AI Crisis in Math: Navier-Stokes, Riemann Hypothesis & Complexity Theory | Lex Fridman Podcast #503

1. 0:00 – 1:56

   Introduction

   1. LFLex Fridman0:00

      All right, so this is a kind of emergency episode.

   2. AKAlex Kontorovich0:03

      [laughs]

   3. LFLex Fridman0:04

      I think it's fair to say, about the state of mathematics and, uh, artificial intelligence, um, a kind of crisis moment in the history of mathematics. Over the last few weeks, machines claimed to have solved some of the hardest problems in the history of math. Navier-Stokes, the quasi-Riemann hypothesis, the unique games conjecture, and, uh, hundreds of other results. So here with me, we have, uh, Steven, uh, Brunton, who's a mathematician, fluid dynamics expert at University of Washington, who can speak to Navier-Stokes, turbulence, all things fluid, and many other topics. Also here is, uh, Ankur Moitra, a mathematician and theoretical computer scientist at MIT, visiting, uh, Institute, uh, for Advanced Studies in Princeton this year, who can speak to many things, unique games, conjecture, and complexity theory in general, and, uh, the mathematics of how the AI learns and all things theoretical computer science. And Alex, a mathematician, number theorist at Rutgers, who can speak to also very many topics, including Riemann hypothesis and, uh, Lean, the automated proof and the automated verifier and checking and so on, and so many other topics in mathematics. All three are brilliant. All are very kind to show up this early morning and, uh, talk to me. I assembled this kind of multi-agent mini swarm of human brains to try to interpret what happened in this very moment. A lot of mathematicians, brilliant mathematicians, including Terence Tao and others, are freaking out. It's a mix of awe, terror, dread, and, uh, few inklings of light, of, uh, hope also.

2. 1:56 – 10:13

   Crisis moment in mathematics

   1. LFLex Fridman1:56

      So, uh, I apologize to start with a, a question about feeling, but what have you been feeling about this historic, uh, moment? Maybe Alex, do you want to go first?

   2. AKAlex Kontorovich2:08

      Yeah, I don't know if I would call it a crisis. There's a lot of opportunity in that crisis. It's a, it's a chance to rethink what it is we're trying to accomplish, and it's the chaos of opportunity that some people are reacting negatively to, some people are reacting positively to, and sometimes those reactions are happening in the same person. And so I think we're entering, if, if we're not already in, a period of some kind of golden age of mathematical knowledge after, uh, millennia of having, uh, a candle that we are walking around the woods, and this is as far as we can see. Now we have a flashlight.

   3. LFLex Fridman2:46

      I, I think we should comment on the magnitude of this moment for people who are not aware. So maybe to quote you, something like, um, something like, "Holy shit, if a human did this, it would be an instant Fields Medal." Maybe you didn't say, "Holy shit," but basically saying, uh, this is an incredible result. So a lot of people wouldn't have expected this amount of progress in this amount of time. So this is truly shocking. So you're now speaking as a person who is dealing with the feelings you've felt over the last couple of days, having found this out. So maybe also it'd be useful to go back to the, uh, the feeling you were having when you first saw the report that in this case, OpenAI dropped. And the two of you happened to be on camera [laughs] loosely on the same day, maybe a few drinks in because it was a conference or a get-together. Maybe, Ankur, you-- can you speak to what you were feeling?

   4. AMAnkur Moitra3:40

      It was an interesting night when the results dropped. Now, it was not entirely unexpected because for weeks there'd been rumors about the two biggest problems in my areas that were in that, uh, in that collection of results, so the unique games conjecture and RL equals L. There were even rumors about matrix multiplication, so really my first reaction was, "Wow, it's all true. Thank goodness it's finally out there because now people can actually go through it and process what's happened and figure out what to do next." Now, as to your earlier question, you know, what are we feeling? Well, for me, it changes on a daily basis, but I think the biggest, most overarching feeling is really feeling like you're getting pulled in two very different directions. Um, my work has always been about asking the right questions, and to that extent, these tools are amazing for me because, you know, they increase my ability to learn about other fields. It's like having a world expert who's there to teach you about things in your own language. Your ability to make connections is just magnified. But, you know, one of the parts of my job that I really love the most is the mentorship aspect. Not just training grad students and postdocs and seeing them do great things, but having them come with me on these journeys. So the part that pulls me in the other direction is the recognition that, you know, the playbook and the blueprint for success that, uh, that we had for so long is now thrown out the window. So it becomes a lot harder to advise students and postdocs about, you know, what are things they should be working on that are gonna make an impact, are gonna make a name for themselves. So I feel very excited, but maybe more intellectually lonely than I felt before, uh, in part because, you know, a lot of times great results come from asking very humble questions, at least for me. And now there's so much anxiety and, uh, dread and feeling that, wow, you would have to do something really momentous to stand out in the sea of AI results, that it feels like we've lost some of that patience in terms of planting the seeds from which the great oak trees grow.

