Transcription
Prime numbers are these abstract, mysterious things. They're the atoms of arithmetic. Lots of the most basic questions about prime numbers remain unsolved today, even if they've been studied by the greatest minds in mathematics for hundreds or thousands of years. If everyone thinks about the problem in the same way and all the standard techniques break down, then you need to do something different.
James, thank you so much for speaking with us today.
Um, not at all. Really glad to have you. Maybe it would be just useful to start if you could just introduce kind of who you are and what it is that you do here.
>> I'm James Maynard. I'm a professor of mathematics and number theory here at the University of Oxford. And I work on questions in pure mathematics, uh, in number theory. So, in particular, questions about the whole numbers, and my particular area of specialism is the distribution of prime numbers.
What exactly are these strange things called prime numbers?
>> Prime numbers are whole numbers which can't be divided into two smaller numbers other than itself and one. So, four is not a prime number because it can be written as 2 * 2. But three is a prime number because you can only write it as two whole numbers multiplied together as 1 * 3 or 3 * 1.
Prime numbers are important to mathematicians because they're the atoms of arithmetic. All whole numbers can be written uniquely as a product of prime numbers multiplied together. And this means often you can take a complicated question to do with whole numbers and break it down into a simpler question about prime numbers. So, this means that in the same way that a chemist might try and understand chemical compounds by understanding the constituent atoms, pure mathematicians try and understand whole numbers by understanding the constituent prime numbers.
But the other thing is that because prime numbers are fundamental objects in mathematics, it turns out that prime numbers do occur in various aspects of nature and in various aspects of musical composition. When you have numbers like two and four that divide each other, things neatly fit together. Whereas when you have prime numbers like five and seven, uh, often they remain out of sync for a very long period of time. And so it's certainly been suggested that certain cicadas' hibernation cycle lasts a prime number of years as a way of staying out of sync with predators. And certain musicians, when they want to have a dissonant musical effect, they deliberately use different prime number beats so that things stay, um, out of sync with one another for a maximally long period of time. And so even these very abstract objects in pure mathematics, uh, we start seeing crop up in, uh, nature and in music and in art because they have, uh, fundamental links to basic ideas like things being in sync or out of sync with one another.
The fascinating thing about prime numbers is that even though they're these fundamental, basic building blocks of whole numbers, they remain very mysterious to pure mathematicians, and lots of the most basic questions about prime numbers remain unsolved today. Even if they've been studied by the greatest minds of mathematics for hundreds or thousands of years.
>> You have solved some things and you, you have had some success. One of those quite early in your career was the twin primes conjecture. I wonder if you could just kind of explain what the twin primes conjecture is and and why it's something that people were so interested in and are so interested in.
>> Yeah, sure. The twin prime conjecture is the claim that there are infinitely many pairs of primes that differ by exactly two. So, for example, three and five are both prime numbers and they differ by two. Five and seven. And if you go on a computer, you find lots and lots and lots of these. So the natural guess is that there should be infinitely many. This problem has been, is a very notorious problem. It's maybe one of the most basic questions you can possibly ask if you're interested in the distribution of prime numbers. How close together do they come? Because they're whole numbers, they're going to differ by a whole number. And because two is the only even prime number, the only pair of primes that differ by exactly one is two and three. Two is the smallest possible gap between prime numbers other than the gap between two and three. And this is claimed that the smallest gap happens infinitely often. This is, you know, a very simple to say problem, but it turns out that it is a fundamental problem on the distribution of prime numbers and questions along this line, um, are very closely connected to important questions that come up in other areas of pure mathematics and in cryptography. So it has wider applications as well. So it's sort of symptomatic of our lack of understanding about the distribution of prime numbers, which means it's one of these high-profile, compelling problems.
There was a big breakthrough in 2013 by Yitang Zhang, who showed for the first time, uh, that there exists a number, he said, 70 million, such that there's infinitely many pairs of primes that differ by no more than 70 million. So, as you go further and further down the number line, typically the gaps between primes get bigger and bigger. We know this, but the twin prime conjecture is asking about these unusual pairs of primes that come unusually close together. And Zhang's big breakthrough was not quite proving the twin prime conjecture, but he could prove this weaker version that that you do get these unusually close together primes. And it's not the case that the prime gaps just inexorably grow as you go further and further along the number line.
