Into the Impossible With Brian Keating - Can AI help us solve the hardest problems in Mathematics? (ft. Terry Tao)

Episode Date: December 30, 2025

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

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While supplies last, ends June 30th, terms at AKA.m.m.S. slash college PC. 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. There are 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 you

This solved more legendary math problems than almost any human alive. Terence Tao won the Fields Medal, the Nobel Prize of 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. We're going to talk about the beauty of numbers, why AI keeps getting the math wrong, what it was like to meet the legendary Paul Erdush 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? I actually don't drink coffee much except in social occasions, actually. Black, no sugar.

Okay. So the reason I ask that, maybe you'll recognize as Erdosh, I believe, said, what did he say about mathematicians and coffee? 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-mathetician is a way of turning co-theums into fee. Interfee. Yeah. That's a very inside joke. That's right. That's a dad joke plus a mathematician joke. That's really bad, but really good at the same time. Well, the reason I bring up Erdash, of course, you actually met him when you were a kid, didn't you? 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 Sackarash. 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 Erdush was always very good at, he was known for, for, um, he was known for, um, um, uh, 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 much of an honor was really. The one thing I remember was that he really treated me like an equal. Like, you know, he didn't condens him as a kid. And he later sent me a postcard that it just had, thank you for your nice hospitality. Here's a math problem, which I didn't solve, actually, but it did get us up later by someone else.

That's amazing. Yeah, he was one of those 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 Erdash number? Yeah, so there's this concept called the Erdush number. So Erdish worked a lot in graph theory, and so this concept is inspired by graph theory.

So Erdich himself has an Erdish number of zero. If you've written a paper with Erdish, you get an Urdish number of 1. If you've written a paper of someone who's written a paper of Urdish, you have a original number 2. So I have an Urdish number 2, for instance. And I think nowadays people is common to Uridians are four or five.

People have made similar numbers in other fields as a bacon as a Bacon number. So if you've starred in a film with Kevin Bacon number of one and so forth. And then there's something called the Erdish Bacon number,

which is the sum of your Urdish number and Bacon number, which is usually infinite, because you either don't have a chain of papers going to Urdish or you don't have a chain of movies going to Bacon. But there are a half dozen people who have like,

combined number like seven or eight. 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 was incredibly productive, but part of the productivity relied on the use of amphetamines. He used to take some, is that true?

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 Erdush, drudgingly did it. And then 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 back then maybe there was less of a stress in work-life balance that you are today.

Now, you had work related to Erdos, right? Erdos's discrepancy theorem or something? Yeah, what is that? Can you explain that from my audience? Okay, so Erdush was famous for posing many, many problems, and I solved a few of them 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, if you pick say 1,000 numbers at plus or minus ones at random, you'd expect 500 of them to be plus ones, 500 minus ones. So the discrepancy is, defines the difference between the number of plus ones and the normal minus ones. A sequence can have low discrepancy if you view it over the whole sequence. But if you look at sub sequences, the sequence can have a higher discrepancy. So, for example, if you take an out, outnake sequence of plus one, plus one, minus one, a thousand times. The discrepancy over the sequence of the sequence of the whole interval is zero because you have 500 plus ones, 500 minus 1, it sums to 0. But if you look only over the even numbers, you just see plus 1, plus 1, plus 1, plus 1, or minus 1, minus 1, then you have a very large discrepancy, I think 500. Some sequences, they can be very well balanced overall, but when you restrict to sub-sequences, they can have a higher discrepancy. So Erdisch was interested in whether you could design a sequence which had what's called bounded

discrepancy over all what I call homogeneous ethnic progressions. So could you create a very, very long sequence of plus ones and minus ones where if you look at any finite segment, say from one to 100, 1 to 1,000, the number of plus ones and minus ones only differ by at most two, same. But also if you look over the even numbers, same thing happens. You look over models of three, same thing happens. So they wanted it as 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 1,164 elements

where the discovery 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 discovery of three.

The sequences became 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 Erlisch asked, do these sequences, if you continue these sequences on forever and ever, must the discrepancy eventually go to infinity?

And this is what I was able to show. Oh, wow. So it does go to Invergivers? Yes. Yes. Extremely slowly, as far as we know, like logarithic or double it logarithic, but it does go to infinity. And I had to use tools from information theory and number theory. I've heard that there are people that use some applications of you.

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 that randomly, true false, true, false, and it'll be too close. The sum would, you know, plus minus would go to zero. Is that, is, is that, any relevance? 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 based to that, but there are other patterns. The most famous is called Benford's Law, which is a very unintuitive law, that roughly speaking, 30% of all numbers in the world start with one, which sounds very weird because numbers can start with one, two, three, or up to nine. But you can take, for example, take all the countries in the world, take their population. And there's about 100 odd countries in the world, about a third of them, the population will start with one, China, for instance.

