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Yeah, so there are other statistical patterns that random sequences have and artificial sequences don’t. I don’t think my load of occupancy work directly based on that, but there are other patterns. The most famous is called Benford’s law, which is very unintuitive law that roughly speaking 30% 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 of take all the countries in the world, take the

population and there’s about about a hundred odd countries in the world. About a third of them that the population start with one. Um China for instance, but or you can take the the wealth of of several millionaires and billionaires or whatever and you’ll also find that most of them start with one or birthdays. The pattern is quite universal, but whereas if you if you pick numbers randomly if you like fudge your accounting books and you you when you pick numbers random artificially, they don’t necessarily

obey that. They’re uniform. So you or you try to make them you think they’re uniform. So so humans actually we’re quite bad at at creating truly random patterns and so yeah, you can distinguish natural patterns from from human generated ones. Interesting. So one thing that I’ve you know, I’ve 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 I

do want to hearken to the work of Jim Simons. He’s most famously known well for being a multi-billionaire establishing philanthropies that support my research and and and hopefully other very highly competent scientists. But 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 I should have you to explain what

a minimal surface is, but as I understand it is you can sort of think of it physically if you had some 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 would obtain would be called a minimal surface. Is that correct? Okay. And then he showed that there are such minimal surfaces in zero dimension 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 right. And he showed it didn’t work and I want to stop 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 induction? >> 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 in scientific method because 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 it’s true for one and whenever you know it’s true for some number n, you you can you know for sure that it implies then the same thing for n plus one, then it implies it’s true for all four numbers. The analogy often given is just a row of dominoes. So if each domino represents one case of what you want 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 it tips over the next domino. Then no matter how long the the string of dominoes is, you you can you can knock over every single domino chain. But it’s really important that your arguments are 100% watertight. You know, if every single domino and like the 97th domino doesn’t tip over the 98th, you know, then it it it it it stops there. So it’s a principle that only works in the world of 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 Simons cone. Yeah, you’re pushing a little bit because this 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 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 one dimensional minimal surfaces are very boring.

They’re 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 it makes sense as a mathematician to first study higher dimensions. It gets you some intuition, which can help guide you with with the

problems that you do care about. Yeah, so it turns out yeah, that in below eight dimensions, I think yeah, all minimal surfaces are smooth. You you can’t tie a soap bubble and create any any kind of knot. Yeah, because there’s there’s always some way to pull it apart and reduce the the the surface tension. Yeah, it’s starting in eight dimensions, he discovered a very surprising fact that singularities do can actually form. Yeah, that that there is there is this yeah, it looks like a cone except

in much higher dimensions and there’s no way to to modify the cone to make it to reduce the surface area. If if you made a cone, if you try to arrange soap into a cone in three dimensions, you could just remove the the do what’s called a surgery. You you remove the vertex of the cone and replace it by two rounded knobs. 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 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 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 75% or something. And if you inscribe a a ball inside a cube, it’s still pretty big 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 actually incredibly tiny. It’s like point zero zero zero one percent. Like balls become extremely poor space filling. Yeah, they 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 if you have some you’re taking a thousand measurements and there’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 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. Right, they they diverge There’s 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 reductio ad absurdum. Can you talk about What is what’s your favorite type of mathematical proof when when you’re onto it, you just get so excited to finish the problem? Proof by contradiction. I think Hardy had a great quote that in chess a chess player may may offer you know, a pawn or a bishop. But a mathematician offers the entire game, you know, so It was a sacrifice. Yeah, so you say it’s okay, we want to prove

this is the conclusion. I’ll 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 it is very unintuitive. The undergraduate students that we teach, they struggle a lot with the notion of of proof by 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, I 1,000 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. They have proven that there is no largest number and in the natural numbers. And this is a proof by contradiction because if anyone ever did claim that a natural number is largest natural number, you just add one and you

have contradicted them. So it is actually a very intuitive proof technique, but but you have to you have to teach it the right way and sometimes kids can teach 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 is 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 all the time, they they also have correspondences which which are really quite 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 the by contradiction that square root of two is irrational. So that’s the Euclid’s original proof or

where does it trace back to before him? Euclid or back I think the Pythagoreans Maybe Pythagoreans. Euclid proved the prime basically by the same sort of ideas though. Infinity plus one. Yeah, yeah, yeah. No, we we have a lot to thank Euclid for actually. I mean he wasn’t the first to write down many of of 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 hadn’t seen before. 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 peaked 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