Soon after his work, um, I came up with a different method for studying this thing, which, uh, was a different proof that you get these primes close together. Uh, but this also showed that you get clumps of primes close together. And then in a, uh, collaborative project for optimizing the gap between primes, the current world record, I think that we have, is that there's infinitely many pairs of primes that differ by no more than 246. So, 246 is still quite a lot bigger than two. Um, but it is showing that you get these prime numbers that come unusually close together.
>> And, and so that, that gap is shrinking all the time, but between pairs of primes. Um, how, how close are we to getting it to that two that, that we, we think the conjecture argues for?
>> So, we're still quite far away. We've been stuck now on 246 for, um, about a decade. So, maybe there's a few, uh, further improvements that are possible, but getting below 100 seems to be, um, out of range of the standard techniques. But even if you take the most optimistic version of the whole method that was used to prove this, there's a big barrier to getting down to two. So, we also had a result that said if you assume the most optimistic sort of input that goes into this whole method and this whole way of studying prime gaps, then you could show that there's infinitely many pairs of primes that differ by no more than six, but you couldn't get two. So, it needs a big new idea to get to the twin prime conjecture itself. But it's, uh, at least proof of principle that even though there are these famous open problems on prime numbers that are well over 100 years old and have been studied by some of the greatest minds in mathematics, modern mathematics can still make progress on some of these ones, even if it's only partial progress.
You must be doing something right because four years ago you won this grand accolade of mathematics, the Fields Medal, which is kind of the, the Nobel Prize for mathematicians under 40, I think it is. And you won it in 2022. I imagine that was a huge moment of your kind of career and, and probably your life as well. Do you still remember kind of where you were when you heard the news and, and what it was the kind of the conversation you had with the people telling you?
>> Yeah. So, we'd recently bought a house and, uh, I was decorating the house. I was actually up a ladder painting at the time, and I came down just to get a drink of water or something like that, and I looked at my phone and I noticed there was an email then from the IMU president, the International Mathematical Union President, saying that he would like to have a Zoom call with me. So, that was the first time I got a strong inkling that maybe I'd won the Fields Medal. I was desperately trying to tell myself that he just wanted me to be on, uh, some committee and this would be something tedious. But that's when I started to get excited because there weren't really many other options as to what it could have been. So that's when I essentially first got the news, and then I had this Zoom call with him, and fortunately, he was very kind, and he cut straight to the chase and, uh, said that, uh, the IMU wanted to award me the Fields Medal, which was an amazing experience. I remember my heart was racing, and because there had been this situation with, um, a previous winner, Peter Scholze, declining the award, he asked, >> Do you accept the the award? And so, um, I remember it's one of the most surreal experiences of my, of my life, I think, saying, "Yes, I accept the Fields Medal." And I was, because my heart was racing, I was somehow totally paranoid in that moment that I was going to accidentally blurt out, "No, I want the Fields Medal," or something like that. So I remember, uh, I must have come across as so weird to him, uh, just saying it so slowly. "Yes, I accept the Fields Medal." And that phrase will stay with me for a long time, but it was a completely surreal experience.
>> Yeah, I'm sure they would have accepted if you accidentally said no and, and they would have said, "You can take it back." If we look back at the history of the Fields Medal winners, there are some really kind of prestigious names there. Terence Tao, Timothy Gowers, Edward Witten, just to kind of name a few. Do you feel like you belong amongst those names yet? Like, has, has the time made you feel more, um, like you're one of those people?
>> It's very difficult to say things like that. I mean, in my day-to-day life, I don't sort of feel like I'm a Fields Medal winner. I just feel like, um, I'm a normal mathematician. And so it certainly seems a bit bizarre to me to sometimes see my name on a list of amazingly famous, uh, mathematicians through history and people that I've looked up a huge amount. So I guess the truthful answer is no. I wouldn't necessarily put myself in, uh, a top tier category, but at the same time, it's not really what I think about in a day-to-day sense. I'm very, very proud of having won the Fields Medal, but my real pride is in my research results. That's the sort of theorems are the bread and butter of mathematics. And so my drive is always to prove more theorems, and I'm very happy with the theorems that I have proven, and I don't spend too much time really comparing myself to other people on the list.
When you said that phrase, "you see yourself as kind of a normal mathematician." I think a lot of people when they think of a mathematician, they might just imagine kind of blackboards full of foreign-looking symbols and conjectures and theories, and it's, it's quite hard to get this image of what it is a, a kind of normal mathematician does. Do you feel as confused as, as maybe kind of the general public does when they look at these equations and things, or, or how do you kind of view your role as being a working mathematician?