Or you can take the net wealth of several millionaires and billionaires, or whatever, and you also find that most of them start with one. Or birthdays, the pattern is quite universal. But whereas if you pick numbers randomly, if you like fudge your accounting books, when you pick numbers randomly, artificially, they don't necessarily obey that. They're uniform, so you try to make them,

you think that uniform is correct. Yeah, so humans are actually really quite bad at creating truly random patterns. And so, yeah, you can distinguish natural patterns from human-generated ones. Interesting. So one thing that I've, you know, And I've been dying to talk to you about for a long time are kind of 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 harken to the work of Jim Simons. He's most famously known for being a multibillionaire, establishing philanthropies that support my research and hopefully other very competent scientists. But one of his lesser-known things that was actually very important in, at least in my understanding of how mathematical induction works, or violates is his work of minimal surfaces where he showed something really fascinating. So I should have you to explain one of minimal surfaces, but as I understand it,

you can sort of think of it physically if you had some shape, 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 obtain would be called a minimal surface. Is that correct?

Okay. And then he showed that there are such minimal surfaces and 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 then you get to eight. Right. Actually, it didn't work.

And I would have stopped that, too, 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? Right. Yeah.

So induction means different things. I think the philosophers and philosophy of science induction rivers is something slightly different where you take facts that you observe from small examples and you, you induce from that, what things, a prediction for what will happen for larger cases. And it's a very basic procedure in the scientific method because you know, you know, you do experiments and then you extrapolate from the experiments. Mathematical induction 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 what's true for one. And whenever you know it's true for some number n, you can, you know for sure that it implies the same thing by n plus one. Then it implies it's true for all for all numbers. The analogy often given is just a row of dominoes. So if each domino represents one case of what you're trying 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 string of dominoes is, you can knock over every single. domino chain. But it's really important that your arguing is that 100% watertight. You know,

if it's between the dominoes and like the 97th domino doesn't tip over the 98th, you know, then 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 100% guarantees. So Simons, yeah, so he discovered what's called the Simon's cone. Yeah, you're pushing a little bit because this is geometry is not completely my area of mathematics. But yeah, yeah, so minimal surfaces most famously are two-pensioned service like soap films, 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. At one dimension, the minimal services are very boring, 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-dimensional, Sometimes it makes sense as a mathematician to first study higher dimensions, it gets you some intuition, which can help guide you with the problems that you do care about. Yeah, so it turns out, yeah, that in up below eight dimensions, I think here, all minimal surfaces are smooth. 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 can actually form, yeah, that there is this, yeah, it looks like the cone except in much higher dimensions, and there's no way 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, the duals sort of surgery. You remove the vertex of the cone and replaced by two rounded nubs, okay, and that would reduce the surface tension. Interesting. You can't do that in higher dimensions. So nowadays, because of data science, actually. We need to understand high dimensional geometry is 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 high dimensions. So just to give you one example, like if you inscribe a circle inside a square, it occupies a fair, you know, a pretty large chunk of the, of the square, maybe like 75% 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 0.00.0.1%. Like, balls become extremely poor space filling. Yeah, there's 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 there's some errors

in them. You know, do you measure the root green square error, which is like trying to place this, your measurement inside some ball, or do you measure the worst of the 1,000 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. Right. There are 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 proof by ridicule out absurdium. Can you talk about What's your favorite type of mathematical proof?

When you're onto it, you just get so excited to finish the film. Proof by contradiction, I think Hardy had a great quote that in chess, a chess player may offer a pawn or a bishop, but a mathematician offers the entire game. You know, so it would say, okay, we want to prove this is a conclusion. I will give you that the conclusion is false. I will just let you run with it. But you do that, and I will show that it gives you. your contradiction. It actually is a technique. So 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, 1,000, and then they'll go a million, a billion, a billion, billion. And they'll go on like this. But at some point, someone will realize, the one of the kids will realize that no matter what number the other kid says, they can just say that number plus one. They have proven that there was no largest number in the natural numbers. And this is a proof of our contradiction because if anyone ever did claim that natural number years, largest natural number years, 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 can teach themselves.

The type of mathematics that I, 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 have a problem which has to do with common torques, nothing to do with the real world, but you find that there is some physical model of it, and you can use ideas from physics.

Of course, as physicists, it does all the time. They also have correspondences, which are really quite amazing. So, yeah, those feel like magic to me. Yeah, I mean, the most famous one, at least in my physicist's mind, is the proof that by contradiction, that square root of two is irrational. So that's what the Euclid's original proof, or what does the trace back to before him? Euclid or Pythagor, I think that's Apathagorean. Maybe Pythagorean.

You could prove the prime basically by the same sort of idea as the infinity plus one, right? 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 notion of proof that complex facts about mathematics

you could deduce from simpler axioms and it was extremely influential way of thinking which it hadn't seen before. So the square root operation, just as a notion, has always fascinated me. You know, for one thing, it seems to occur in physics, you know,

quite regularly, and I'll get in some examples that peak 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. It's peak pollination season, and my business is scaling fast.