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 and momentum called Poisson brackets. And as soon as you take them from the classical world to the quantum world, um instead of commuting, being equal to zero, they become equal not to zero, but times a fundamental constant times the square root of -1. And it’s

just so baffling to me that once you introduce the concept of a square root and imaginary number, um 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 LLMs, they use LU decomposition and we have a spinners that we have spinner representation and our square roots. Is there something special about the square

root or could like in other words, why don’t we say like cube roots or fifth roots or hundredth roots? Why is the square root something that’s that’s so prevalent in physics, for example? Yeah, so um 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. All right, so real numbers have lots of wonderful properties. Um I would the algebra of real numbers works really well, you know, for things

like um addition is is commutative, x + y is y + x and x * y is y * x, multiplication is commutative and so forth. But they have one flaw, which is that not every polynomial equation has roots. Um so uh if you take um the equation x² + 1 = 0, in in the real numbers, x² + 1 is is never zero because x² is always positive. So it’s what’s called it’s it’s not what’s called algebraically complete. But it’s very close to being complete. Um so if you take a polynomial which is of odd degree, like a cubic, x³

• 3x + 1, it must have a root because a 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 um large and negative, it becomes negative. And because um the reals are continuous, um polynomials are continuous, um therefore at some point in between, you must hit zero to get from negative to positive. So sort of a half of all the polynomials in the world, um the um um you can solve them in the reals and half you can’t.

So um in with with the benefit of hindsight, this really suggests that you should you should make the reals sort of twice as big in order to get this really useful property of algebraic completeness. And so um as it turns out, there are these numbers called complex numbers, which um are twice as big as the reals are. So the real numbers are one-dimensional and the complex numbers are two-dimensional and they have wonderful wonderful properties. They have lots of very nice geometric structure and nice algebraic

structure, ultimately coming from this algebraic completeness. Um and so yeah, um 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 that you you didn’t have previously. Uh so once you know that you’re looking for a a new number system that’s twice as big as as 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 -1. So that’s kind of the in retrospect um 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 asks my guests to judge their books by their covers, but today I’m asking myself to judge my own book by its cover. >> [music] >> My newest book, Focus Like a Nobel Prize

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correct? Is >> Yeah, yeah, uh we don’t have roots of a polynomial. We’ve we’ve learned that the notion of number is very flexible. Um I mean, people get upset when when they they learn that what they what you know, it it 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 uh we occasionally do that with with with math, too. Like a the number one

used to be prime about a hundred years ago. Oh, really? Oh, I I I mean, I still consider it prime, but I still consider Pluto a planet. Uh yeah, yeah, but it um it’s it’s because um in all mathematicians inside it any any in any field of study, when you first study a subject, um you don’t really know what concepts are the most fundamental and important and and which ones are not. So you make a guess um based on on your experience. So maybe you think that numbers that don’t have any small factor uh are important, so

you call them prime numbers or maybe that the the the 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 astronomy as why um the new notion of a planet is is better, but um but for example, one of the the 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 other than rearranging the factors. So 12 is 2 * 3 * um 2 or or 2 * 2 * 3. So but other than interchanging the um the order, that’s the only way to break up a number into 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 12 is also 1 * 2 * 2 * 3

um and you just add too many exceptions to this really important um fact in in number theory. So we made a decision to therefore redefine Uh I know 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 a um Yeah, it’s it’s it’s a human convention. Um but it’s it’s but we we update our human conventions to match reality better over time. You I’ve done work in prime pairs, is that right? Yes. Yeah, so primes are one

of the oldest subjects in mathematics. As we Euclid had the first theorem almost ever. Um and it was about prime numbers almost more than 2,000 years ago. And so it’s very um frustrating and annoying that even the most basic questions about primes, we still cannot answer definitively. Um we can we have good guesses. Uh like so for almost all questions about primes, we can predict the answer, uh but we cannot get the 100% mathematical standard of proof for for many of them. And one of the most basic

questions, which is at least 300 years old, is called the twin prime conjecture, that um uh there should be um so Euclid showed there’s infinitely many primes. The primes never end. You can always find primes bigger than any number you wish. But uh 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. Uh for example, 11 and 13. Um well, okay, two and three are closest, but um but after two, all primes are odd. So the the

closest you can get is 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. You know, sometimes um the prime gaps are large and sometimes they’re small, but every so often, they they they come close to each other and you get a a twin. And they seem to occur infinitely often, you know, as we we we can find trillions and trillions of these 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. Um and um random sequences with a 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 the them being um uh being odd. Now, so I mean, it’s it’s a it’s a very likely hypothesis, but we can’t prove it. Pseudo-randomness

meaning that it could be derived from some algorithm, but not in all cases or something. What’s pseudo-random versus random distinction? Um random means not the primes, if you forgotten all 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 um a set by rolling dice or flipping coins, you would get a different set. Pseudo-random 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 uh end in five as end or end in seven and in six. Like the the digits should be equally distributed. Now, the primes that they’re not perfectly pseudo-random because they they do avoid certain they do have certain patterns, like that they tend to be odd, for instance. But there’s there’s ways of excluding those sort of obvious biases. And once you exclude them, um