>> The whole point of doing interesting research is that you're trying to understand things that have never been understood before. And so I spend virtually all my day trying to think about objects that are completely confusing to me and I really don't understand. And occasionally there'll be small moments of insight where I gain a little bit of understanding. But it means that the vast, vast majority of the time, I really don't understand what's going on at all. So I think certainly if you want to be a mathematician, you have to embrace that uncertainty and that lack of knowledge.
The other thing that I think I find maybe is most different to the reality of being a research mathematician compared to say when I was a child and my perception of what a mathematician might be like, is how there's a huge amount of creativity at research mathematics that you really need to come up with new ideas, and that's the core of the subject. But also that there's a very experimental aspect of trying to understand these new things that haven't been thought about before. That people often separate mathematics from say lab-based sciences, where in the lab sciences or the wet sciences, you're doing experiments the whole time, and you're looking at the data, and then you're getting feedback, and then you're gradually building up an understanding from these experiments. But actually, often mathematics is somewhat similar, that, uh, I'm often playing around with mathematical objects and doing essentially mathematical experiments that may be theoretical, but I'm looking at a simple case and trying to work out things that are happening, uh, explicitly in one simple case and trying to extract from that and guess patterns and notice things to try and build up an intuition of what's really going on, which would then allow the creativity to try and come up with a good explanation as to how to handle the truth of these mysterious objects. As you move closer and closer to research, actually, lots of the sciences look more and more similar rather than being further apart.
I've also seen you talk about the way that daydreaming and your letting your subconscious work in the background really helps you mathematically. Could you explain a bit about what you do to kind of evoke that state? There are certain things you do where you think, "I'm just going to stare out the window now and let my brain wonder." Like, how do you view kind of the role of letting your brain work in the background for maths?
To really put myself in the right mindset to understand something, I feel like I need to immerse myself in a problem. So, I need to really have, um, several days in a row where I'm thinking about that problem and I'm living and breathing that problem. And so, you know, when I have a shower in the morning, when I go to the toilet, when I go to bed at night, it, the problem's in my head and I'm still thinking about it slightly. When I'm trying to also evoke my subconscious to think about things, uh, one thing that I do quite often is, um, I like to just go for a walk. That somehow the act of walking is distracting enough for my active brain that it allows my active brain to calm down a little bit and let my subconscious work. It's certainly a very common thing that maybe one day I'll go into work and I'll really want to prove some small result in my research. And I'm sure that, you know, if I work hard enough, I should be able to do it. It's not some big conceptual leap that's required. And I try really hard all day and I completely fail to make any progress. And so I walk home feeling very frustrated. And the thing that I really want to do is just stop thinking about this thing for the day, to be able to, uh, look at it afresh. And almost inevitably on the walk home, I realize how it all works out because that somehow builds up a familiarity and intuition that I don't have if I'm just thinking about it very actively and hard. So maybe I look quite odd to my colleagues, but you'll quite often find me doing laps in the mathematical institute here where I just walk around and I'll do a few walks around the building and just drinking a glass of water or something like that. And I find that's a good way of stepping back from the active thinking and trying to just absorb the things that I've been mulling over in my head.
>> I imagine some of your colleagues probably know when they see you kind of walking out in in the square, not to disturb James, kind of he's thinking about something very important.
>> Yes. I think sometimes I have a slightly glazed look over my eyes. So sometimes one of my colleagues might say hi to me and I won't notice them at all and I'll just walk straight past them. So I apologize. It's not me trying to be rude. It's because I'm lost in my thoughts.
>> I'm sure lots of people would be very interested to kind of know what those thoughts are like and what, what it's kind of like to be in your head when you're doing mathematics. What is actually happening inside your head when you, when you think about a problem?