   5. LFLex Fridman5:49

      Terence Tao, Fields Medal winner, wrote, "I am horrified by the many person-years of ongoing patient and deliberately slow research efforts, particularly by graduate students and postdocs, toward many motivating problems in mathematics being casually disrupted or destroyed by such a release." Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution, and I mourn the path not taken and the opportunities lost in the frantic race to develop the technology. So, I mean, if we're being realistic, it's more dramatic for a, uh, PhD student and a postdoc than, than maybe the way you've articulated. I mean, you're, uh, you're saying it quite calmly, but if I was a PhD student right now, I'd be in a state of, like, shock. Like, my whole life-- I was going through some of this. I know a lot of my friends are programmers. They're going through this state of shock in last December when basically most of the programming world switched to AI writing most of the programs. So I would approximate it to a feeling of kind of a loss of meaning, kind of terror, like dark stuff. Um, we'll keep exploring that. Uh, Steve, what do you think? What were you-- what are you feeling?

   6. SBSteven Brunton7:15

      I would say most of my feelings are pretty hopeful. Um, I think I'm an engineer. I like solving, you know, problems that make, uh, technology more realistic for humans, artificial hearts and airplanes. And our ability to solve real applied problems is going to get much, much better. Our scope and reach is gonna increase, and so for that, I'm extremely hopeful. Um, but there is a lot of uncertainty, and I think, you know, some of this fear of meaning, of losing control, of, um, you know, just being kind of cut out of the process that we all love being a part of, um, it is a little scary. And I don't think any of us would be surprised at either of the kind of two paths that might, might take place over the next five or 10 years, where we really are just dwarfed by, uh, you know, oth-other intelligence, or it kind of levels off, and this becomes an incredible tool that we use, um, for our curiosity, for our human problem-solving. Uh, and that uncertainty is massive. Those are two very different worlds. Uh, and so uncertainty is, is difficult for humans. I think it's also difficult for those of us who have chosen math. We chose math, you know, probably because we liked rules and formalism and, and, you know, uncertainty is uncomfortable. I think the humility aspect is really important, though. This is a really good opportunity for us to remember why we're doing this. Like, what is human purpose? It's not to be famous. It's not to have a Fields Medal. You know, solving interesting problems, helping other people, being curious, just wanting to understand the world, uh, those are really good things to kind of get back. Like, what do we need to be satisfied humans? Um, and I think that's a really interesting moment for us to think about. Like, I've been thinking about it a lot lately. Um, it is more difficult for young people, right? Like figuring out how to, to find their way. Um, but I actually have more faith in them than, [chuckles] than, than us in a sense.

   7. LFLex Fridman9:19

      Do you think there's something to what Terry says about it's harder to be motivated when you know a problem has been solved, to explore that space when a p-problem has been solved like this by an AI system?

   8. SBSteven Brunton9:36

      Yes and no, in the sense that I do very much agree that the path is as important as the destination. Like, if you go on a road trip, the road trip is, you know, is usually just as fun as, as getting there. At the same time, if we found out that another country had invented, uh, teleportation or time travel, I guarantee we would reinvent it in the next five years. Like, if you knew it was possible and you could reverse engineer a little bit, we would be curious, we would dive into it, and we would figure it out. So knowing there's a solution also is an incredible motivation to get there

3. 10:13 – 26:24

   Will AI "solve" mathematics?

   1. SBSteven Brunton10:13

      much faster.

   2. LFLex Fridman10:14

      Let me ask the question this way. If this moment marks the moment of the trajectory where AI basically, quote-unquote, "solves math" or kills math, can you make the case that that is in fact a trajectory, and then can you make the case that that, that is not a trajectory? Can you steel man both paths? That this is a path towards which, like, mathematics as we know it is done completely, so that's gonna be a thing done by machines, and then what humans do is something fundamentally different, or there still will be something that looks similar to our mo- to, to our current mathematics. Can you make a steel man both cases, Alex?

