there could be really unusual undiscovered patterns in the prime numbers that we weren't previously aware of. And if they existed, they could present a vulnerability to crypto systems. There have been a few other crypto systems where similar patterns have been discovered. I think not for a prize, where you're elliptical curves and other things where people actually had to migrate to a different crypto system because of these weaknesses.

• 28:26Brian Keating?

  Yeah, so that brings up what kind of an inversion, maybe a contradiction of what Wigner said, he commented on the unreasonable effectiveness of mathematics and the physical sciences or the natural world. But what you said just made me think about the kind of inverse of that, which is the unreasonable effectiveness of physics in the mathematical world. In other words, you mentioned cryptography and it said that quantum computers can perhaps factor and break these previously considered to be uncorrect. So yeah, my question is, what is it about quantum computers that could then illuminate or elucidate things in number theory and pure mathematics from the physical world to the, you know, the quantum world to the mathematical world? Do you see that as, you know, sort of a viable topic?

• 29:09Speaker 03

  So quantum computers are a fascinating topic. Yeah, they interface with paths in various ways. So one is actually just the actual software engineering of creating good quantum algorithms. So it requires a very different type of software mindset, you know? So classical computers, we have this sort of sequential way of thinking where you just sort of, you have these bits of memory and you flip them and if you do this, you do that. And we have decades of experience with it. Yeah, for a quantum computer, the state is not a bunch of ob zero one bits, but it's a wave function and the operations you assign to them, you have to multiply them, but you're only allowed to multiply them by matrices, really, really large matrices. Except that your basic operations, your matrices are mostly the identity and there's

• 29:53Speaker 03

  only, you only change a few corner bits at a time, but you want to couple them together in a very efficient way so that you can do really complicated operations. Yes, so quantum computers are both exponentially more powerful than classical computers, but also exponentially more limited.

• 30:10Brian Keating?

  Yes.

• 30:11Speaker 03

  So because they can handle superposition of quantum states simultaneously, in principle, there's this exponential speedup. And for certain applications like factoring, and I think quantum chemistry.

• 30:22Brian Keating?

  Molding Lagrangians.

• 30:23Speaker 03

  Right. They are, at least in principle, very, very powerful. But quantum mechanics is also very restrictive. The number of things you can do to quantum state, you can only do linear operations and only do time reversible operations.

• 30:37Brian Keating?

  Non-destructive.

• 30:38Speaker 03

  Yeah. So this has, yeah, so this requires you to develop theories of reversible computing. Error correction is also much, much more annoying.

• 30:49Speaker 03

  So yeah, so there's the software challenges. Maybe once quantum computers become a reality, they could be used to do large scale computations of a type that we haven't done before. Whether they have a practical impact on the actual theory of mathematics, I don't know of any examples off the top of my head, but.

• 31:08Speaker 00

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• 31:38Speaker 03

  Certainly classical complexity theory has been very influential. Historically, mathematicians only cared about whether something was true or false or provable or disprovable. And with the advent of computers, people also started asking questions of how computable is an object like. So if they could prove that something exists, could you go further and actually say, is there an algorithm to compute? And is the algorithm exponential time or polynomial time? So a much finer grained notion of truth actually than just it's true or it's false. But how easy is it to actually compute? Um. And it um that has led to very productive mathematics.

• 32:18Speaker 03

  Sometimes just the the effort to not just show something exists, but actually find it. Creates um yeah, creates new techniques. Yeah, complexity theory has offered as as sort of um given a much more nuanced um understanding of of of how true a statement is. Or how um and yeah, this has led to to um a better understanding of you just prove something is is true and you don't. But uh you may not have any insight. So what was what was the key ingredient that made it work? Or if you had two different proofs, which one which proof is better? But maybe one proof leads to a faster algorithm than the other. And so you can say, oh, that that proof actually is is stronger.

• 32:57Brian Keating?

  It's more efficient.

• 32:58Speaker 03

  Yeah, um hum. So it yeah, it it indirectly is sort of uh provides much more insight is is into the um into the the proofs that mathematicians want.

• 33:08Brian Keating?

  Has AI actually enabled new discoveries in mathematics or new proofs that otherwise would not have existed?

• 33:14Speaker 03

  Slowly, it's beginning to um. I mean, by itself, uh so the um the big weakness um of these AIs right now is that they can begin to produce output that looks like, say, a human mathematician reasoning their way through a problem. But um it's not grounded um that um it's it's it's it's probabilistic, okay? They often make mistakes and um much like, you know, if if I were to get a student to solve a problem on a blackboard and they're nervous and they they just say the person comes to mind, they might get it right or they might get it wrong. But if it's a weak student and they don't have sort of um fundamental knowledge of of what they're actually doing, once they go off the rails, they can go really um off off the rails.

• 33:56Speaker 03

  And and this is something which uh is a fundamental problem with the current large language models. But if you use them as a component of um a more rigorous and grounded reasoning system. So if you if you converse with a a large language model to for them to make suggestions, but you've you understand with the output and you can verify it. So um then people have had some success talking about their math problems with a large language model. I'm sure one will produce will produce. Some suggestions, some of which the human expert can dismiss as as not viable. Some of us would be thinking, oh, I thought of that already. But one or two is, oh, that's actually something which I should have known, I should have come up with myself, but I just didn't realize.