um they should it’s expected that there’s there’s there’s no test that can distinguish them from random 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 your 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 conveys no information about um 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 study prime numbers is that we occasionally get a shock that um I mean, it hasn’t really happened in number theory in in century in in decades, at least. But but 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. Uh I think not for primes, but maybe other curves and other things where people actually had to migrate to a different uh crypto system because of these weaknesses. >> Yeah, so that brings up well kind of an inversion, maybe a contradiction of

of what Wigner said. He 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’s said that quantum computers can perhaps factor and break these previously considered to be un 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? So quantum computers are a fascinating topic. They yeah, they interface with maths in various ways. So one is actually just the actual software engineering of 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 you do this, you do that. And and and we have decades of experience with yeah, but for a quantum computer the state is is not a bunch of of zero one bits, but it’s it’s yes, it’s a wave function and the operations you you assign 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 only a you only change a few corner bits at a time. But you want to couple them together in a in a very efficient way so that you can do really complicated operations. So quantum computers are both exponentially more powerful than classical computers, but also exponentially more limited. >> Yes. So because they can handle superposition quantum states simultaneously, in principle there’s this exponential speed up. And in in for certain applications like factoring and I think quantum chemistry this this

is >> Holding the ground here. They are at least in principle very very powerful. But quantum mechanics is also very restrictive. The the number of things you can do to a quantum state you can only do linear operations. And only do time reversible operations. >> Yeah, so yeah, so this has um Yeah, so this requires you to develop theories of reversible computing. Error correction is also much much more annoying in in in and and so yeah, so so there’s there’s software challenges. Maybe once

computers become a reality they will 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 certainly classical complexity theory has been very influential. Historically mathematicians only cared about saying whether something was true or false or provable or or disprovable. And with the advent of computers people also started asking questions of how

computer how how computable is is an object. Like so if they 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 grain notion of of truth actually than than just it’s true or it’s false, but how easy is it to actually compute? Mhm. And it that has led to very productive mathematics. I mean sometimes just the the effort to not just show something exists, but actually

find it creates yeah, creates new techniques. Complexity theory has offered sort of given a much more nuanced understanding of of how true a statement is or how um And yeah, this has led to to a better understanding of if you just prove something is is true and you don’t but uh you may not have any insight into what was what was the key ingredient that made it work. Or you had two different proofs, which one which was 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. It’s more efficient. It’s yeah. So it yeah, it it it indirectly sort of provides much more insight into the into the 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 um I mean by itself so the the big weakness 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 it’s not grounded that it’s it’s it’s it’s probabilistic. Okay, they often make mistakes and 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 first thing 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 off the rails. And 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 system. So if you if you converse with a large language models to for them to make suggestions, but you you understand what their output is and you can verify. So then people have had some success talking about their math problems with a large language model. Large model 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. So one thing where 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 may be say a dozen ways already to attack this problem. And you the human working on the problem

maybe you can remember six of them, but you the other six I don’t come to your mind. And you can use the large language models to prompt you to remind you of of the of the missing six. They may also hallucinate three more that don’t exist. So you you do need you can’t trust them. Supervise them. Yeah, you have to you have to verify. There is hope in the future you know, so there’s this separate math technology to to automate 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 that you can verify and to filter out the hallucinations. Has it been able to reproduce a Wiles you know proof for Fermat’s Last Theorem or your work in Navier-Stokes? I mean has it been able to actually just just simply quote unquote reproduce what a natural intelligence person like you did? This is issue it can but but often because of what’s called contamination. So

if if a result is like taught in textbooks somewhere then it it is implicitly in the training data that these AIs 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. 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 that they they really sort of thought it up. If you ask the AI to explain the the chain of thought

they can often give complete nonsense like it it it’s clear that they just didn’t know. Yeah, I mean I 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 3,000 years which JPL up the road here has has access to and computer. And then we said well if you observe this and this planet, you know, basically could you first discover this anomalous precession of the perihelion of 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 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, I know. I I think I think that’s a very promising use case of of

AI. I mean yeah, I mean that you know, neural networks in general I mean that they’re designed to make patterns to to detect correlations and things. Yeah, so there have been a few examples in mathematics where for example in in knot theory a neural network not not the fancy LLM so much, but like a more old school neural network was used to to detect correlations between different types of knot invariants that was not believed people did not existed before or after. Yeah, and then well once once I mean initially this

type of correlation was was just sort of a black box relationship. So So knots have these loops in the space which some are some can be untangled some can’t be. They come with all these numbers they’re called knot invariants. Um And so um the neural network found that that by by feeding it a database of like 1 million knots that there was one knot invariant called the signature which could be predicted with really high accuracy from a whole bunch of other invariants called the hyperbolic invariants. Um