   3. AKAlex Kontorovich11:02

      Yeah. So, you know, you said two things, and I'd like to separate them. One is solve math, and the other is destroy math. So, uh-

   4. LFLex Fridman11:08

      [chuckles] Right

   5. AKAlex Kontorovich11:08

      ... let's make sol- let's make that precise. What does solve math mean? If you think back to when chess engines first came out and people were like, "Oh, this is gonna be amazing. We'll play centaur games." So this is a human being plus a computer system against another human being and a computer system, and that lasted like 10 seconds-

   6. LFLex Fridman11:26

      [chuckles]

   7. AKAlex Kontorovich11:26

      ... and then the computers got so good that if you did anything but choose the top move that the computer recommended, you would lose.

   8. LFLex Fridman11:33

      Mm-hmm.

   9. AKAlex Kontorovich11:33

      And so the human being's role in that game became, "Yes, accept the first move. Yes, accept the first move." That's no longer fun, and we don't see many games of centaur chess anymore, right? And so, uh, in fact, I just had a, a conversation with, uh, a colleague named Thomas Gilberto where we axiomatized our argument for what the future might look like, and one of the axioms that I'm willing to even accept for the sake of argument is what I call the solved axiom, and the solved axiom says, let's assume at some future point, we're nowhere near there now, and I don't expect us to be, uh, there Necessarily, we'll see what, uh, you know, where the capabilities taper off if, if they do. But let's assume the solved axiom, which says any mathematical problem that can be solved with human interacting with AI could also have been solved by AI alone. That's the solved axiom. So that, that would mean that really there's nothing new for a human being to con-- you know, you can say to an AI system, "Okay, I have problem X, I want to think about problem X, but the only way that I will make progress, any progress that I'll make on problem X will just be press a button." The, the AI thinks about it. It doesn't necessarily solve it, doesn't mean that it'll solve every problem. The Riemann hypothesis is not solved, okay? That's what I really want the solution to. I just didn't think in my lifetime I would see any kind of progress towards it that we saw, uh, a few days ago. So it's not that AI systems are solving every problem. They're solving every problem that could be solved by humans interacting with AI. That's what that axiom would be. So I'm saying if we are ever in that world, does that necessarily destroy mathematics? And my strong argument, uh, is no, I don't think, uh, it destroys mathematics.

   10. LFLex Fridman13:18

       If that axiom is true, what's the role of humans?

   11. AKAlex Kontorovich13:21

       So I don't think that mathematics, pure mathematics, not engineering mathematics, not, uh, things that will, uh, cure cancer and, and solve, you know, jet-- everybody has a jet pack or whatever other technological advancement you might imagine. Pure mathematics, you know, we know that we're proving theorems just because we're curious about them. I don't think the ex-- the solution to theorems living in ones and zeros in data centers is valuable to human beings. I think the value of pure mathematics is that it lives in the heads of human beings.

   12. AMAnkur Moitra13:52

       So just to, you know, add to this, I mean, I, I'm glad you came right out with the question that's on everyone's mind.

   13. AKAlex Kontorovich13:57

       Mm-hmm.

   14. AMAnkur Moitra13:57

       So thank you for that. Um, I would caution against overfitting to one objective function for this, right?

   15. AKAlex Kontorovich14:04

       Yes.

   16. AMAnkur Moitra14:04

       Because, you know, we used to think of ourselves as just the smartest things around. You know, human mathematicians can solve, you know, reach the highest treetops. But now we're faced with the fact that many of the problems that we thought were, you know, quite a ways out have been solved by AI. But maybe one of the consolations we can take is actually the fact that we're very computationally limited compared to the way that AI is solving it. You know, given the same finish line of proving the same theorem, I can't really hold that many papers or ideas in my head at one time. So in order for me to reach that same finish line, maybe I have to develop more generalizable and abstractable ways of getting there that then help you not just solve that one problem, but, you know, explain connections or other directions, uh, you want to get into, right? So it's always amazing when you look back on papers you've written and you can see these threads, invisible sometimes, that connect different areas of work, because developing systems of ways of understanding problems helps you for surrounding problems, not just to get a lean, formalized, you know, one theorem down.