• 34:36Speaker 03

  So one thing where AI models are already beginning to be useful is in like literature review type tasks where there is a cluster problem. And in the literature, there may be, say, a dozen ways already to attack this problem. And you, if the human working on the problem, maybe you can remember six of them, but you forgot the other six and it don't come to your mind. And you can use the these language models to prompt you, to remind you of the of the missing six. They may also hallucinate three more that

• 35:04Brian Keating?

  Exactly, right.

• 35:04Speaker 03

  don't exist. So you you do need, you can't trust them.

• 35:07Brian Keating?

  Supervise them.

• 35:08Speaker 03

  Yeah, you have to, you have to verify. There is hope in the future, you know, so this this separate math technology to to order to have the software that can automatically verify certain types of proofs.

• 35:17Brian Keating?

  Right.

• 35:18Speaker 03

  And so the hope is that if you force the large language models to only output in some verified, some language that you can verify and to filter out the hallucinations.

• 35:28Brian Keating?

  Has it been able to reproduce a Wiles, you know, proof from that last theorem or your work and now your Stokes? I mean, has it been able to actually just just simply quote unquote reproduce what a natural intelligence person like you do?

• 35:39Speaker 03

  This is issue, it can, but but often because of what's called contamination. So if a result is like taught in textbooks somewhere, then it is implicitly in the training data that these AI train on. And so they're basically memorizing the same way, again, that a student at the board may just reproduce from memory a proof that they saw in a textbook. And so AI has basically read all the textbooks in the world. It's hard to to discern when that happens, whether it was training data or whether they they really sort of thought it up. If you ask the AI to explain their their chain of thought, they often give complete nonsense like it did. It's clear that they just didn't know.

• 36:18Brian Keating?

  Yeah, I mean, I've found that even we tried with my student Evan Watson, we tried to, we just gave it the information about the orbit of Mercury over the past three thousand years, which JPL up the road here has has access to in Kibbutz. And then we said, well, if you observe this in this planet, you know, basically. Could you first discover this anomalous procession of the perihelion and mercury? And then could you predict it, you know, and it was just completely unable. It required us, we had to first discretize everything, make everything Euclidean, which then totally ruins it. Right. So I've proposed, and I want to get your take on it, kind of a joke. I call it the Keating test, but it's basically the Turing test. We'll know when it's true and actual AI can come up with new and unknown, heretofore unknown, you know, predictions that can be verified by humans like you.

• 37:01Speaker 03

  Yeah, yeah, no, I think, I think that's a very promising use case of AI. I mean, um, yeah, I mean, that, you know, neural networks in general, I mean, they're, they're designed to make patterns, to detect correlations and things. Yeah. So there have been a few examples in mathematics where, for example, in not theory, um, a neural network, not a fancy LM so much, but like a more old school neural network was used to detect correlations between different types of not invariance that was not believed, people did not suspect existed before.

• 37:32Brian Keating?

  The layer chart.

• 37:33Speaker 03

  And then once, once, I mean, initially this type of correlation was, was just sort of this black box relationship. So, um, so, so knots have these loops in space, which, um, you know, some are, some can be untangled and some can't be. They come with all these numbers, they're called not invariance.

• 37:50Speaker 03

  And so, um, the neural network found that, that by feeding a database of like one million knots, that there was one not invariant called the signature, which could be predicted with really high accuracy from a whole bunch of other invariance called hyperbolic invariance.

• 38:03Brian Keating?

  No.

• 38:04Speaker 03

  So, but this, this neural network was this black box. You just fed in these 20 numbers as your hyperbolic invariance and it was spit out. The signature should be plus three and like 90% of the time it was correct. But once they had this black box, they could analyze it because I suppose, I suppose I change this input. I, I just, I modified this hyperbolic volume or whatever. How much does this change the output? And so they, it's like a box with 20 dials that they could, they could, they could play with. And basically by running experiments, they could see that, that three of these inputs were actually really important. And the other 17 were, were, were very peripheral. And by doing those types of analysis, they, they actually got some insight as to what the relationship was. And they could actually make a formal mathematical. Prediction, which they could then prove. So, you know, once you have these neural

• 38:47Speaker 03

  network models of, you could actually probe them. So in your astronomy example, maybe your neural network might not be able to tell you exactly what the new law of physics would have to be, but it can say, well, I can at least predict the orbit of Mercury over the next thousand years and here's my, my model. And then, and then you can, you can just try to tweak it. And I suppose I changed my, the, the, the period of Mercury or, or the, or the mass, whatever, what, what happens to it. And maybe you can work out laws of nature experimentally. It gives you a new paradigm to, to access reality in traditional experiment or theory.

• 39:21Brian Keating?

  Yeah, I've used that example to kind of, there's a lot of AI doom people that think we're, you know, AI is going to run amok and turn us all into paperclips and all sorts of nonsense. But, you know, because it seems to have this feature that you mentioned, that it's sort of averaging over all of human knowledge. And so it'll have errors and it'll have mistakes, but it's bounded by the amount of human knowledge that's used in some level in its training set. But then there's something magical about it. And I wonder, the mathematics, I mean, I'm in the presence of greatness, right? So the mathematics though, aren't that, I mean, it's matrix multiplication at a massive scale, high dimensions and, and huge volumes. But is it really that complicated and intrinsically?