>> There's the active thought side where often there'll be a problem and I won't really, I often have a feeling that you gradually build up an intuition as to when you're getting to the heart of a problem. So normally, whatever problem, uh, I'm faced with, I won't have any idea of how to solve it, but there should always be an easier version of that problem which I also can't solve. And so, um, normally my first instinct is to try and find simpler versions of the same problem that are isolating some of the difficulties of the problem. And, you know, there'll be some standard techniques for transforming this problem into related problems, and I'll be looking at them, and you gradually get a feeling as to whether these transform problems are moving closer to something that's getting to the heart of what's going on, or is moving you further away. So I'm guided a lot by intuition, but I'm always trying to distill the difficulties and come up with the simplest possible version of the problem, uh, that is an authentic representation of some of the difficulties that are going on in the original problem. So that's a very active, conscious thing that I'm doing. I'm exploring using the techniques that I know, trying to distill difficulties. Then once I've got what I think is a good model problem that's getting to the heart of the difficulty of the main problem, even if it's a massively simplified case, that's the thing that I then really need to think about. And, you know, I've maybe tried all the standard techniques and they don't work, which means that's the interesting part. That's the fun part. That's when you have to do something new. And so that's when I'm going through this long process of not really understanding the objects involved, but trying to experiment and play around and get a feeling and build up an intuition. And often it's this, uh, the active part of me building up an intuition is working out explicit cases, doing some experiments, maybe with pen and paper, maybe with a computer, but I feel then the heavy lifting of building up this intuition, uh, a lot of it comes from my subconscious, and it's very difficult to know how this happens. But I feel very much like I'm a child playing with a toy for the first time, and at first I have no idea how it works, and so I'm looking at it, trying to look at it from different angles, trying to work out on earth, uh, this toy is supposed to fit together and how you're supposed to play with it. And gradually over time, you, you know, start to get an idea of how to play with it. But it, but maybe I start off playing in a very clunky way and a very awkward way. And then I'll suddenly realize, oh, it's much better if I turn it upside down, then, uh, and then it works, uh, much more simply. And I'm always looking for simplicity. I think that I can't hold too many complicated ideas in my head at one time. So, I'm always trying to come up with the simplest explanation for something or the simplest way of thinking about things. And I think that helps me focus very much on what the core difficulty is and where you need to have a new idea and what sort of ideas might be kind of good ideas and which ones are leading you in the wrong direction.
Do you get really annoyed and frustrated when you can't solve a problem? Like, do you remember times when there's been something you've been trying to solve and you just like can't stop thinking about it and you're like, "Why, why can't I get to grips with this?"
>> So much of my day is spent not solving problems. I don't get annoyed at that at all. At any given time, I have a whole list of problems that I like to think about, most of which I'm completely stuck on, and most of which I'm never going to solve. So I think it's very important as a research mathematician to just embrace the fact that most problems that you want to think about and most problems that are interesting to you won't be able to solve. The only times I get frustrated is when there's a problem that I'm sure I can solve. I'm sure it's not terribly difficult, but somehow I'm not quite doing things in the right way. When there's a problem that requires an interesting idea, then those are the fun problems, and that's the fun bit. But I, I completely accept the fact that in all likelihood, I won't be able to solve.
>> One of those problems that, that you mentioned that is very old, but arguably the biggest in, in studying prime numbers, is the Riemann Hypothesis. And it's this hypothesis that if anyone can prove it either false or, or, or correct, then they'll win a million dollars as part of this Millennium Prize, this great mathematical competition of prizes that were set out at the turn of the century. Would you be able to explain a little bit about what the Riemann Hypothesis is and why it's such a huge focus of attention for mathematicians?
So the Riemann Hypothesis is sometimes dubbed the most famous and most important problem in mathematics, and it's describing, uh, the distribution of prime numbers. So we're thinking about how many primes are there less than or equal to some large number. So this is maybe the most basic question you can ask on the large scale distribution of a sequence that you're interested in. And Gauss, a very famous mathematician, when he was 16 years old, um, had been studying lots of tables of prime numbers, just numerical tables. And he made a statistical guess saying that the number of primes which are less than some large number x should be very close to some simple analytic function that it's very easy to understand. And you can then test this numerically, and Gauss's guess is amazingly good. That, uh, it's sort of as good as you could possibly hope for. It's certainly not given a formula for exactly the number of primes less than or equal to x. But it's saying that this very complicated question about the number of primes less than or equal to x, it's very arithmetic, is very close to a very simple thing. And then there's going to be a a leftover correction term which is necessarily going to be very complicated and have all the arithmetic in. And we know that this complicated correction term can be understood in terms of the so-called zeros of this mysterious function, the Riemann zeta function. So there's a complex analytic function whose zeros govern the behavior of prime numbers. In some sense, they're dual. If you knew everything about all the zeros of the Riemann zeta function, you'd know everything about the distribution of prime numbers. But the key point of the Riemann Hypothesis is that, um, if you knew a bit about the distribution of the zeros, in particular their real parts, then you would understand a huge amount about the distribution of primes. And in particular, this would give a really good explanation as to why Gauss's guess for the number of primes less than or equal to x is such a good approximation to the truth. So the Riemann Hypothesis is the claim that all the non-trivial zeros of this Riemann zeta function have real part equal to 1/2. And so they all lie on this magic line in the complex plane with real part equal to 1/2. And it points to this very subtle hidden structure in the distribution of prime numbers that the number of primes is governed by these constants, the zeros of the Riemann function, and they should have this magical symmetry of lying on this line with real part equal to 1/2. And, and we've seen lots of computational efforts to try and prove the Riemann Hypothesis incorrect, kind of calculating where the zeros lie, and it's up to some enormous number now, and it would seem from a naive point of view that we, we could say we've calculated so much of the Riemann function and we've seen that it, it does appear to kind of hold it, even up for for very large numbers. Why can't we just kind of take that as, as it looks good enough? We're up to something like trillions of zeros now. Why can't we just say that, yeah, that the hypo, the Riemann Hypothesis is doing what it's saying? Why, why do we have to kind of prove it?