So but this this neural network work was this black box. You you just fed in these 20 numbers as your hyperbolic invariant and it would 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 they were like okay, suppose I suppose I change this input. I just modify this hyperbolic volume or whatever. How much does this change the output? And so they it’s like a box of 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 what were very peripheral. And by doing those kind of types of analysis they actually got some insight as to what the relationship was and they they could actually make a formal mathematical prediction which they could then prove. So you know, once you have these neural network models of of you he actually um them. So, in in your astronomy example, maybe um you a

new network might not be able to to tell you exactly what the um the new law of physics would have to be, but it could say, “Well, I can at least predict uh the orbit of Mercury over the next thousand years, and and and here’s my my model.” And then and then you can you can just try to tweak it. Now, suppose I I change my my the the the the period of of Mercury or or the uh or the mass, whatever, what what happens to it? And maybe you can work out laws of nature experimentally. Okay, it gives you a new

paradigm to to access reality than traditional experiment or theory. Yeah, I’ve used that as an example. 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 paper clips 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 and so it’ll have errors and it’ll have mistakes, but it’s bounded by the amount of human knowledge

it’s used at 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 most matrix multiplication at a massive scale, high dimensions, and and huge volumes, but is it really that complicated intrinsically? So, the mathematics to train and run a um 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. I basically need to know how matrix multiplication works and a little bit of calculus. Oh, but uh the um the area where uh we don’t have a good method of theory is is how to evaluate um uh how to predict the performance um of of this model. So, um the the mystery is not so much how how they run. Okay, we we we we can we know how to make a large language model and how to 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. And we don’t know in advance we we don’t have good rules of um even good heuristic rules of thumb for predicting uh which in advance which tasks are good for which ones or not. We can only just make empirical experiments. Part of the reason is that um the data that we train on, like so um we um so the data on on on one level is just strings of of zeros and ones. Um and um mathematically we understand sort of very very random data. So, like if if if complete noise completely random zeros

and ones, we have the math that probability theory which explains we can analyze this situation very well. And then we have very very structured types of data, like a sequence of all ones or all zeros or just alternating 1010 in a very periodic fashion. Very structured data we understand very well. But, the type of data that is natural data, like um like English text, you know? So, you can digitize that as strings of zeros and ones, but very specific zeros and ones. >> Right. Um but not so specific that that

that they’re completely predictable. Um but they still seem to be somewhat predictable. Um and yeah, so we don’t have good um mathematics for partially structured objects. It’s analogous to physics with in physics actually. So, in physics we have continuum mechanics, which is the one where uh everything’s sort of averaged out and and um and we have a good theory there. And then we have atomic level physics where you just have a you 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. >> Emergent. It’s emergent. >> Yeah, it’s emergent um and uh we don’t have good mathematics for this. Um you know, I mean, in principle you could break down the atoms, but you can’t you can’t possibly analyze Yeah, it’s not it’s not mathematically impossible, but in practice it might be physically impossible. You 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 that I want you to take it your impression on in one of his math books and and treatises. Uh but um but he said, you know, that the the book of of knowledge of nature or the universe is written in the language of mathematics. And it kind of, you know, was echoed later, you know, by Wigner and and so forth as we’re going to discuss. But, is it really a language? Yes, it has a vocabulary and has a syntax and you

know, but in the same text, you know, you know, Shakespeare and math, if they’re truly, you know, at root some proto er language or something like that, then they should have more combinations and or or similarities, I would think. But but again, I I want your opinion. Do you think of math as a language or is it is it more much more than that? Well, certainly when mathematicians talk to each other or to other scientists, I mean, they have to use math as as as 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 analytic mathematics as efficiently as possible. Language natural language is not always about efficiency. Um you know, I mean, you you you also want to convey nuance and and emotion and and and at at art or or just uh um express frustration or or whatever. Um so, it it it it isn’t um driven purely by by by efficiency. Um but mathematics pretty much is partly because we over

time we try to to do more and more ambitious mathematical tasks. And if we didn’t op- optimize our math language in this fashion, um it it uh we would not be able to do more complicated tasks. And the same is true in the sciences, you know, we we we keep updating our our laws of of of of nature so that we can make um more um complex predictions. When you optimize a language for efficiency, you’re basically just trying to compress uh a description of the universe into um as as as as minimal and elegant a a a form as

possible. And so, when you’re doing that, you are somehow getting to the essence of uh of how the universe actually works, you know? So, assuming the universe does operate by some laws of nature, which maybe you don’t know yet, but we’d like to believe that they are these are you know, the 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 the whole history of science has been sort of