• 39:57Speaker 03

  So the mathematics to train and run a, a large language model or, or any other modern AI is not that complicated. Yeah. So an undergraduate math major would have all the prerequisites. Basically, you need to know how, how matrix multiplication works in a liberal calculus. But the, the area where we don't have a good method of theory is, is how to evaluate, how to predict the performance of these models. So the, the mystery is not so much how, how they run. Okay. We, we can, we know how to make a large, a large language model and how to, to train it and how to run it. But what is surprising is that it works really well for certain tasks and it doesn't work well for others.

• 40:39Speaker 03

  And we don't know in advance, we, we don't have good rules of, even heuristic rules of thumb for predicting which, you know, which tasks are good, which are not. We can only just make empirical experiments. Part of the reason is that the data that we train on, like, so we, we, so the data on, on one level is just strings of zeros and ones. And

• 41:02Speaker 03

  Mathematically, we understand sort of very, very random data. So like a complete noise, completely random zeros and ones. We have the mathematical probability theory, which explains how we can analyze this issue very well. And then we have very, very structured types of data, like a sequence which is all ones or all zeros, or just alternating 1, 0, 1, 0 in a very periodic fashion. Very structured data, we understand very well. But the type of data that is natural data, like English text, you know, so you can digitize that as strings of zeros and ones, but very specific zeros and ones.

• 41:32Brian Keating?

  Right.

• 41:33Speaker 03

  But not so specific that they're completely predictable. But they still seem to be somewhat predictable. And yeah, so we don't have good mathematics for partially structured objects. It's an analogy in physics, actually. So in physics, we have continuum mechanics, which is where everything's sort of averaged out. And we have a good theory there. And then we have atomic level physics, where you just have, you can look at individual molecules and particles. But at the MISO scale, there's lots of intermediate structures, like cells, for example, biological cells.

• 42:03Brian Keating?

  Virgin, it's emergent.

• 42:04Speaker 03

  Yeah, it's emergent, and we don't have good mathematics for this. You know, I mean, in principle, you could break down atoms, but you can't possibly analyze.

• 42:12Brian Keating?

  Yeah, it's not mathematically impossible, but in practice, it might be physically impossible. We mentioned, you know, inevitably, when we talk about LLMs, you know, the middle L is language. We'll get to my friend Galileo. I brought a book, I want you to get your impression on, one of his math books and treatises. But he said, you know, that the book of knowledge of nature, the universe, is written in the language of mathematics. And it kind of, you know, was echoed later, you know, by Wigner and so forth, as we already discussed. But is it really a language? Yes, it has a vocabulary and it has a syntax, you know, but in the same text, you know, Shakespeare and math, if they're truly, you know, at root, some protoer language or something like that,

• 42:54Brian Keating?

  then they should have more combinations or similarities, I would think. But again, I want your opinion. Do you think math is a language or is it much more than that?

• 43:04Speaker 03

  Well, certainly, when mathematicians talk to each other or to other scientists, I mean, they have to use math as a language. I think the difference between mathematical language and natural language is that mathematical language sort of has evolved over time to describe its, you know, to to describe the underlying mathematics as sufficiently as possible. Language, natural language is not always about efficiency. Um, you know, I mean, you you also want to convey nuance and and emotion and and and art or or just express frustration or or whatever. Um, so it it it it isn't driven purely by by by efficiency. Um, but mathematics pretty much is partly because we, over time, we

• 43:46Speaker 03

  try to do to do more and more ambitious mathematical tasks. And if we didn't optimize our math language in this fashion, um, it it we will not be able to do these more complicated tasks. And the same is true in the sciences. You know, we we we keep updating our laws of of of nature so that we can make more complex predictions. When you optimize a language for efficiency, you're basically just trying to compress a description of the universe into as as as minimal and elegant a a form as possible. And so when you're doing that, you are somehow getting to the essence of of how the universe actually works, you know. So presumably, the universe does operate by some laws of nature, which maybe you don't know yet, but we'd like to believe that

• 44:28Speaker 03

  there are these are, you know, these are simple predictable laws, and it it isn't just some big chaotic, there isn't some agent that's just making things up as they go along. Uh, and you know, and and the whole history of science has been sort of validating that that that that belief, you know, that that the naturalistic, you know, um, people to to philosophy. And mathematics has been trying to do the same thing, to mathematical theories, trying to find the the the most elegant, minimal inputs that would explain lots and lots of of mathematical phenomena. So maybe that that's why they sort of converge over time. And this is why we now so observed that the types of mathematical language and formalism that is good for mathematics, for example, um, the language of curved space to describe all kinds of geometries.

• 45:10Speaker 03

  Happens to coincide quite well with with the language that describes the universe, like Einstein's use of that same language to describe spacetime.

• 45:18Brian Keating?

  Figure of space, yeah, exactly. So one of the, you know, questions I love to ask mathematicians that have been on from Jim Simons and Stephen Shrogatz and many others is whether or not you believe that math is invented or discovered. So there's four options. You could say invented, discovered, both, or neither. So where do you come down on this classic, classic?