   17. LFLex Fridman15:11

       But the interesting thing here is those threads, what you're saying is sometimes invisible, are quite visible to the AI systems now. It seems like that's perhaps the actual very reason why they're able to make so much progress, is they can extract the threads and pull on them because they have the view of the so many papers.

   18. AMAnkur Moitra15:31

       I think it, it depends on what you mean by threads. You know, for me, the reason you prove results is so that you better understand the world around you. Mathematics, to me, is a vehicle for abstraction and understanding things. And, um, you know, it's when you have a major result, what's important is not just the truth of the statement, but what you've learned en route to doing that. Usually, people's first question is, "Okay, you proved something fantastic, but how did you do it?" And then people, once they digest that new idea, that new definition, that new way of looking at things, there are lots of other problems nearby that fall. So somehow you're building structures that help you attack not just one problem, but a whole sea of problems. And, you know, it's not that we need fully formalized, you know, results to do that. Sometimes we reason in very sloppy metaphors that, you know, help us conceptualize the world around us.

   19. LFLex Fridman16:27

       And how much of this is, uh, something Alex mentioned offline, attention is all you need. How much of this is the human ability to shine a light on part-- on particular areas that, quote-unquote, "matter," so the human ability to ask the right questions, the important questions?

   20. AKAlex Kontorovich16:42

       That's always been the case. So within this-- Before AI, there's a-- you can think of, like, here's the axioms, and just abstractly from the axioms, you can get to logical deductions, right? There are theorems that we could prove from the axioms. And there's a much, much, much larger space of theorems that we could prove than the space that human beings ever actually care about. And so, you know, so by, almost by definition, it's like a tautology that the theorems that we'll get excited about are the theorems that we care about as humans. So maybe AI can convince us that there's some interesting structure. Maybe it can say, "Hey, look over here. Let me give you this new definition or something." It hasn't done that yet.

   21. LFLex Fridman17:19

       And sometimes the, the mechanism by which we select the things that we care about, it could have an application. So Navier-Stokes is connected to fluids and all, um, uh, natural systems. But it could also be a thing that some kind of mixture of beauty and difficulty [chuckles] that just pulls at us. Maybe there's an intuition. Uh, you know, the prime numbers are, are full of these really complicated problems, uh, that maybe don't have an obvious application in the, in the immediate sense, but it pulls at us somehow and maybe, uh, as a symptom of the pursuit of the hard thing, uh, great applied mathematics and engineering and solutions to real world problems comes out.

   22. AMAnkur Moitra18:06

       So just, you know, as an example, right? A lot of times when I'm working on a problem- I might have some like bigger goal that I want to accomplish, but usually what I do is I try to find the simplest, most embarrassing open problem that I still don't know how to solve. You know, that aspect of like digesting it down to something, "Boy, how can we not know this?" It, you know, it feels like it's on the right track because then when we can solve that example, maybe we're assembling the set of tools we need to make attacks on other problems. So I, I do think we're gonna have to become a lot more deliberate about, you know, what do we care about? How do we use these tools from a metacognition standpoint? You know, what exactly are we delegating to them and what are we not delegating to them? And we're really at the beginning stages where we're just trying to piece it together as we go along. You know, on the one hand, looking for a proof is kind of like exploring this tree of possibilities. You might have some intuitions based on some other proofs you've seen about, you know, which directions to go in. Maybe I should go down this branch, maybe I should go down this branch. But sometimes we encounter, you know, points on the tree which have really high branching factor. We just don't know which way to go down. We don't even have a guess about if we went down them, what would we discover? And now in this centaur world of math, we suddenly, you know, have something else intelligent that we can interact with, right? And what is the right way to use that tool to search for the beautiful theories, I think will be challenging and evolving, but it's important to not lose sight of the fact that we don't just care about theorems. So I, you know, I'm a mathematician, I'm a theoretical computer scientist, but I've also been reading a little bit more broadly, you know, about social sciences just to sort of make my peace with all of the changes that have been happening. One of the things that really stuck in my mind that I liked is there's something called, um, the doorman fallacy, which is very interesting. So the idea is that you take a complex job and you boil it down to its simplest, most visible task. So what does a doorman do? I guess they hold the door for people as they're coming and going, right? And when we lived in New York, we really enjoyed talking to our doorman. Uh, you know, they were wonderful people, but, you know, if you digest it down to just that simple job, you can think, "Oh, um, I can, uh, replace them just by a sliding door that detects when people are going or coming," and you miss out on all the aspects of their job that, you know, add a lot of value and are harder to articulate. And I think that's a good way to think about mathematics in the age of AI, too. What does a mathematician do? You know, the simplest and most visible part of the job is we prove theorems. And, you know, theorems about conjectures that were open for hundreds of years, like Fermat's Last Theorem, these are very visible things that we've spent a lot of time popularizing, but I don't think that that's all that a mathematician does. I like to think about it as a mathematician is someone who thinks deeply, and a byproduct of thinking deeply is, you know, proving old conjectures, but it's also building new theories, building new ways of looking at old things, making new conceptual connections. And these are parts that, um, you know, maybe AI will swallow up too, but we're not there yet at this moment. So even though this is a momentous time, I don't think it's the right time to hit the panic button. I think it's the right time to really recenter in terms of what our values are and have these conversations as a field, and remember that what we do really has a plurality of different aesthetics and goals, and that when one thing becomes cheap, like problem-solving for well-defined things, it makes the other things that are orthogonal to it all that much more valuable.