validating that that that that belief, you know, this naturalistic um if you want to to uh philosophy. And mathematics has been trying to do the same thing to mathematical um theories, trying to find the the uh the most elegant minimal um inputs that would explain lots and lots of of mathematical phenomena. So, maybe that that’s why they so converge over time. And this is also how why Wigner 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, happens to coincide quite well with with the language that would describe the universe, like like Einstein’s use of that same language to describe uh space-time. space, yeah. Exactly. Um so, one of the, you know, questions I love to ask mathematicians that have been on from Jim Simons and Steve Strogatz and many others is whether or not you believe that math is invented or discovered. So, there’s four options. You could say uh invented, discovered,

both, or neither. So, where where do you come down on this classic classic divide? Um definitely both. So, I mean, we um I think there is an innate um mathematical um structure which we are trying to to discover. And um but in order to do that, we have to invent mathematical language, and and and initially it’s not a very good language. We we we are focusing on the wrong things. Um but over time, as you said, to to try to make our language more efficient and and more powerful, it sort of naturally

converges to the um ideal Platonic ideal of mathematics. And that’s that certainly feels like discovery. Um but it’s it’s done through human um human means. So, yeah, it’s it’s both invention and discovery. >> Yeah, that’s what Jim Simons told me. Uh when we look at uh the future of education, you’re not only a Fields Medalist and mathematician and father of everything else that you do, but you’re a teacher and you’re educator. Um talk to me about your vision for the future.

What’s your philosophy of teaching? Yeah, so um it needs to evolve um quite a bit from from many reasons. Yeah, so the world has become infinitely more complex and unstable and unpredictable. Um and now they know and and and now with AI, you know, humans used to be sort of uh the monopoly on cognitive tasks, like um you know, um and and now uh AI so, one of the problems with AI actually, I mean, the way the subject develops is is not so much that they’ll they’ll overtake human um like research

level mathematics or any other discipline in the near future, but already undergraduate level um mathematics, for instance, um many of the homework assignments that we we we assign right now, they can be done by by AI. Yeah. So, we have to reinvent uh the way we um uh we teach. So, um one thing that will become more important is um students will need to have much more training in how to validate um information that they see. You know, so in the past we had like a small number of authoritative um

uh sources of information, our textbooks and and your teacher or something. And um and uh you know, we didn’t have social media and and and the internet and all kinds of information of and now and now AI of all sorts of information of really variable quality. On the other hand, the um in the past, um when you had information that was low quality in content, it was also low quality in presentation. You know, so you could tell that like a really well-produced textbook would likely have more um

accurate content than than, you know, something written in crayon or something. But but now our our ability to produce high-quality presentation has far outpaced our ability to produce high-quality content. So, um you can now have have um you know, YouTube videos or or or or or textbooks that look flawless, okay, and now AI-generated output, but have got lots of fundamental mistakes. So, um yeah, we we need to to encourage uh critical thinking, you know, um I I already see teachers experimenting with things like, “Here is

a a question that I would have assigned, uh but I’ve given it to to ChatGPT, and this is this is the answer that they give. It’s wrong. Please critique it.“ I and and and correct Interesting. Um and these are I think more the skills and more interactive, you know, so not treating knowledge as a as a passive thing to be acquired by authority, but something that you always have to question. And struggle with. Interesting, you know, that kind of reminds me of John Preskill at Caltech uh 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 to throw more qubits at the problem. And I wonder, do will we throw more AIs at the problem? This kind of flipped it. You threw natural human brains at an AI to prove which are. But will we be in a place where AI could police itself? And so what what would it take to trust them? It’s it’s good to to make them more reliable, but I think um Well, maybe if if we use a very different architecture from

the current AIs. It So by by nature they are inherently unreliable, but we we have we have ways to use unreliable tools. Random number generators are the the most unreliable device technology we have, but they’re extremely useful for all kinds of things. I’ll give you a quick example. Yeah, that’s I think as long as you pair these AIs with with good verification. And and you and you only use the AIs to the extent that that you can verify the outputs and and and no further. Then they can be a

great tool. I see them more as complementing human scientists and mathematicians. So um because there are not there are so few human scientists in the world and and we don’t only have so much time 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 would be good to have some someone or something look at them. And so I think AI actually their best use case is not to to to target them on 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 this way. Okay. Um But not legally,

yeah. No, not legally. Yeah, or ethically. Okay. But but AI I think that that’s that’s where the the big the real value lies. What’s your highest priority task right now? Well, [clears throat] research-wise what I’m interested in most nowadays is is new workflows to um um modernize mathematics and make it more um more collaborative, more more accessible to public, and to integrate in these new tools like AI. The way we’ve done we’ve done mathematics has not changed fundamentally in centuries. You know, we

you you see our blackboards in my office. You know, we still work with pen and paper. We use computers a little bit. Um but not so much. And our collaborations are still very small. We work with two or three people. You know, in the sciences of course you know, with thousands thousands. In large part because we don’t know how to incorporate contributions to the general public. I mean, there’s a barrier to entry first of all. A lot of what we do is very technical. Um but you know, we need to