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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 physicist. 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 LLMs, they use L-U-D composition, and we have spinners that would have a spinner representation in our square roots. Is there something special about the square root?

root or like, in other words, why don't we say the cube roots or fifth roots or a hundredth root? Why is the square root something that's so prevalent in physics, for example? Yeah, 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, X, Y, and so things like addition is commutative,

X plus Y is Y plus X, and X times Y is Y times X, multiplication is commuted 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 1 equals 0, in the real numbers X squared plus 1 is never zero because X squared is always positive. 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 an odd degree, like a cubic, XQ plus 3x plus 1, it must have a root because a cubic polynomial or an odd degree polynomial, when you make X very, very big,

it becomes very large and positive. And when you make X very large and negative, it becomes negative. And because the reels are continuous, polynomials are continuous. Therefore, at some point in between, you must hit zero to get from negative to positive. Sort of half of all the polynomials in the world, you can solve them in the wheels, and half you can't. So, with the benefit of hindsight, this really suggests that you should make the real so twice as big 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 they were there. 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, and not a nice algebraic structure, ultimately coming from this algebra completeness. 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 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 vector respect. how you might have predicted. Yeah, I mean, this is not historically how complex numbers were discovered, but this could be sort of one explanation. Hey, everybody, I'm usually the one that asked my guest

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and my publisher's got an 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? Yeah, yeah.

But they don't have roots of polynomium. 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 to taught in school and then not have that changed. 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. Oh, really? I mean, I still consider it prime, but I still consider a planet. Yeah, yeah. But it's because in all mathematicians and science, in any people 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 the new notion of a planet is better.

But for example, what we've learned is that one of the really important properties of primes is what's called the fundamental theorem of arithmetic. 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. But other than interchanging the order, that's the only way to break up a number to primes, just like there's only one way to break up a chemical compound into atoms.

So primes are like the atoms of mobication. But if you made one prime, then you would lose, you have to give up the fundamental theorem of arithmetic, because now 12 is also 1 times 2 times, 2 times 3. and you just add too many exceptions to this really important fact in number theory. So we made a decision to therefore redefine crime numbers. Oh, when it was. Okay.

But just like with Pluto, there's no consequence to life on Earth. It's more sort of... Yeah, it's a human convention. But we update our human conventions to match reality better over time. I've done work in prime pairs, is that right? Yes. Yeah. So primes are one of the oldest.

subjects in mathematics, Euclid had the first theorem almost ever, and it was about prime numbers more than 2,000 years ago. And so it's very frustrating and annoying that even the most basic questions about primes, we still cannot answer definitively.

We have good guesses. So for almost all questions on the primes, we can predict the answer, but we cannot get the 100% mathematical standard of proof for many of them. And one of those basic questions, which is at least 300 years old is called the twin-pime conjecture that there should be, so you could show there's

infinitely many primes, the primes never end. You can always find primes bigger than any number you wish. But we cannot find, we cannot say the same yet for prime twins. So these are pairs of primes that are differ by the closest they can, which is two. For example, 11 and 13. Well, okay, two and three are closer, but after two, all primes are odd. So the closest you can get is two. And so we can observe that every so often, you know, the primes, they don't seem to obey a pattern. Sometimes the prime gaps are large, sometimes they're small, but every so often they come close to each other and you get a twin, and they seem to occur infinitely often, you know, as though 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. And random sequences, if you have a random sequence of the same density as the primes, they hit form twins infinitely often. But the primes are not random. We believe that what's called pseudo-random, that they have no obvious pattern besides the ones that we can obviously see,

such as them being odd. So, I mean, it's a very likely hypothesis, but we can't prove it. Pseudorandumness, meaning that it could be derived from some algorithm, but not in all cases or something. What's pseudo-random versus random distinction? Random means not deterministic. That there's no single...

So the primes, If you forgotten on the primes were, and you have to regenerate them by a computer program, you would generate exactly the same set. Whereas if you were generating a set by rolling dyes or flipping coins, you would get a different set. Sudorandum are sets which are either random or deterministic, but statistically they are indistinguishable from random noise.

So for example, a random number should just, if you have a random sequence of numbers, there should be just as many numbers that end in five or in seven and six, like the digits should be equally distributed. Now, the primes, they're not perfectly pseudo random, because they do avoid certain patterns, like they tend to be odd, for instance. But there's ways of excluding those obvious biases.

And once you exclude them, it's expected that there's no tests that can distinguish them from random numbers. This is important for cryptography, actually, because there are many crypto systems, like the ones we use to encrypt web traffic, cryptocurrency, you know, financial transactions, where data, like sensitive data,

like a passwords or credit card, numbers are encrypted using mathematical routines that rely implicitly on crimes having no pattern. And so 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 conveys no information about your personal data. And we really hope that that's true. I mean, 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 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 prior, very elliptic curves and other things, where people actually had to migrate to a different crypto system because of these weaknesses. Yeah, so that brings up what kind of an inversion, maybe a contradiction of what Wigner said. You commented on the unreasonable effectiveness of mathematics in the physical sciences or in 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 in pure mathematics from the first. physical world, you know, the quantum world to the mathematical world. Do you see that as, you know, sort of a viable topic? So quantum computers are a fascinating topic. Yeah, they interface of maths 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, of a quantum computer, the state is not a bunch of zero one bits, but it's a way function and the operations you're assigned to them, you have to mop them, you're only allowed to mop them by matrices. Really, really large matrices, except that your basic operations, your matrices are mostly the identity. And there's 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.

Quantum computers are both exponentially more powerful than classical computers, but also exponentially more limited. Yes. So because they can handle superposition of quantum space simultaneously, in principle, there's this exponential speed up. And for certain applications like factoring, and I think quantum chemistry, this is... Modeling. Magrongians.

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. Non-destructive. Yeah.