   23. LFLex Fridman21:49

       That's actually really beautiful. The doorman fallacy, going to Perplexity here, is the mistake of reducing a person's job to its most obvious task and assuming that automating that task replaces the value of the whole role. A hotel decides, quote, "We're paying someone to open a door. An automatic door can do that more cheaply." But in fact, the doorman also welcomes guests and recognizes returning customers, handles luggage and hails taxis, provides security and discourages unwanted visitors, and signals prestige and makes guests feel cared for.

   24. AMAnkur Moitra22:23

       That was my favorite part.

   25. LFLex Fridman22:24

       Yeah. [laughs]

   26. AMAnkur Moitra22:25

       I looked up what a doorman does, and boy, we have a lot of commonalities with mathematicians.

   27. LFLex Fridman22:29

       [laughs]

   28. AMAnkur Moitra22:29

       If I don't signal brand value, I don't know what I'm doing.

   29. LFLex Fridman22:32

       [laughs] That's funny. But I, I, I think the task for us humans in general, but also mathematicians, is to come up with that list, like actually make more explicit exactly what it is that humans do. And I haven't heard any of you mention, uh, beauty. I apologize if I've mentioned beauty excessively already. Uh, but there's something that mathematicians are, from my outsider sort of fan perspective, are kinda expositors, scholars of, of, of, of things beautiful, broadly defined.

   30. AKAlex Kontorovich23:08

       That's exactly right is, uh, until five seconds ago, there was a near perfect correlation between the ability to solve new deep theorems and what we actually want as human mathematicians. And, uh, and what we want is exactly what you said, beauty and understanding and depth of thought, and, uh, the fact that you can press a button and get what used to be our, our proxy, uh, without any effort, without any deep thought, and without any, uh, necessarily knowledge. Like, we have, uh, people saying, you know, "Do a breakthrough," and then putting whatever they find on, on Twitter and, uh, and now, you know, what are mathematicians supposed to do with that?
4. 26:24 – 1:05:56

   Can AI create new math?

   1. AMAnkur Moitra26:24

      to do that.

   2. AKAlex Kontorovich26:25

      By the way, and so to-- Sorry, to go back to your point, I don't actually believe the solved axiom. I don't think we're there. I can prove a theorem, but I don't understand why the proof works. There's some magic that happens over here, and until I generalize somehow the phenomenon and understand it more deeply, then I'll see why the, the proof works the way it does. Maybe that's one of the reasons why I don't actually believe the solved axiom, because the things that we do are much more exploratory and what constitutes a good question beyond just, "Here's the Riemann hypothesis, make some progress on it."

   3. SBSteven Brunton26:58

      So I think you also brought up a really important point, this, like, generality and generalization of our concepts. So related to beauty and mathematics, I think one of the gold standards has always been how simple can your result be and how much does it generalize? And, you know, out of this infinite space of theorems that could be proven, uh, finding some, some structure that is simple and generalizes, like that is, I think, still the gold standard. And unless I'm mistaken, humans are still asking the questions. So, like, we, you know, we prove, uh, existence uniqueness of Navier-Stokes with AI, but humans are guiding this. Humans are still asking the questions. And, like, I would love to see the next breakthrough if you had, uh, a machine that can, uh, prove any theorem you pose. I want it to start telling me what questions I should be asking.