synthesize proofs where every single step has to be verified. So if we had thousands of people you’d have you’d have to verify a thousand little components. It it it it wasn’t feasible until very recently. Also because of all these factors it’s we don’t collaborate as much with the other sciences as as we ought to. Especially the the the new sciences which are so data-driven and and connect with the real world in new ways. You know, you know, in with like social network analysis or

whatever. So that that is I think by the direction [clears throat] in which my research is going. It’s it’s almost more the sociology of mathematics actually than than the than the technique. And more recently I’ve been I’ve been interested in trying to secure funding for for my theoretical research that has become very unstable in in recent years. We’ll talk about that in a bit. So your friend Sergio Kleinerman asked me a question related to what you just brought up, sociology of science. And and he he

wondered how was stressful for you to be, you know, reputed the the the best mathematician on Earth, the Fields Medalist, a very young and extremely successful mathematician. Did that did that affect you? Was that was that a challenge for you with that mantle that that weight on your shoulders perhaps? Or maybe not. Okay. I I I do remember the the year of like it was like 2006. My life did change in many ways. Like so suddenly 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 was sought after. So that that was a sea change. I mean, I was only some of these already, but so that took some getting used to. But I think one thing that helps ground mathematicians a little bit is is that you know, I mean, as a few mathematicians your main task is you have these problems you want to solve and you want to prove theorems that solve these problems. And your proof um has to be correct and every step has to be validated. And 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 detailed steps for the proof. And if you if you don’t have the proof, you don’t have 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. You know, I mean, that there are countless problems that I would love to solve. You know, that shouldn’t have been conjectured or talked about. But

but 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 that I can’t solve than the problems I have solved. So I think that you know, so that keeps you somewhat honest. What about the get out the old trope that you know, mathematicians do their best work by age 30? Were you 50s 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 10 years or 20 years to do your And he he did stop at age 30, but that was because he worked at Pensley for 10 years, not because he hit an arbitrary age. All right. You’ve heard this trope. What do you make of it? Yeah, so different mathematicians have different career tracks. So I definitely was I was more typical in that sense. And I had um I got to um I skipped I skipped a grade at that accelerated um I skipped several grades. Um And so yeah, I I did all my work when I

was younger. But there are other mathematicians who started quite late. They were they didn’t become interested in mathematics until college when they switched and became quite good. My advisor when I was at Princeton, my pre-senior advisor Yau started. I would meet with him every week. And I would discuss the problems that he would assign 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 it didn’t work.

I all this energy time. And he would just sort of look at what I wrote on the 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 he goes by his cabinet and he’ll push out this this one paper. A preprint. He’d say, “Read this. This will solve your problem.” Um so yeah, there was it was a different way of doing mathematics that I I didn’t I couldn’t

see how he put because now I would go home and read it and it would solve my problem. I I would then hit another obstruction like the next week. But but you know, I spent hours on these problems. Um And he just thought about it for for 10 seconds. He just knew from experience what to do. It’s the wisdom. Just wisdom. Yeah, wisdom. So I think as you get older you you approach find different ways to do mathematics which um it it may not be as flashy you know, in terms of more brute force. It may not

be as as efficient, but actually it can be more it can be more productive. Yeah, I can now pull the same trick on my own graduate students. So and so it’s kind of kind of satisfying. You see the full circle there. But you can do You can do second order. You can say what my advisor told me to do. My grand advisor. Um let me ask you a question related to pedagogy. Um so it’s obvious you know, from what we’ve already talked about with Wigner that math is really important for physics. Um 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? Oh, definitely. I think this one great thing about 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 physical analogies and you use insights from from various subfields of physics to to help you. I mean, there’s there’s a very direct connection. So if you if you study classical differential equations, then very naturally you you should know some physics because physics has so many examples of of of

of great great differential equations. And having intuition about say how fluids work or how how waves work can really really help you. And I think as in general just the more you know in other areas, you know, I mean, sometimes I I find benefit from taking economic terms. Like if you want to prove that X is less than Y, one way to think about it is that if you own Y amount 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 but maybe

you can trade in Y for Z and then use Z to to buy X. So like if you have to if you place up in a mindset of you have some bizarre and and you can there are certain merchants where you can trade X for Y. And but you want to negotiate. You want to get a good price for for these things. And you you don’t want to trade X Y for Z if if it’s if it’s a bad deal. Um 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 mathematical you can think of as as games. So in in analysis there are lots of the famous which say things like for every epsilon there’s a delta such that blah blah blah. It’s called epsilon-delta type type proofs. And they um undergraduates are often very um um I I hate those cuz even yeah, there’s a it they’re quite complicated. But they’re just games. Like if if you’re used to to games like chess and so forth like you if you know if your opponent moves here how do you