So this has, yeah, so this requires you to develop the use of reversible computing. Area correction is also much, much more annoying. So, yeah, so there's software challenges. Maybe once quantum computers become a reality, they could be used to do large-scale computations or 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. Your summer starts now with Memorial Day deals at the Home Depot. It's time to fire up summer cookouts with the next grill, four-burner gas grill, on special buy for only $199.

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Historically, mathematicians only cared about whether something was true or false or provable or provable or disprovable. And with the advent of computers, people also started asking questions of how computable is an object. 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 is true or it's false,

but how easy is it to actually compute? And that has led to very productive mathematics. I mean, sometimes just the effort to not just show something exists, but actually find it, creates new techniques. Complexity theory has offered as sort of given a much more nuanced understanding of how true a statement is.

And yeah, this has led to a better understanding of, you just prove something is true and you don't, but you may not have any insight. So what was the key ingredient that made it work? Or if you had two different proofs, which proof is better, but maybe one proof leads to a faster algorithm than the other. And so you can say, oh, that proof actually is stronger.

It's more efficient. It's more efficient. So it indirectly sort of provides much more insight into the proofs that mathematicians want. Has AI actually enabled new discoveries in mathematics or new proofs that otherwise would not have existed? Slowly it's beginning to. I mean, by itself, so the big weakness of these AIs right now is that they can begin to be, produce output that looks like, say, a human mathematician reasoning their way through a problem, but it's not grounded that it's probabilistic.

They often make mistakes, and much like, you know, if I were to get a student to solve a problem on a blackboard and they're nervous and 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 fundamental knowledge of what they're actually doing, once they go off the rails, they can go really. of the rails. And this is something which is a fundamental problem with the current large language models. But if you use them as a component of a more rigorous and grounded reasoning systems, so if you converse with a large language model for them to make suggestions, but you understand

the output and you can verify it. So then people have had some success talking about their math problems, a large language model, a large language model will produce some suggestions. some of which the human expert can dismiss 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. So one thing where AI models are already beginning to be useful is in like literature review type tasks where there is a class of problem.

And in the literature, there are maybe say a dozen ways already to attack this problem. And if the human working on the problem, maybe you can remember six of them, but you forgot, the other the sixth and I don't come to your mind. And you can use these language models to prompt you, to remind you of the missing six. Then they also hallucinate three more that doesn't exist. So you do need, you can't trust them. Supervised them. Yeah, you have to verify. There is hope in the future, you know, so there's this separate method technology to, to order to have, the software that can automatically verify certain types of proofs. Right. And so the hope is that if you force the large language models to only output in some verified, some language, um, some

that you can verify and to filter out the hallucinations. Has it been able to reproduce, say, while, you know, proof of formats last theorem, or your work, now of your Stokes? I mean, has it been able to actually just simply, quote-unquote, reproduce what a natural intelligence person like you do? There's this issue. It can, 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 discern when that happens whether it was training data or whether they really sort of thought it up. If you ask the AI to explain their chain of thought, they often give complete nonsense. Like it's clear that they just didn't know.

Yeah, I mean, I've found that even we tried with my student, Evan Watson. And we tried to, we just gave it the information about the orbit of Mercury over the past 3,000 years, which JPL up the road here has access to and can predict. And then we said, well, if you observe this in this planet, you know, basically could you first discover this anomalous procession of the peritialine 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 will know when it's true when actual AI can come up with new

and unknown heretofore unknown, you know, predictions that can be verified by humans like you. Yeah, yeah, no, I think that's a very promising use case of AI. I mean, yeah, I mean, you know, neural networks in general, I mean, they're designed to make patterns to detect correlations and things.

Yeah, so having a few examples in mathematics where, for example, in Nod theory, a neural network, not a fantasy LM so much, but a more old-school neural network, was used to detect correlations between different types of knot invariance that was not believed people did not suspect existed before.

The error jar. And then once, I mean, initially this type of correlation was just sort of this black box relationship. So, so knots have these loops in space which, you know, some are, some can be untangled, some can't be.

They come with all these numbers. They call not invariants. And so the neural network found that by feeding a database of like 1 million knots, that there was one non-invariant called the signature, which could be predicted with really high accuracy from a whole bunch of other invariants called hyperbolytic invariants. But this neural network was this black box. You just fed in these 20 numbers as your hyperlink variance, and it will spit out.

The signature should be plus 3, and like 90% of the time it was correct. But once they had this black box, they could analyze it. They could say, okay, suppose I changed this input. I just, I modified this, high-pollary volume, or whatever. How much does this change the output? And so it's like a box of 20 dials that they could play with.

And basically by running experiments, they could see that three of these inputs were actually really important, and the other 17 were very peripheral. And by doing those types of analysis, 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 network models of, you could actually probe them. So in your astronomy example, maybe a neural network might not be able to tell you exactly what the new law of physics would have to be, but you can say, well, I can at least predict the orbit of Mercury over the next thousand years, and here's my model. And then you can just try to tweak it. Now, suppose I changed the period of mercury or the mass, whatever, what happens to it. And maybe you can work out laws of nature experimentally. It gives you a new paradigm to access reality in traditional

experiment or theory. Yeah, I've used that an example that kind of, there's a lot of AI doom people that think we're, you know, AI's going to run amok and turn us all into paper clips and all sorts in the 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 I'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 common.