• 45:38Speaker 03

  Definitely both. So, I mean, we, I think there is an innate mathematical

• 45:45Speaker 03

  structure which we are trying to to discover. And but in order to do that, we have to invent mathematical language. And initially, it's not a very good language. We are focusing on the wrong things. But over time, as I said, to try to make our language more efficient and more powerful, it sort of naturally converges to the ideal, Platonic ideal of mathematics. And that's that certainly feels like discovery.

• 46:10Speaker 03

  But it's done through human, human means. So, yeah, it's both invention and discovery.

• 46:14Brian Keating?

  Yeah, that's what Jim Simons called it. When we look at the future of education, you're not only a fields medalist, a mathematician, and father, and everything else that you do, but you're a teacher and you're an educator. Talk to me about your vision for the future. What's your philosophy of teaching?

• 46:30Speaker 03

  Yeah, so it needs to evolve quite a bit for many reasons. Yeah, so the world has become infinitely more complex and unstable and unpredictable.

• 46:41Speaker 03

  And now, you know, and now with AI, you know, humans used to be sort of have a monopoly on cognitive tasks like, you know, and and now AI. So one of the problems of AI, actually, I mean, the way the subject develops, it's not so much that they overtake human, like research level mathematics or any other discipline in the near future, but already undergraduate level mathematics, for instance, many of the homework assignments that we teach, we assign right now, they can be done by AI.

• 47:10Brian Keating?

  Yeah.

• 47:10Speaker 03

  So we have to reinvent the way we.

• 47:14Speaker 00

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• 47:31Speaker 00

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• 47:45Speaker 03

  We teach, so one thing that will become more important is students will need to have much more training in how to validate information that they see. You know, so in the past, we had like a small number of authoritative sources of information or textbooks and your teacher or something. And, and you know, we didn't have social media and the internet and all kinds of information of now, now AI of all this information of really verbal quality. On the other hand, the, in the past, when you had information that was low quality in content, it was also low quality in presentation. So you could tell that a really well-produced textbook would likely have more accurate content than, than, you know, something written in crayon or something.

• 48:25Speaker 03

  But, but now our, our ability to produce high-quality presentation has far outpaced our ability to produce high-quality content. So you cannot have, have, you know, YouTube videos or, or, or textbooks that look flawless. And now AI-generated output, but have got lots of fundamental mistakes. So, yeah, we, we need to, to encourage critical thinking. I already see teachers experimenting with things like, you know, here is a question that I would have assigned, but I've given it to, to chat GPT. And this is, this is the answer that I give. It's wrong. Please critique it and, and, and correct it.

• 49:02Brian Keating?

  Interesting.

• 49:03Speaker 03

  And these are, I think, are more the skills, more interactive, you know, so not treating knowledge as, as a passive thing to be acquired by authority, but something that you always have to question.

• 49:12Brian Keating?

  And struggle with. Interesting. You know, that kind of reminds me of John Preskill at Caltech, talking about quantum computing and quantum supremacy and stuff with. And one of the ways to overcome some of the issues with error correction in quantum computing is to throw more qubits at the problem. And I wonder, do, will we throw more AIs at the problem? You know, this kind of flipped it. You see through natural, you know, human brains that an AI to prove what's wrong. But will we be at a place where AI could police itself? And so would, what would it take? To trust them.

• 49:39Speaker 03

  It's it's good to to to make them more reliable, but I I think um. Or may maybe if if we. Use a very different architecture from the the current AIs.

• 49:50Speaker 03

  So by by nature, they are inherently unreliable, but. We we have, we have ways to use unreliable tools. You know, random number generators are the the most unreliable.

• 50:01Speaker 03

  Device technology we have, but they're extremely useful for all kinds of things. So crypto.

• 50:04Brian Keating?

  Yes.

• 50:05Speaker 03

  I think as long as you pair these AIs with with good verification and and you. And you you only use the AIs to the extent that that you can verify the outputs and and and no further. Then there can be a great tool. I see them more as complementing human scientists and mathematicians. So. Because there are so few human scientists in the world and and we don't only have so much time to to work on on research. We tend to focus on sort of high value, high priority. Isolated problems, but in mathematics and the sciences, there are millions and millions. There's a long tail of lots and lots of of less well known problems, which. Should require some attention, but. And they're not the most difficult or important, but it'll be good to have some someone or or something look at them.

• 50:50Speaker 03

  And so I think AI actually, their best use case is not to to to target them on the on the most high profile problems, but actually. On the millions of medium difficulty problems. And you know, they they may fail and they may only do, they only solve 10% of these million problems, but that's 100,000 problems solved. So scale is is the big advantage. You know, you cannot scale a graduate student.

• 51:10Brian Keating?

  You're right.

• 51:11Speaker 03

  This way, okay.

• 51:13Brian Keating?

  But not legally.

• 51:14Speaker 03

  No, not legally, uh, yeah, or ethically, okay, but but AI, you know, I I think that that's that's where the the big, the real value lies.

• 51:21Brian Keating?

  What's your highest priority task right now?

• 51:23Speaker 03

  Well, uh, research wise, uh, what I'm interested in most nowadays is is new workflows to.