counter that move and so if you think I every time someone gives you an epsilon you need to find a delta to to counter it and you think in these sort of game theoretic terms sometimes that can can provide you a useful mindset. Yeah and you can use it with biology social sciences every academic discipline has guys who have studied this. 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 geometrical so it’s

the English version is Galileo’s it’s by Galileo Galilei the operations of the geometric and military compass. This is not for finding north and south but instead it’s for finding really doing calculations since it’s really an an early version of a slide rule. So this is the 1649 second edition the 1601 first edition is several times our salaries at the University of California. So I didn’t afford being able to afford that but what’s so amazing in addition to Galileo’s actual

signature which we can zoom in on there and he didn’t have this paper 7 years after he died but he had a stock pile. He was a minor celebrity and he never left Italy he he never got outside of Italy. Just to observe for 40 years. It is it is isn’t it beautiful? I find it like a treasure. I’ll I’ll bring that 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 to think about is he goes in in this I think I marked it in in this posted note let’s take a look at that page Terry. He talks about it’s basically an instruction manual. So nowadays we get the device we get a iPhone it doesn’t come with an instruction manual right there 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 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 lines we could change every kind of currency to every other in a very easy and speedy way. We first set up the instrument taking that twice the price of money you want to exchange and fitting this crosswise to the price of money which to exchange is the rate. We illustrate this by an example if this everything is clearly understood. Suppose you wish to exchange Florentine gold scudi into mention the tax since

the price of value of the tax is six lira for scudi is necessary we want to calculate currency of scudi given a scudi is twice a 160 in the price of tax 124. I’m so glad we all have to do the same work. >> [laughter] >> Because I said I get to do the fun. Yeah exactly. I think it’s so funny because you know nowadays a scudi is worth nothing. I mean it might be worth a couple dollars or whatever but if Galileo had just you know put away a couple first editions of this book you know where his heirs

they’d be worth billions of dollars but we mentioned you know this this notion of currency conversion and you know my my friend Eric Weinstein and I know well known the same uh I worked on you know gauge theory and applied it so what do you make of this? I think that’s okay okay yeah that currency exchange like is a very good example. Yes so so gauge theory has this reputation of being a really um abstruse 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 so um yeah so so currency is one of the examples 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 it as a number but but it is not actually it’s not a number or it’s not a number but you can measure it by numbers. Gauge theory is about quantities which can be measured by numbers but or by coordinates XYZ but there’s choice of of of which units to use or which axis to use. And and maybe

if you’re a different location on the earth or you may have to use different units and so as you go from one country to the next you know you you you your units may change it’s a you use you need some way to convert as you as you go from one location to the next. Similarly so you know in in in the world of the electromagnetic fields which as we teach in high school we teach that these fields are vectors it’s you know there’s there’s some there’s some triple numbers at every point we call E and one for B

but actually they’re not numbers that they are and they’re directions in some in some abstract in some abstract space and so as you go from as you move from one place to another they these numbers to change in a certain way and they have to gauge theory is just about how to how to to manage these conversions and one day you may decide that I’m going to price my my currency here not not in pounds but in in lira and so that doesn’t change how wealthy you are 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 there’s a natural curvature if you go around in a loop and you and you follow and you just transport whatever you vector whatever it is along the gauge sometimes you end up with what back where you started and sometimes you don’t there’s there’s a there’s a correction and the correction is and the correction doesn’t matter actually what units of currency or what

was the gauge it’s a gauge invariant. I so for example if if you have a certain amount of dollars and you you travel to Europe and you convert to euros and you show up back to to the euros and convert back to the dollars because of of exchange fees and so forth you might not have exactly the same amount of money you started so in a sense that is some curvature in it’s not exactly curvature but it’s a bit like curvature in in the um in in the hyper currency bundle of of of of of of the world. Yes I think that currency

actually is a nice metaphor. Yeah okay so yeah and it’s surprising that you get from different symmetry laws and and so forth that you get properties that are unexpected and things emerge where they wouldn’t be expected. One sort of commonality of fellow Fields medalist I believe the only physicist to win it is Edward Witten on the Institute for Advanced Study. And of course he’s known for contributions to string theory you know string theory is our 60 years old in the house. So fellow Fields medalist