Well, there's various different reasons depending on your viewpoint. So, as a mathematician, I'm really fascinated by the why. So, why is Gauss's guess so good? And why do the zeros lie on the line with real part equal to 1/2? And in some sense, if you told me, uh, that this morning someone had proven the Riemann Hypothesis, I would obviously be super excited. But I wouldn't be so excited because the Riemann Hypothesis is true. I know it's clearly going to be true. I believe the Riemann Hypothesis very strongly. The reason I'm excited is that any proof of the Riemann Hypothesis would develop all of these new tools that would be able to do much more than would surely be able to do much more than just prove the Riemann Hypothesis. They would give us a whole new set of techniques and a whole new set of tools for looking at prime numbers. And so this would be a huge advance in our machinery. And so as well as proving the Riemann Hypothesis, it would undoubtedly open up a new area of investigating prime numbers and it would reveal deep insights into the structure of prime numbers. And so that's the thing that would really excite me. And so the Riemann Hypothesis, I feel, is true for a very good reason, that it's not coincidence that the zeros all lie on this special line. And what I really want to know is not that they lie on the line, it's why they lie on the line, um, and what's the structure behind all of this that's governing it. So that's my perspective as a pure mathematician.
One also has to be a little bit careful about how far, uh, you take computational evidence. There was a famous conjecture that the difference between Gauss's guess and, uh, the number of primes up to x always had a particular sign. And this was true for all, um, numbers that you could possibly test on a computer. But it's been shown that as soon as you get to really big numbers, so numbers that look like one with 300 zeros after it, then actually this is false. And the sign flips infinitely often in the difference between Gauss's guess and, uh, the truth. And sometimes these theoretical things that seem far beyond any computational regime in the basic question crop up in lots of different other places in mathematics. And so even constants that seem astronomically huge in one context can have much more pronounced effects in related problems. And so it's very important to know the difference between, uh, something that we believe to be true and something actually being true. And, you know, that's what mathematics is all about. We love proof.
>> And why is the Riemann Hypothesis something that is so hard to prove either way? Why, why is it the focus of a Millennium Prize problem and, and why have the greatest minds of mathematics for hundreds of years struggled to crack it?
It's difficult to say why a problem is difficult, but it certainly points to a very subtle truth to do with the distribution of primes that we know that if you look at other sequences that are maybe somewhat similar to the primes, if you looked at the distribution of twin primes, you'd expect again them to have a approximately regular distribution, but with fluctuations. But the fluctuations would look rather different and would look like random noise. And the reason that the fluctuations in the primes don't look like random noise is because there's this very unusual hidden structure caused by the zeros of the Riemann function. And so this points to something that's a fundamental truth about the primes, uh, but one that's very subtle. >> And, uh, it's this subtlety that it's only a small difference between random noise, uh, that I think is maybe one of the difficult things. But the main reason is just we don't really have a good idea as to why the Riemann Hypothesis is true at the moment.
>> And I know that you've done work, not solving it obviously, but kind of chipping away at the, maybe the, the, the foundations. Um, and you had this really remarkable result a couple of years ago now with mathematician Larry Guth, where you were kind of limiting some counterexamples to the, the Riemann Hypothesis. And I wondered if you could just explain what that work was that, that you were doing with Larry Guth and, and how much closer it gets us to solving it overall.