I mean, it's matrix multiplication at a massive scale, high dimensions and huge volumes, but is it really that complicated and intrinsically? So the mathematics to train and run a large language model 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 matrix multiplication works in a little bit of calculus. But the area where,

we don't have a good methodical theory is how to evaluate, how to predict the performance of this model. So the mystery is not so much how they run. We know how to make a large, language model and how 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. And we don't know in advance, we don't have good rules of even heuristic rules of thumb for pretty, for pretty much. predicting which in which tasks are good, which are not. We can only just make empirical experiments. Part of the reason is that the data that you train on,

so we, so the data on one level is just strings of zeros and ones. And mathematically, we understand sort of very, very random data. So like if complete noise, completely random zeros and ones, we have the mathematical probability theory, which explains, we can analyze this situation 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 one zero, one zero 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 zeals and ones. 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 analogy of physics, actually. So in physics, we have continuum mechanics, which is one where everything's sort of averaged out and we have a good theory there. And then we have atomic level physics, where you can look at individual molecules and particles. But at the meso scale, there's lots of intermediate structures,

like cells, for example, biological cells. It's emergent. It's emergent. And we don't have good mathematics for this. I mean, in principle, you could break down atoms, but you can't possibly analyze. Yeah, it's not mathematically impossible, but in practice, it might be physically impossible. We mentioned, you know, inevitably, when we talk about L-L-L-Ms,

you know, the middle L is language. We'll get to my friend Galileo. I brought a book. I want you to take 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. But in the same text, you know, Shakespeare and math, if they're truly at root some proto-er language or something like that, then they should have more combinations or similarities, I would think. But again, I want your opinion. Do you think of math as a language or is it much more than that?

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 describe the unlike mathematics as efficiently as possible. Language, natural language is not always about efficiency. I mean, you also want to convey nuance and emotion and art or just express frustration or whatever. So it isn't driven purely by efficiency. But mathematics pretty much is partly because over time we try to do more and more ambitious mathematical tasks. And if we didn't optimize our math language in this fashion, we would not be able to do these more complicated tasks. And the same is true in the sciences.

We keep updating our laws 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 minimal and elegant a form as possible. And so when you're doing that, you are somehow getting to the essence of how the universe actually works. 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 these are simple, predictable laws. And it isn't just some big chaotic. There isn't some agent that's just making things up as a. go along. And the whole history of science has been sort of validating that belief, you know,

that naturalistic, you know, philosophy. And mathematics has been trying to do the same thing to mathematical theories, trying to find the most elegant minimal inputs that would explain lots and lots of mathematical phenomena. So maybe that's why they sort of converge over time. And it's why Bignell observed that the types of mathematical language and formalism that is good for mathematics, for example, the language of curved space to describe all kinds of geometries happens to coincide quite well with the language that describes the universe, like Einstein's use of that same language to describe space time. It's for curve space, yeah, exactly.

So one of the questions I love to ask mathematicians that have been on from Jim Simons and Steven Trugats 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 debate? Definitely both. So, I mean, we, I think there is an innate mathematical structure, which we are trying to discover.

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 certainly feels like discovery. But it's done through human means.

So yeah, it's both invention and discovery. Yeah, that's what Jim Simon's telling. When we look at the future of education, you're not only a field mentalist, a mathematician, and father and everything else that you do, but you're a teacher and your educator. Talk to me about your vision for the future. What's your philosophy of teaching?

Yeah, so it, needs to evolve quite a bit for many reasons. So the world has become infinitely more complex and unstable and unpredictable. And now with AI, humans used to be sort of a monopoly on cognitive tasks like, you know, 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 overthink human like research level mathematics or any other discipline in the near future. Already undergraduate level mathematics, for instance, many of the homework assignments that we assign right now, they can be done by AI. Yeah.

So we have to reinvent the way we... Ambition comes in all shapes and sizes. At First Citizens Bank, we roll with your goals because we're built for what you're building. Fit for your ambition for Citizens Bank. 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. So in the past, we had like a small number of authoritative source of information or text books

and your teacher or something. And, you know, you didn't have social media and the internet and all kinds of information of now AI, all that's information of really variable quality. On the other hand, 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 something written in crayon or something. But now our ability to produce high-quality presentation has far outpaced ability to produce high-quality content. So you cannot have YouTube videos or textbooks that look flawless and now AI-generated output, but have got lots of fundamental mistakes.

So, yeah, we need to encourage critical thinking. I already see teachers experimenting of things like, you know, here is a question that I would have aside, but I've given it to chat GPT, and this is the answer that they give. It's wrong. Please critique it and correct it. Interesting. And these are, I think, are more of the skills, more interactive, so not treating knowledge as a passive thing to be acquired by an authority. But something that you always have to question. And struggle with.

Interesting. Yeah, that kind of reminds me, John Preskill at Caltech, talking about quantum computing and quantum supremacy and so forth. And one of the ways to overcome some of the issues with error correction in quantum computing is just throw more cubits at the problem. I wonder, will we throw more AIs at the problem? This kind of flipped it. Through natural human brains out of an AI to prove what's wrong,

but will we be in a place where AI could police itself? So what would it take to trust them? It's good to make them more reliable, but I think, well, maybe if we use a very different architecture from the current AIs, so by nature they are inherently unreliable. But we have ways to use unreliable tools. Rather, number generators are the most unreliable device technology we have, but they're extremely useful for all kinds of things.