Edward Witten Institute for Advanced Study is the first physicist make it the only physicist to win a Fields Medal. He’s worked on extensively quantum gravity and string theory etc. What do you make of the current status of it and you know mathematical nature of it that seems to only be able to solve things in very high dimensional spaces for which we have no evidence. What what’s your outlook as an outsider perhaps? So it’s it’s I mean physics as you know of course history of we have to re-evaluate our

conception of the universe and nature of reality several times already and so was the Copernican revolution that deals with one beyond the center of the universe you know there’s Einstein’s relativity that that space time had to be curved and then of course there’s quantum mechanics that that reality is is is is actually is should be described by wave functions and quantum fields in a way that this theory is now a victim of its own success right because because we we can now explain like all 99% of

all observable phenomena by these theories except that at really tiny scales or at the origin of the universe you know it it doesn’t work and the mathematics isn’t consistent um and so we have to replace it by something so in particular the idea that space time is a smooth manifold does not seem to be compatible with quantum observations that that they look like those. 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 as to the right mathematics. You can just plug in any biological theory and hope that this will explain so yeah for many for many decades string theory was the leading contender it’s a very elegant theory I mean so I’m not an expert in but my my and I understand it it it it has not quite lived up to the expectations not not uh or provide at least not a unique canonical theory that would fit the data. Maybe it is it promises too flexible gives you

too many possible ways. And that that brings up you know question I meant to ask you in mathematics there’s Gödel’s incompleteness theorem which sets a bound on 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 as the sine qua non as a as a definition of good science is as falsifiable. Do you think I I’ve often joked that physics is that mathematician envy. Yeah a lot of

people say sociology has physics envy but I think that 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 we can’t as I said we can’t even prove we you can prove 1 + 1 = 2 it takes what I don’t know how many years and things 200 pages but but we can’t prove anything in physics. Where does that leave us in the epistemological search for truth? I think you just always have to keep separate

the real world and our models of the real world so um I mean physics has um provided us with mathematical models which which are which you can prove things so um relativity for example Einstein’s equations are completely precise mathematical equation and you can you can specify initial conditions so if they’re really if you can specify the initial conditions of space time you can you know there is one mathematical solution unless there’s singularities okay but um you know and you can prove things about

that and and and so the models are are you can you can prove falsifiable and and I mean that um that they have that they have to that they are on on the stairs of of a mathematical construct. Where the physics physics comes in is how that model interfaces with reality. So you know you know even if it doesn’t quite match up even if it’s technically falsifiable by by experiment doesn’t actually mean that the theory is destroyed. You know Newtonian gravity is still a very useful theory in what

technically is is the more modern and accurate. It’s good enough for you know modeling you know planets and comets and so forth. >> I think as saw much as you don’t conflate your model with the reality. You can have both your mathematical cake and eat it too. Yeah. 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. Um I I was talking to some statisticians and uh 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. Um and um I they were reporting some some results

which were amazing that they were they were getting uh they were able to reconstruct an image using much fewer measurements than traditional um uh uh imaging and um they’re hoping to to to to use this for medical imaging. Um and I I talked to them and I I I solved their their their little um linear algebra problem. In fact, I first was trying to disprove it cuz I couldn’t believe how good the results were. But I I I I’m trying to do that. I figured out how how it worked. Um and this technique we we published it

and it it it became very widespread um and in fact nowadays um um uh most of the um the big manufacturers of medical MRI machines um they use our technology uh um methods um uh which is now called compressed sensing to um uh to speed up MRI scans by like a factor of 10 or so. Yeah, you you would 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 if given a 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. Yeah, as I said, you know, there could be ways you you could use this to to uh detect forward maybe um or or uh uh or filter out noise and and try try to you know get better um signal acquisition algorithms. Um so we it’s our ecosystem. Um I think you uh in order for the more applied scientists and engineers to to get the the ambient ideas from the literature in

order to solve their problems, they need the people from the more basic sciences to uh to ask questions more in a curious curiosity-driven way. Um and uh maybe things that we do don’t directly have uh a practical impact, but this uh yeah, this is unreasonable effectiveness. You know, like if if you don’t have these people asking these questions, the people downstream who are actually trying to make things um practical application um things a reality, they can waste um they they can they can um spend a lot more time and

maybe a lot more money, you know, trying to invest, you know, um Actually, so to give one example, um uh Shannon developed this this theory of communication complexity um over a century ago. Just theoretically, if you could 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 the digital revolution. Yeah, uh later when when um

the um um when we needed, you know, when we had everyone had cell phones and we needed to transmit huge amounts of data simultaneously and we we wanted to make sure that that cell phones didn’t interview each other. All this um uh mathematical work uh was really important and um it may not have directly told you how to build the phones, but it could it it did things like it it provided the theoretical limit. It’s it’s called the Shannon bound. Like like exactly how much information you could cram into a

certain amount of spectrum. Um 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 um uh um information you could communicate from that and you can you can do budgets budgeting and planning. Um and uh yeah, and there’s still lots of engineering that needs to be done. Um but uh mathematics can tell tell you what’s possible. So yeah, you you need this basic science and it’s much cheaper

to do that when it’s all 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. Yeah, in which case it’s wasteful and yeah. 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 [music] 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 [music] to check out my recent conversation with Stephen Wolfram, one of the deepest thinking mathematicians of all time. Don’t forget to like, comment, and subscribe. >> [music]