• 51:31Speaker 03

  Modernize mathematics and make it more more collaborative, more more accessible to the public and to integrate in these new tools like AI. The way we've done mathematics has not changed fundamentally in centuries. You see, I have blackboards in my office. We still work with pen and paper. We use computers a little bit, but not so much. And our collaborations are still very small. We work with two, three people. In the sciences, of course, thousands. In large part because we don't know how to incorporate contributions to the general public. There's a barrier to entry, first of all. A lot of what we do is very technical, but we need to synthesize proofs where every single step has to be verified. So if we had thousands of people, it'd verify a thousand little components, it wasn't feasible until very recently.

• 52:16Speaker 03

  Also, because of all these factors, we don't collaborate as much of the other sciences as we ought to, especially the new sciences, which are so data-driven and connect the reward in new ways, like social network analysis or whatever. So that is, I think, the direction in which my research is going to do it. It's almost more the sociology of mathematics actually than the technique. And more recently, I've been interested in trying to secure funding for mathematical research that has become very unstable in recent years.

• 52:47Brian Keating?

  We'll talk about that in a bit. So Rachel friend Sergio Kleinerman asked me a question related to what you just brought up, sociology of science. And he wondered how it was stressful for you to be reputed the best mathematician on earth, the Fields medalist, a very young and extremely successful mathematician. Did that affect you? Was that a challenge for you with that mantle that weighed on your shoulders perhaps? Or maybe not?

• 53:12Speaker 03

  Okay, I do remember the year that it goes down, 2006. My life did change in many ways. So somehow I got invitations like embassies and I would meet with people who I would normally meet. And I got asked to be on all these committees. Suddenly my opinion was swapped after. So that was a sea change. I mean, I was on some of these already, but so that took some getting used to. But I think one thing that helps ground mathematics. A little bit is that, you know, I mean, as a mathematician, your your main task is you have these problems you want to solve and you need you want proof theorems that that solve these problems. And your proof has to be correct and every step has to be updated.

• 53:53Speaker 03

  And it doesn't matter how famous you are or how much reputation you have, you can't just say, I've proven something, trust me, okay? You have to supply the details of the proof. And if you don't have the proof, you don't know the proof. So I think this naturally provides some check on on just sort of how high your ego can go just from these awards. Because, you know, I mean, the countless problems that I would love to solve, you know, the shrimp and have conjecture we talked about, but but the hundreds of problems that I would love to solve. And I just know I don't know how to solve. And so I know more problems I can't solve than the problems I have solved. So I think that, you know, so that keeps you somewhat honest.

• 54:31Brian Keating?

  What about the, you know, the old trope that mathematicians do their best work by age thirty, where you're in our fifties now, you and I? What what do you make of of that statement? Jim Simons used to tell me, he didn't really believe it. He thought that actually a clock starts, you know, at a certain moment and then you have ten years or twenty years to do your. And he he stood stop at age thirty, but that was because he worked at Pennsley for ten years, not because he had an arbitrary age.

• 54:54Speaker 03

  Right.

• 54:55Brian Keating?

  You've heard this trope. What do you make of it?

• 54:57Speaker 03

  Yeah, so different mathematicians have had different career facts. So I definitely was had was stereotypical. That's why I had, I kept it then, I skipped the child at that, I accelerated.

• 55:11Speaker 03

  I skipped several grades. And so, yeah, I did what I worked when I was younger. But there are a lot of mathematicians who started quite late. They were good. They didn't become interested in mathematics until college, when they switched, became quite good. My advisor, when I was in Princeton, my associate advisor, I started, I would meet with him every week. And I'll discuss the problems that he'd assigned me to work on. And I'd spend hours trying all kinds of crazy things and I'll report all these things I tried didn't work. I tried this to do work, it's like all this energy at time. And he would just sort of look at what I wrote in Blackboard and and just think for a few seconds, he said, you know, The difficulty you're having is exactly the same difficulty that so and so had in this paper. So if you go spy on cabinet and you push out this, this one paper, a preprint to

• 55:55Speaker 03

  read this, this will solve your problem. So yeah, there was, there was a different way of doing mathematics. I, I didn't, I couldn't see how you put, because I would go home and read it and it would solve my problem. I would then hit another fracture in the next week. But no, I spent hours on these problems. And he just thought about it for 10 seconds. And he just knew from experience what to do.

• 56:18Brian Keating?

  It's the wisdom.

• 56:19Speaker 03

  Wisdom. Yeah, wisdom. So I think as you get older, you, you approach, you, you find different ways to do mathematics, which it, it, it may not be as flashy, you know, it does have more brute force than it will be as, as if it has, but like it can be more, it can be more productive. I mean, yeah, I can now pull the same trick off my own guidance students. Okay. And so it's like grand satisfying because you see full circle there.

• 56:43Brian Keating?

  You could do second order. You could say when my advisor told me, you're the grand advisor. Let me ask you a question related to pedagogy. So it's obvious, you know, from what we've already talked about with Wigner, that math is really important for physics. Do you believe that there's a, an experimental or physics minimum amount of knowledge that a mathematician should have? I've, I've asked this of theoretical physicists. It's much more closely related to experimental physics, but do you believe that there's a certain amount of connection to the real world that a mathematician can benefit from?