I think as long as you pair these AIs with good verification, And you only use the AI to the extent that you can verify the outputs and no further. Then there can be a great tool. I see them more as complementing human scientists and mathematicians. Because there are so few human scientists in the world, and we don't only have so much time to work 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 of not. of less well-known problems, which should require some attention, and they're not the most difficult or important, but it'll be good to have someone or something look at them. And so I think AI actually, their best use case is not to target them on the most high-profile problems, but actually on the millions of medium difficulty problems. And, you know, 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 their big advantage. You know, you. You cannot scale a graduate student.

Right. This way. Okay. But not legally. No, not legally. Yeah, or ethically. Okay. But AI, you know, I think that's where the real value lies.

What's your highest priority task right now? Well, research-wise, what I'm interested in most nowadays is new workflows to modernize mathematics and make it more collaborative, 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 our 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 so very small. We work with two, three people, in the sciences, of course, you know, thousands. Thousands, in large part because we don't know how to incorporate

contributions in 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, we had to verify a thousand little components, it wasn't feasible into very recently. Also, because of all these factors, it's, 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, you know, like social network analysis or whatever. So that is, I think, by the direction which my research is going into it. It's almost more the sociology mathematics, actually, than the, then the and this technique. And more recently, I've been interested in trying to secure funding for mathematical research that has become very unstable in recent years. We'll talk about that in a bit. So, Richard O'Brien, 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, you know, 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 weight on your shoulders perhaps, or maybe not. Okay, I do remember the year of that goes back in 2006. My life did change in many ways. So suddenly I got invitations like embassies, and I would read with people who I would want to meet. And I got asked people on all these committees. Suddenly my opinion was sort of up there.

So that was a sea change. I mean, I was, I mean, some of these already, but that took them getting used to. But I think one thing that helps ground mathematicians a little bit is that, I mean, as a few mathematicians, your main task is you have these problems you want to solve and you need more proof theorems that told these problems. And your proof has to be correct. And every step has to be validated. And it doesn't matter how famous you are or how much of a reputation you have. You can't just say, I've proven something. Trust me.

Okay. You have to supply the details of the truth. And if you don't have the proof, you don't have the proof. So I think this naturally provides some. check on just how high your ego can go just from these awards. Because, you know, I mean,

the count of those problems that I would love to soul, you know, that shouldn't have conjecture we talked about, but the hundreds of problems that I love to solve, and I just know I don't know how to solve. And so I know more problems

that I can't sell than the problems I have sold. So I think that, you know, so that keeps you somewhat honest. But what about the, you know, the old trope that, you know, mathematicians do their best work by age 30,

were you, we're around 50s now, you and I. What do you make 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 10 years or 20 years to do, and he did stop at age 30, but that was because he worked at Pensley for 10 years, not because he had an arbitrary age. You've heard this show. What do you think about it?

Yeah, so different mathematicians have had different curious facts. So I definitely was stereotypical. I had Egypt and I had a gifted child at that I got several grades and so yeah

I did all over the work when I was younger but there are other mathematicians who started quite late they didn't become interested in mathematics

until college when they switched became quite good my advisor when I was in Princeton my PC advisor Eli Stein I would meet within every week and I would discuss

the problems that he'd assigned me to work on And I'd spent hours trying all kinds of crazy things, and I'll report all these things I tried didn't work. I tried this, it didn't work. All this energy and time. And he would just sort of look at what I worked in Blackboard and just think for a few seconds.

It's that, you know, the difficulty you're having is exactly the difficulty that so-and-so had in this paper. So if you go Spionicampeter, and you push out this one paper, a pre-print. To read this, this will solve your problem. So, you know, there was a different way of doing mathematics. and I didn't I couldn't see how he pulled

because I would go home and read it and it would solve my problem I would then hit another structure in the next week but you know I spent hours on these problems and he just thought about it for 10 seconds

and he just knew from experience what to do it's the wisdom just with stuff yeah so I think as you get older you you approach you find different ways to do by fact which

it it may not be as flashy it was a brute force than what you as a fact but like it can be more productive

I can now pull the same trick off my own guy just doing it's like it's like a kind of satisfying because you see the whole circle there you could do second order you could say

what my advisor told me really about 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 an experiment or physics minimum amount of knowledge that a mathematician should have. I've asked us a theoretical physicist. 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? Oh, definitely. I think one of mathematics is that there are so many ways to approach mathematics. So you can be a very visual repetition. And so you see pictures. You can be a very symbolic mathematician, and you just view it as a gain of vulnerability numbers or symbol.