>> Yes. So, we don't know how to prove the Riemann Hypothesis true, which means that there could be potential counterexamples. We can't rule out the possibility of there being zeros off this magical line. But it turns out for lots of questions to do with the distribution of primes, it wouldn't matter if there was just one zero that lay off the line, provided there weren't too many zeros. And so as a workaround for the Riemann Hypothesis, um, for lots of questions to do with the distribution of primes, it's sufficient to say, even if the Riemann Hypothesis might be false, there can't be too many counterexamples to the Riemann Hypothesis. The technical name for lots of these results is zero density estimates. And so that's, uh, what I proved with Larry Guth. I improved, um, our zero density estimates. And so our work was saying that even though we can't prove the Riemann Hypothesis, uh, we can show you that there aren't many counterexamples to the hypothesis with a better quantification of how few the number of counterexamples is. This was very satisfying to me because this area of number theory had been stuck for a very long time. There were lots of questions on the distribution of primes which had been stuck on the possibility that there were lots of zeros with real part equal to 3/4. So we thought that there should be no, obviously we think that there should be no zeros with real part equal to three quarters, that would follow from the Riemann Hypothesis, but there was this bizarre potential conspiracy of there being lots of zeros with real part equal to three quarters that was limiting progress on lots of different questions on the, uh, distribution of primes. And my work with Larry managed to show that there couldn't be that many, uh, possible counterexamples with real part equal to 3/4, which, astonishingly, allowed us to get improvements on lots of different questions about the distribution of primes.
>> And, and obviously that the Riemann Hypothesis is this huge goal and, and probably will be a lifetime or several lifetimes of work for different mathematicians. How much closer does, do you think that your work kind of moves the needle or, or, or gets us closer?
>> Our work is somehow in the wrong direction for proving the Riemann Hypothesis itself. It's more a workaround for the Riemann Hypothesis. So the Riemann Hypothesis is like this big mountain that we don't know how to scale, and if we could scale it, we could get to all this fertile land on the other side. My work with Larry is more a workaround that we can, uh, go a little bit around the side of the mountain, and we don't have to go over the top to get to some of the nice fertile land. Yes, I think if all you cared about was proving the Riemann Hypothesis, uh, my work with Larry is not the right direction to go on. Somehow it's not getting to the heart of what's really going on with the Riemann Hypothesis. Like I mentioned before, you can, you somehow build up a feeling for what's really the meat of a problem. And my work with Larry is saying important things about the distribution of primes, but it's not really getting to the meat of the Riemann Hypothesis itself. So I view it much more as a workaround for the Riemann Hypothesis. That even though we don't know how to improve the Riemann Hypothesis, that doesn't necessarily block our progress on understanding the primes themselves.
You've said before that you think that the Riemann Hypothesis probably requires some big new idea, um, and, and revealing some new connections. Do you have any kind of feeling or intuition about where that idea might come from? Do you think it's going to be completely left field and kind of alien mathematics that no one's seen before, or, or do we have some feeling about what it might be?
I guess that's a kind of the literal million-dollar question, but, um, yeah, I don't have a good feeling at all. It's certainly the case that I feel all the techniques that I know and love don't really fully get to grips with the Riemann Hypothesis. So, I don't have a feeling for any terribly good place to start if I just wanted to sit down and prove the Riemann Hypothesis itself. We have been able to prove the Riemann Hypothesis or analogues of the Riemann Hypothesis in some other settings. And so people have hoped that maybe you could adapt those ideas to prove the Riemann Hypothesis. But, uh, my understanding is that so far all attempts to do anything like that have failed, and my feeling is that again, there's some fundamental new input that's required and new insight to deal with the Riemann Hypothesis itself. So, um, one of the things that I think is so hard about the Riemann Hypothesis is that we don't really even have a plausible pathway of stepping stones that might lead to a proof of the Riemann Hypothesis, which is why it's most likely that it's going to require some deep insight, but probably some deeper insight that's coming from a slightly different or unexpected field that maybe hasn't even been developed yet.
If we imagine, kind of tomorrow, we see a paper uploaded online that that purports to kind of prove the Riemann Hypothesis, and we can consider it solved. How different does, does the world look? I mean, we, we've spoken about our understanding and these connections between mathematics, but what would kind of change if we, if we were able to say this problem is solved?