• 57:13Speaker 03

  Oh, definitely. I think a long difference in mathematics is that there are so many ways to approach mathematics. So you, you can, you can be a very visual mathematician. And, and so you see pictures, you can be a very symbolic mathematician and, and you just view it as a game of manipulating numbers or symbols. Or you can be a very physics oriented mathematician and you always use disco analogies and you use insights from, from various subfields of physics to, to help you. I mean, so there's some very direct connections. If you, if you study classical equations, then very naturally you, you should know some physics because physics has so many examples of, of, of, of quick, of quick transfer equations. And having intuition about, say, how fluids

• 57:56Speaker 03

  Work or how how waves work really, really helps me. And I think it's in general, just the more you know in other areas, you know, I mean, I sometimes I find them from thinking economic terms. Like, if you want to prove that X is less than Y, what I think about it is that if you own Y amounts of stuff, then can you buy X? All right. And sometimes if you don't have a, you know, sometimes you can't do it directly, but but maybe you can trade in Y per Z and then use Z to to buy X. So like, um, if you have, uh, if you put yourself in a mindset of of you, you have some bizarre and and you can, you can, you can, there are certain merchants where you can trade X for Y. And but you want to negotiate, when you want to get a good price for these things, then you you you don't want to trade X, Y from Z if it's a bad deal.

• 58:37Speaker 03

  That kind of mindset can actually could be be very helpful in in seeing sort of the right route, how to get from from from from X to Y. Sometimes it it you can take some types of metagrams, you can think of as as games. So in in analysis, there are lots of the same as which say things like for every epsilon, there's a delta, blah, blah, blah. It's got epsilon delta type type proofs and and they undergraduates are often very, um. I hate those because even, yeah, they're they're they're quite complicated. It does the games. Like, if if you're used to to games like chess and something else, like you, if, you know, if your opponent moves here, how do you counter that move? And so if you think, like, every time someone gives you an epsilon, you you need to find a delta to to counter it.

• 59:21Speaker 03

  And if you think in these sort of game theoretic terms, um, sometimes that can can can provide you an, or give you a mindset. So, yeah, I mean, you you can use intuition in biology, some social sciences, no, every academic has a has guys.

• 59:37Brian Keating?

  Yeah, that reminds me of this, uh, book that I've been wanting to show you and we we we did take a look at it before we started recording. So this is called the Compasso Geometrico. So it's the English version is Galileo's, it's by Galileo, Galileo. The Operations of the Geometric and Military Compass. This is not for finding north and south, but instead it's for finding, um, really doing a calculation. Since it's really an an early version of a slide book. So this is the sixteen forty nine. Second edition, the sixteen oh one first edition as several times our salaries at the University of California. So I didn't afford be able to afford that. But what's so amazing, in addition to Galileo's actual signature, which we can zoom in on there, I don't know how

• 1:00:21Brian Keating?

  in this paper, seven years after he died, but he had a stockpile. He was a minor celebrity. And he never left Italy. He never got outside of Italy.

• 1:00:28Speaker 03

  Just the observed for forty.

• 1:00:30Brian Keating?

  It is, it is. Isn't it beautiful? I find it like a treasure. I'll bring it up in just a bit. But here's an example of it. So it had segments. It had, it had, it was made of metal and it had indications on it. It could do angles and so forth, but it could also do calculations. And one of the calculations, kind of funny to think about, is he goes in, in this, I think I mark it in this post in one. Take a look at that page, Terry. He talks about, it's basically an instruction manual. So nowadays we get the device, we get an iPhone. It doesn't come with an instruction manual, right?

• 1:01:02Speaker 03

  Yeah.

• 1:01:03Brian Keating?

  And you're expected to be able to use it. So at some point he starts talking about, you know, comparing lengths of lines. But I think on this page here, he goes, real for monetary exchange. So you just mentioned this. You want to read that? That would be cool.

• 1:01:14Speaker 03

  By the means of the same arithmetic lines, we can change every kind of currency to every other in a very easy and speedy way. We first set up an instrument, taking that price, the price in money you want to exchange, and fitting this crosswise to the price of money which the king just made. We illustrate this funny example, but this everything is clearly understood. Suppose you wish to exchange Florentine gold scooty into vintage attacks. Since the price of value of the cut is six liter four sooty is necessary. We work out all kinds of sooty, given I scoot as price of one sixty sooty, the price of the cats one twenty four. I'm so glad we ought to do this in what?

• 1:01:46Brian Keating?

  Because I get into the fun. Exactly. I think it's so funny because, you know, nowadays the scooties worth nothing. I mean, it might be worth a couple of dollars or whatever. But if Galileo had just, you know, put away a couple of first editions of this book, you know, where his heirs, they'd be worth billions of dollars back. But we mentioned, you know, this, this notion of currency conversion. And, you know, my friend Eric Weinstein, and I know I'm known the same, I've worked on, you know, gauge theory. Yeah, yeah, applied. But so what do you think of this? I understand. That's. Okay, okay. Yeah. Currency exchange actually is a very good example. Yeah. So so gauge theory has this reputation of being this, it's really.