or you can be a very physics-oriented mathematician, and you always use physical analogies, and you use insights from various soft fields of physics to help you. I mean, so there's some very direct connections. If you study class of different equations, that very naturally you should know some physics because physics has so many examples of great transatl equations and having intuition about, say, how fluids work

or how waves work, really, really helps me. I think it's in general just the more you know in other areas sometimes I find I've found Benzroom from taking economic terms like if you want to prove

that X less than Y one way to think about it is that if you own Y amounts of stuff can you buy X all right and sometimes if you don't have a you know

sometimes you can't do it directly but maybe you can trade in Y per Z and then use Z to buy X so like if you put you up in the mindset of you have some bizarre

and you can there are certain merchants where you can trade X for Y. But you want to negotiate when you want to get a good price for these things. And you don't want to trade X, Y, from Z if it's a bad deal. That kind of mindset can actually be

very helpful in seeing sort of the right route how to get from X to Y. Sometimes you can think some types of metal algorithms you can think it as games. So in analysis, there are walks of the same as which say things like

for every epsilon, there's a delta, such as blah, blah, blah. It's got Upsilon, Dota, type of truce and they undergraduates are often very I hate those because they're yeah they're quite complicated in terms of games like if you're used to games like chess and something

so like you if your opponent moves here how do you counter that move and so if you think every time someone gives you an epsilon you need to find a delta to counter it and you think in these sort of game theoretic terms sometimes that can provide you and

you for mind So, yeah, you can use interest in biology, social scientists, every academicism has guys this has to do its sense. Yeah, that reminds me with this book that I've been learning

to show you, and we did take a look at it before we started recording. So this is called the Compaso Geometrico. So it's English version, is Galileo's, it's by Galileo, the operations of the geometric and military compass. This is not for finding north and south,

but instead it's for finding really doing a calculation since it's really an early version of a slider book. So this is the 1649 second edition, the 1601 first edition, as several times our salaries at the University of California. So I didn't afford be able for that. But what's so amazing, in addition to his actual signature, which he can zoom in on there, out of the aisle, didn't out this paper seven years after he died, but he had a stockpile. He was a minor celebrity.

Now, he never left Italy. He never got outside of Italy. just the preserve for 40. 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 I had segments. It had, it had, 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 it in this hosted note. Once I 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, and you're expected to be able to use it. So at some point, he starts talking about, you know, comparing links of lines, but I think in this page here, he goes, Rule for Monetary Exchange. So you just mentioned this. You want to read that?

That would be cool. By the means of the same arithmetic lies, we can change every kind of currency to every other in a very easy and speedy way. We first set up an instrument, taking left twice the price and money you want to exchange and fitting this cost-wise to the price of money in which they can just be made.

You invest rate this one example, everything is clearly understood. Suppose you wish to exchange Florentine Gold Scootty into reaching Decax, since the price of value of Decat is $6.004. It's necessary. We've got called currency's

of Saudi. Given a scooters price of 160 sodi, the price of Decat's 1.24. I'm so glad we ought to do a day and walk. Because I get into the phone. Yeah, exactly. I think it's so funny because, you know,

nowadays the Scootie is worth nothing. I mean, it might be worth a double dollars or whatever. but if Galileo had just put away a couple first editions of this book, you know, whereas errors, they'd be worth billions of dollars. But we mentioned, you know, this notion of currency conversion. And, you know, my friend Eric Weinstein, and I know, I'm known the same. I worked on, you know, Gage theory applied. So what do you make of this?

How I understand? That's okay, okay, yeah. Co-Ce Exchange Act is a very good example. So Gage theory has this reputation of being such a really obstruce area of physics and athletics, but it comes down to many questions. quantities in the real world are scalar, but they don't have a natural unit. So, yeah, so currency is one of the top. So, you know, 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 number, but it is not actually itself a number. Or it's not a number, but you can measure it by numbers. Gage theory is about quantities which can be measured by numbers, but or by coordinates, XYZ, but there's a choice of which units to use or which axes to use. When you need to build up your team to handle the growing chaos at work, use Indeed sponsored jobs. It gives your job posts the boost it needs to be seen and helps reach people with the right skills, certifications, and more. Spend less time searching and more time actually

interviewing candidates who check all your boxes. Listeners of this show will get a $75-sponsored job credit at Indeed.com slash podcast. That's Indeed.com slash podcast. Terms and conditions apply. Need a hiring hero? This is a job for Indeed sponsored jobs. And maybe if you're a different location on the Earth, you may have to use different units. And so as you go from one country to the next, you know, your units may change. So you need some way to convert as you go from one location of the next.

Similarly, so, yeah, in the rule, the electromagnetic fields, which, as we teach in high school, we teach that these fields are back to the there's some triple numbers at every point we go E and one for B. But actually, they're not numbers. there are directions in some abstract space. And so as you go from, as you move from one place to another, these numbers will change in a certain way. So gauge theory says about how to manage these conversions.

And one day you may decide that I'm going to price my currency here, not not in pounds, but in lira. And so that doesn't change how wealth you are, but it does change the gauge. And so there's a mathematics of how this gauge works. and which things are gauging invariant, what things are not. So, progdamas international curvature, if you go around in a loop and you follow and you just transport

whatever the vector, whatever it is along your gauge, sometimes you end up to work back where you start and sometimes you don't. There's a correction, and the correction doesn't matter actually what units of currency or what's your gauge, it's your gauging variant. So if you have a certain amount of dollars and you travel to Europe and you go to euros, and you show back to the U.S. and the dollars

because of exchange fees and so well. You might not have exactly seen what my is hard of. So in a sense, that is some curvature. It's not exactly curvature, but it's a bit like curvature in the currency bundle of the world. So actually, currency actually is a nice metaphor. Yeah, okay, you said. 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 then things emerge where they wouldn't be expected. And one sort of commonality of fellow Fields Medalist, I believe the only physicist to win it is Edward Whitton, the Institute for Advanced Study. And of course, he's known for contributions to stream to your, string theories are 60 years old than us. So fellow Fields Medalist, Edward Whitten, Institute for Advanced Study, is the first physicist, make the only physicist to win a Fields Medal. He's worked extensively in quantum 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?