So there's a huge number of statements, uh, in mathematics that are conditional on the Riemann Hypothesis because people believe the Riemann Hypothesis and they know that there's all this fertile land on the other side of the Riemann Hypothesis. There's many theorems that mathematicians prove that say, "If the Riemann Hypothesis was true, then you'd have this really nice consequence," that can be for prime numbers, but it can also be for things that are seem pretty far removed from prime numbers. Although this seems like a very abstract problem, very much in the realm of pure mathematics, one of the amazing things about the Riemann Hypothesis is that it has a cascade of consequences if it's true. And so would have lots of different applications to, um, mathematics, but also outside of mathematics to questions in computer science as well. So it's very much an academic question, but at the same time, it's put on this pedestal because it points to this really deep mystery within pure mathematics.
So lots of cryptographic algorithms, when you're buying things online, use lots of advanced mathematics, and in particular, use prime numbers. And to be able to use prime numbers, they need to know whether a number is a prime number or not. And there's various different ways of testing whether a number is a prime number. But one of the, um, most common algorithms that's very efficient and is used a lot in practice, we don't have a theoretical proof that it definitely works all the time. But we would know that it works very efficiently all the time if, uh, the so-called generalized Riemann Hypothesis is true, which is a slight extension of the Riemann Hypothesis. However, as I mentioned to you before, the real value to a proof of the Riemann Hypothesis, I think, is that I'm completely convinced that any proof of the Riemann Hypothesis would give a fundamental new insight, uh, into prime numbers and how prime numbers behave. And most likely, this would be very consistent with all the ways we think about prime numbers. So I wouldn't worry that internet cryptography is suddenly going to be broken on through the Riemann Hypothesis, but it would give us a whole load of new tools to understand the distribution of primes, and I think that would be the really, um, revolutionary thing. It would open up all kinds of new avenues in mathematics, but also much, much wider than mathematics. And so maybe we could start having real mathematical proofs of lots of these open questions in computer science that boil down to questions about primes, for example.
Given that you said that whatever is involved in proving the proven Hypothesis will probably come from some corner of math that we don't know about now, or at least we're not aware of, it will probably take a bit of time to verify. And we've seen cases in the past of people producing proofs and it taking months or even years to kind of be fully looked at by the mathematical community and going to be true. But we've also seen now the rise of kind of computer assistants for, um, checking proofs. And I wondered from from your perspective in the mathematical community, do you think that these tools are kind of useful, these proof assistants and, and checking things with the computer?
My feeling is that mathematics is, uh, at the beginning of a period of flux where things are going to change quite a lot. So at the moment, I think most mathematicians, um, uh, may be interested in formal verification of proofs and using tools from AI and things to help them, particularly to strip out some of the tedious bits of their work so they can focus on the important creative, uh, bits. At the moment, all of these tools are in their infancy. So, most mathematicians aren't coding up a formal verification of their proofs, and most mathematicians at the moment are not really using AI tools in a fundamental way to really further their kind of key research. But all of these tools are developing very quickly. And so, it's very interesting and uncertain as to how it will develop over the next 10 years. It's certainly plausible that as these tools develop, and my guess would be, uh, that they will become more and more embedded in the working mathematician's, uh, day-to-day life. And so, um, and my hope is that this will allow us to focus on the really important things in mathematics, the creative reasoning, and it will mean that people aren't hung up on some, uh, technical detail that sometimes can be tedious to write out explicitly, sometimes requires technical knowledge in another area of mathematics that you're not so familiar with. So my hope is that, um, these AI tools can strip out the, um, routine but maybe complicated aspects of math research mathematics and allow people to focus on the really creative things that at the moment it still seems AI struggles with. I imagine that proof assistants will also, and formal verification will become a bigger and bigger thing. So that it'll be much more common that when I have a claim's proof, I might well have a formal verification that this is logically sound. It will still require, uh, human mathematicians to look at my proof and try and extract out what are actually the important ideas in the proof, which is always been the most important aspect. But it means that you won't have to worry about, uh, checking line by line that I haven't made a logical slip.
We've seen some remarkable progress in the past few months of, um, mathematicians using AI tools to, to generate whole proofs, um, and solutions in a seemingly kind of end-to-end way. Uh, specifically with Erdős problems, like these hundreds of problems that were posed by the mathematician Paul Erdős before he died. And these aren't from what experts say the most difficult problems, but they are still problems that you would probably need to be a research mathematician to solve. And I wondered, is there anything in these AI systems, AI tools that we've seen in the past few months that have made you want to look at them and use them in your work, or is it for you still like a pen and paper at the end of the day?