• 1:02:20Speaker 03

  Um, obstruse area of physics and mathematics, which, but it comes down to, to, to many quantities in the real world are scalar, but they don't have a natural unit. Um, so, um, yeah, so, so currency is one example. So, you know, if, if I have a certain amount of wealth, I can measure it in dollars or, or euro or whatever. And so you can refer to a, to a number, but, but, but, um, it is not actually itself a number or it's not a number. But you can measure by numbers. Gauge theory is, is about quantities which can be measured by numbers, but, um, or by coordinates X, Y, Z. But there's a choice of, of, of which units to use or which axes to use.

• 1:02:53Speaker 00

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• 1:03:23Speaker 03

  And, and maybe if you're a different location on the earth, or, um, you may have to use different units. And so as you go from one country to the next, um, you know, your, your, your, your units may change. So you, you, you need some way to convert as you, as you go from one location to the next. Similarly, um, so, yeah, in, in real world, the electromagnetic fields, which, uh, as we teach in high school, we teach that these fields are vectors. But, you know, um, there's, there's, there's some triple of numbers at every point, we call E and on for B. But actually they're not numbers that they are, um, they're directions in some, in some abstract, um, um, in some abstract space. And so as you go from, as you move from one place to another, the, these numbers will, to change in a certain way. Uh, and yet so gauge theory is just about how, how to, to manage these conversions.

• 1:04:08Speaker 03

  And one day you may decide that I'm going to price my, my currency here, not, not in pounds, but in, in naira. And so, um, that doesn't change how wealthy you are, um, but it does change the, the gauge. And so there's a mathematics of how this gauge works and and which things are gauge invariant, which things are not. So for example, the natural curvature, if you go around in a loop and you follow and you just transport whatever your vector, whatever it is along with your gauge, sometimes you end up back where you started and sometimes you don't. There's a correction and the correction is and it doesn't matter actually what units of currency or what was your gauge or is it gauge invariant. So for example, if you have a certain amount of dollars and you travel to Europe and you go to euros,

• 1:04:52Speaker 03

  then you show back to the euros and go back to the dollars because of exchange pieces or you might not have exactly the same amount of money. So in a sense, that is some curvature. It's not exactly curvature, but it's a bit like curvature in the in the currency bundle of the world. Yes, currency actually is a nice metaphor.

• 1:05:15Brian Keating?

  Yeah, okay, so yeah, I mean, it's surprising that you get from different symmetry laws and so forth, that you get the properties that are unexpected and things emerge where they wouldn't be expected. And one sort of commonality, a fellow Fields medalist, I believe the only physicist to win it is Edward Witten in the Institute for Best Study. And of course, he's known for contributions to string theory. String theory is our 60 years old of us. So fellow Fields medalist Edward Witten, Institute for Best Study, was the first physicist, maybe the only physicist to win a Fields Medal. He's worked extensively on gravity and string theory, et cetera. What do you make of the current status of it and the mathematical nature of it? That seems to only be able to solve things in very high dimensional spaces for which we have no evidence.

• 1:05:58Brian Keating?

  Where's your outlook as an outsider perhaps?

• 1:06:01Speaker 03

  So it's, I mean, physics, as you know, of course, history of we had to reevaluate our conception of the universe and the nature of reality several times already. And so it was the Copernican revolution that leads one bigger. This is simply universe. You know, this is Einstein's relativity that Einstein had to be curved. And then of course, this quantum mechanics that that reality is actually this should be represented by wave functions and quantum fields in a way that this series now victimizes on success, right? Because we can now explain Like all 99.9% are observable phenomena by these theories, except that at really tiny scales or the origin of the universe, you know, it doesn't work. And the mathematics is inconsistent. And so we have to replace it by something.

• 1:06:46Speaker 03

  So in particular, the idea that spacetime is a smooth manifold does not seem to be compatible with observations that people think. So we need something else to replace it. The problem is that there's infinitely many candidates for what to replace it with. Despite mathematics being reasonably effective, ask you the right mathematics. You can't just conk in any biblical theory and hope that this will explain.

• 1:07:07Speaker 03

  So, yeah, for many, for many decades, string theory was the leading contender. It's a very elegant theory. I'm not an expert, but I understand it has not quite lived up to the expectations of providing. At least not a unique canonical theory that would fit the data. Maybe it's too flexible, gives you too many possible ways.

• 1:07:30Brian Keating?

  And that brings up, you know, a question I've been meaning to ask you. In mathematics, there's Gödel's incompleteness theorem, which says it's bound on what's possible to extract from a given system of axioms. And possibly bound what's possible to prove, as I understand. In physics, we don't have that, right? We don't have any proof. We can't prove that gravity is always 9.8 meters per second here, right? So it's provisional and subject to new data, right? So and that's part of the beauty of it. But the closest we seem to have is what Popper, you know, suggested as the sine qua non as a definition of good science is that's falsifiable. Do you think, I've often joked that physicists have mathematician envy. A lot of people say sociology has physics envy.

• 1:08:09Brian Keating?

  But I think because we can't prove stuff, so is there always going to be this limit to, you know, what is capable of being asked of of a physical theory? Because we can't, as I said, we can't even prove, you can prove one plus one equals two. It takes, what, a backlog identity and it takes 200 pages. But, but we can't prove anything in physics. Where does that leave us in the epistemological search for truth?