Where's your up look as an outsider perhaps? So it's, I mean, physics, as you know, of course, history of, we had to re-evaluate our conception of the universe and the nature of reality several times already. So it was the Copernican revolution that deeds won't be. This is the simple universe. You know, there's Einsteinian relativity that, that Skastain had to be curved. And then, of course, there's quantum mechanics that reality is actually, it should be, gripped into power wave functions and cornfields, in a way that this series now are victim's own success, right?

Because we can now explain, like, 9.9% are all observable phenomena by these theories, except that at really tiny scales or the origin of the universe, no, it doesn't work. And the mathematics is inconsistent. And so we have to replace it by something. So in particular, the idea that space times a smooth manifold, does not seem to be compatible con-obstrations. So we need something else to replace it.

The problem is that there's infinitely many candidates for what to replace it with. Despite methodics being unreasonably effective, as to the right mathematics, if I just conk in any biblical theory and I'd hope of this will exclaim. So, yeah, for many decades, string theory was the leading contender.

It's a very elegant theory. I'm not an exoperative, and I understand it has not quite good enough the least the latest patients or providing. At least not a unique canonical theory that would fit the data. Maybe it's too flexible.

It gives you too many possible. Right. And that brings up a question on the meaning to ask you. In mathematics, there's girdles incompleteness theory, which sets a bound on what's possible to extract from a given system of axiens 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 there. 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 it's falsifiable. Do you think, I've often joked that physicists have mathematician envy. Yeah, a lot of people say, you know,

sociology has physics envy. 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 prove one plus one equals two. It takes what our Barlok identity is and it takes 200 pages, but we can't prove anything in physics. Where does that leave us in the epistemological search for truth? I think you just always say to keep separate the real world and our models of the real world. So, I mean, physics has provided us with my Bible models, which are, but in which you can't prove things. So relativity, for example, where Einstein's equations are a completely precise mathematical equation, and you can, you can get

starts by initial conditions still. If they're really, if you specify the initial conditions of space time, you can, you know, there is one neptychical solution, and a system delay at least doesn't, okay, but, you know, and you can prove feelings about that, and, and so the models are, you can both file so by and, I mean, they, they are, they are on, on the status of a mathematical construct. Where the physics comes in is how that model interfaces, with reality. So, you know, even if it doesn't quite imagine, even if it's technically volesify by experiment, it doesn't actually mean that the theory is destroyed. Newtonian gravity is still a very useful fear.

You know, with tech and crazy, it's good enough for, you know, modeling of planets and comets and so forth. I think as long as you don't conflate your model with the reality, you can have both your mathematical cake needed to. Very good. Just as we were wrapping up, Terry grabbed the chalk and gave us a lot of lightning talk about how he helped to crack a brutal image analysis problem that was vexing physicians trying to get the best quality images 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. 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 results which were amazing that they were getting, they were able to reconstruct an image using much fewer measurements than traditional imaging. And they were hoping to use this for medical imaging.

And I talked to them and I solved their little linear algebra problem. In fact, I first was trying to disprove it because I couldn't believe how good the results were. But on trying to do that, I figured out how it worked. And this technique, we published it, and it became very widespread. And in fact, nowadays, most of the big manufacturers of medical MRI machines,

they use our technology methods, which is now called compressed sensing, to speed up MRI scans by, like, a factor of 10 or so. You still never know. I mean, there's a lot of work I do, for example, these days is how to tell, given us some sequence of numbers, whether it has patterns or whether it's structured or whether it's random, and what kind of tests can you apply and which tests are sort of better than others in various ways.

You know, as you said, you know, there could be ways you could use this to afford maybe or filter out noise and try to get a bit better. signal acquisition algorithms. It's our whole ecosystem. I think in order for the more applied scientists and engineers to get the ambient ideas from the literature in order to solve their problems, they need the people from the more basic sciences to ask questions more in a curiosity-driven way. And maybe things that we do, but they don't directly have a practical impact.

But this is unreasonable effectiveness. If you don't have these people asking these questions, people downstream who are actually trying to make things practical application, things of reality, they can waste, they can spend a lot more time and maybe a lot more money, you know, trying to invest, you know. So to give one example, Shannon developed this theory of communication complexity over a century ago. Just theoretically, if you could only send a certain number of bits of messages per second,

how much information can you send it? And what's the best way to compress this data? And there's this whole practical theory that was developed actually long before the digital evolution later 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 interview 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. It was called the Shannon Brown. like exactly how much information you could cram into a certain amount of spectrum. And so because of that, you could plan. 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 do budgets, budgeting and planning. And there's a lot 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 it's still mathematics. you do by pen and paper, rather than deploy a billion dollars getting to realize that it doesn't have the capacity that you need. Or it has too much. Right. In which case, it's wasteful.

Yeah. Awesome. Okay. I know if you enjoyed this conversation with Terry, you're going to 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 Wolff from one of the

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