So absolutely, I'm fascinated. I feel it's part of my job to stay abreast of these sorts of developments. So I certainly spend quite a bit of time playing around with the extent to which AI can help me in my research. AI is already, I think, a fantastic tool for literature search. It's been the case that it's uncovered references and papers that I hadn't been aware of, uh, beforehand, particularly more historical things or things that are slightly further away from my main research expertise. I am trying to use and incorporate AI more and more, and I'm sure that this is just going to become a bigger thing. So far, in terms of my core research problems, AI hasn't been terribly useful at helping me. It's still been that pen and paper has been more successful so far for me, but this is a rapidly moving field, and so, um, I imagine that AI will be more and more useful in the years or decades to come. So yes, absolutely. This is, uh, something that I think is, um, an exciting development. It's very uncertain exactly how far it will go and in what direction and how it will change mathematics, but I'm sure it will change mathematics in lots of different ways.
>> But there's nothing that you've seen that suggests that in a year's time we'll wake up and and see that an AI has proved the Riemann Hypothesis, kind of by itself.
I would be very surprised if that happens. But it certainly challenges me to think about, um, what is real creativity? Chess went through this quite a long time ago, that originally it was perceived that, um, people who are very good at chess had this creative human ingenuity, uh, that was out of the reach of machines. Even if people knew that theoretically you could crunch through all possible combinations, um, beforehand and have a computer that was perfect at chess, because that would take so long, it would take the heat death of the universe to crunch all those possibilities, it was felt that there was something kind of fundamental about human ingenuity that allowed chess players to be much better at computers, up until the point when computers started beating even the best human chess players. And it wasn't just that, uh, computers made no mistakes in chess. It was also that, uh, particularly some of the top computer chess programs nowadays, they come up with moves that we would think of as very creative moves. It's a very open question. Uh, I'd be a little bit skeptical as to whether exactly the same thing is going to happen in mathematics. I feel that the search space in mathematics is, um, orders of magnitude bigger when you're really trying to come up with new arguments. But it does make me reflect on to what extent are ideas that I come up in my work, that come up in my work, to what extent are they real creativity on my part, or to what extent is this a mesh of things that I've seen before, and to what extent could, uh, these things that I call creativity come as an emergent phenomenon of a computer system that's just very well trained on having seen lots of good mathematical ideas elsewhere.
So currently, you're, you're still leaning towards a side that there is something special about human creativity and intuition, uh, that, that isn't quite in computational systems right now.
>> So I, I don't think there's anything kind of fundamentally different that, on one basic level, we're, our brains are a processing engine, and so in principle, a computer could perfectly model the sorts of thought processes that our brains go through. And so I don't think that there's necessarily an inherent reason why human thought is, uh, obviously going to overpower and be better than currently designed computer systems. Uh, but the big advantage that we have, and our brains have, is through hundreds of millions of years of evolution in terms of how to train all the connections that we make. Whereas, um, computer systems and large language models, which are the popular AI systems at the moment, they're all based on specific architectures. And my understanding from talking to people who know quite a lot more about this is that this means that large language models have been exceptionally effective at doing some things, but there's some other relatively basic things which they really don't perform terribly well. And I certainly know of various people who have suggested that it needs a different, um, AI architecture behind the AI algorithms if you're going to really make AI programs that are overcoming some of these hurdles and limitations of the current models. And so I think my guess is that a computer could ultimately do whatever humans do, but maybe we don't have the right form of computer programs at the moment. But this is wild speculation on my part.
Oh, >> of course. Is it possible to describe what you're currently working on and what's really kind of caught your attention now that you're spending your walks outside thinking about?
One of my main research projects at the moment is a long-running project with one of my collaborators, Kevin Ford, where we're testing the limits of, um, the techniques that we have at the moment to detect prime numbers. So we're trying to understand as precisely as possible the techniques that we use to study prime numbers, the ways in which they're limited. And so there's certain questions that these techniques fundamentally can't address. And then the hope would be, once we understand these limitations, we can start adding in new ideas to precisely address the kinds of questions that seem out of reach of the current techniques as a way of pushing forward the kind of techniques. And so that's one of the main research problems that I think about. And so if you see me pacing later today, it's reasonably likely that that's what I'll be thinking about.
>> Well, I hope it's a successful walk.
>> Thanks, James. Thank you so much for speaking with us today and giving us all your thoughts. Really appreciate it.
Not at all.