• 1:08:31Speaker 03

  I think you just always have to keep separate the real world and our models of the real world. So, I mean, physics. Has provided us with with mathematical models which which are which you can prove things. So relativity, for example, Einstein's equations are completely precise mathematical equation. And you can you can specify initial conditions still. If they really didn't specify the initial conditions of spacetime, you can, you know, there is one mathematical solution. And the system delivers, okay. But, you know, and you can prove fields about that. And and and so the models are are you can you can both follow by and I mean that that they are. They have that they are on on the status of of a mathematical construct.

• 1:09:14Speaker 03

  Where the physics physics comes in is how that model interfaces with reality. So, you know, even if it doesn't quite match, you know, even if it's technically also by by by experiment. Doesn't actually mean that the theory is destroyed. Newtonian gravity is still a very useful theory. Even with technical reasons, it's the mantra is accurate. It's good enough for, you know, for modeling, you know, planets and comets and so forth. I think as much as you don't conflate your model with the reality, you can have both your mathematical cake. You need it too. Yeah.

• 1:09:43Brian Keating?

  Very good. Just as we were wrapping up, Terry grabbed the chalk and gave us a lightning talk about how he helped to crack a brutal image analysis problem. That was vexing physicians trying to get the best quality images out of their MRI machines. Terry and his colleagues cracked this mystery using what he calls compressed sensing. Using math to reconstruct physical images from far less data than ever before. The result, MRIs that run up to 10 times faster. Enjoy, you're in for a treat.

• 1:10:11Speaker 03

  I was talking to some statisticians and engineers about an image acquisition problem, which they had converted into this sort of math puzzle about how to solve a certain system of linear equations. And they were reporting some some results which were amazing that they were they were getting. They were able to reconstruct an image using much fewer measurements than traditional imaging. And they're hoping to to to to use this for medical imaging. And I I talked to them and I I I solved their their their little linear algebra. In fact, I first was trying to disprove it because I couldn't believe how good the results were. But I I I on trying to do that, I figured out how how it worked.

• 1:10:52Speaker 03

  And this technique, we we published it and it it it became very widespread. And in fact, nowadays, um, um, most of the the big manufacturers of medical MRI machines, they use our technology, um, methods, um, which is now called compressed sensing, to um, to speed up MRI scans by like a factor of ten or so. You still never know. I mean, there's um, um, a lot of work I I do, for example, these days is is how to to tell, given us some sequence of of of numbers, whether it has patterns or whether it's structured or whether it's random, and and what kind of tests can you can you apply and and what uh, which tests are sort of better than others in in various ways. You know, as I said, you know, there could be ways you could use this to to uh,

• 1:11:36Speaker 03

  detect fraud maybe, um, or or uh, uh, or filter out noise and and try to try to, you know, get better signal acquisition algorithms. It's our ecosystem. Um, I think you, uh, in order for the more applied scientists and engineers to to get the ambient ideas from the literature in order to solve their problems, they need the people from the more basic sciences to to ask questions more in a curiosity-driven way. And maybe things that we do, if they don't directly have a practical impact, but this, yeah, this is unreasonable effectiveness, you know, like, if you if you don't have these people asking these questions, the people downstream who are actually trying to make things um, practical application,

• 1:12:18Speaker 03

  things a reality, they can waste, um, they can, they can, they can spend a lot more time and maybe a lot more money, you know, trying to invest, you know.

• 1:12:27Speaker 03

  Like, to to to give one example, Shannon developed this this theory of communication complexity over a century ago. Just theoretically, if you can only send a certain number of of bits of of messages per second, how much information can can you send and what's the best way to compress this data? And there's this whole practical theory that was developed actually long before the the digital revolution. Later, when when. The when we needed, you know, when everyone had cell phones and we needed to transmit huge amounts of data simultaneously and we want to make sure that cell phones didn't interfere with each other. All this mathematical work was really important. It may not have directly told you how to build the phones, but it did things like it provided the theoretical limit.

• 1:13:09Speaker 03

  It's called the Shannon bound, like exactly how much information you could cram into a certain amount of spectrum. And so because of that, you could you could plan, you could you could buy, purchase a certain amount of spectrum and you would know sort of theoretically how much information you could communicate from that. And you can you can do budgets, budgeting and planning. And there's still lots of engineering that needs to be done, but mathematics can tell you what's possible. So, yeah, you need this basic science and it's much cheaper to do that when you're still mathematics and you do it by pen and paper, rather than deploy a billion dollars and realize that it doesn't have the capacity that you need or has too much. Right. In which case it's wasteful. Yeah. Awesome.

• 1:13:50Brian Keating?

  Okay, I know if you enjoyed this conversation with Terry, you're gonna want to catch part one of our interview. We talked about the dramatic cuts that Terry faced at UCLA thanks to the Trump administration's policies in the middle of 2025. And you'll also want to check out my recent conversation with Stephen Wolf from one of the deepest-thinking mathematicians of all time. Don't forget to like, comment and subscribe.

• 1:14:14Speaker 04

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• 1:14:44Speaker 05

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