   4. AKAlex Kontorovich27:49

      Do you think it's possible that that machine just does a for loop-

   5. SBSteven Brunton27:53

      [laughs]

   6. AKAlex Kontorovich27:55

      ... and, uh, finds all the simple and generalizable, basically all the E equals MC squared?

   7. SBSteven Brunton28:01

      That's happening today, actually. So, like, there are absolutely research programs that are systematically, automatically discovering kind of minimal physical laws, so this is less mathematical theorems, more physics, uh, governing laws and equations.

   8. AKAlex Kontorovich28:15

      So observing natural systems and trying to see in those natural systems-

   9. SBSteven Brunton28:19

      Yep

   10. AKAlex Kontorovich28:19

       ... if there's simple physical laws that, uh, model how they operate.

   11. SBSteven Brunton28:23

       Yeah, distilling it down to, like, the basics. Um, so you can kind of recapitulate known theories. If you observe balls dropping, you can, you know, deduce gravity and F equals MA. If you observe fluids, you can deduce Navier-Stokes. Um, and then taking that to data we haven't written down simple equations, weather, turbulence, um, solar flares, things like that. Um, and I'm assuming similar things are happening in math where, you know, there are probably these, like, minimal pieces, um, that can be deduced.

   12. AKAlex Kontorovich28:54

       Do you think there's a, a large space of simple and generalizable things in math that could be discovered automatically?

   13. SBSteven Brunton29:01

       So my version of the solved axiom is that anything that I do as a human mathematician, I've tried on several occasions to sit down and say, "Okay, you know, LLMs will never do this that I do." And I'm like, "Well, why not? I can ask a question. It can-- You know, I can ask it a question."

   14. AKAlex Kontorovich29:18

       Mm-hmm.

   15. SBSteven Brunton29:18

       I can ask it what questions should I ask.

   16. AKAlex Kontorovich29:20

       Mm-hmm.

   17. SBSteven Brunton29:21

       Right? There's, there's sort of it, uh, because it's such a generalized system, uh, I'm not saying that it's good at all of those things now, uh, but my version of the solved axiom is anything that I can contribute as a human being, it could also have contributed on its own. And so what I really mean by that is in terms of value, what do I bring to the community? It can't be I asked a great question. It can't necessarily be in this world of the solved axiom, which again, I don't believe.

   18. AKAlex Kontorovich29:50

       Mm-hmm.

   19. SBSteven Brunton29:50

       But even in that world, it, it should be something other than I asked a great question, I proved a great theorem. And even in that world, I think you need mathematicians, people with human mathematical expertise.

   20. AKAlex Kontorovich30:02

       Ankur, do you believe in the solved axiom? Where on the spectrum of, like, how close we can get to it, and in that spectrum, what's the role of the human, do you think?

   21. AMAnkur Moitra30:13

       I guess I, I don't really believe in the solved axiom. I think- You know, maybe it's half true in some ways, and figuring out which half of it is true will be very interesting and challenging. So I think there'll be major surprises and more to come in terms of what AI can solve. But, you know, for me, when you're facing a new technology that's really, really good, the danger is that when you try to conceptualize why it's working, you do so by anthropomorphizing it. So, you know, maybe one of the earlier lessons here is from deep learning. So there are these amazing examples. You know, you can take, uh, neural nets, and you can do human-level or better object recognition. But in general, the way that it succeeds is less robust than an, how a human does, uh, because there are all these examples like adversarial examples, where you can add very slight patterns to natural images that can completely fool these object detection systems, but not a human. And, you know, I had a former colleague at MIT, Alex Madry, they had some really nice work, which I really enjoyed. It was, um, I think the title was something like, uh, Adversarial Examples Are Features, Not Bugs. So they did the following cool experiment. So, uh, you take a bunch of natural data, maybe pictures of cats and dogs. You train a deep network to identify them very well. You add some adversarial perturbations to them that, you know, make to the AI system a cat look like a dog and vice versa. So it's kind of think like a small little pattern that changes how the object detector works. And then you create a totally fake data set. You know, it's got pictures of cats with dog patterns on top, and you label it as dog, and vice versa for cats. And amazingly, when you train a deep network on this with all the wrong labels, it's gonna learn something that does well on natural data. So somehow, when we recognize what a cat and dog is, we do it, you know, in a way that we can articulate in terms of, uh, you know, its, its legs and its tail and its shape in general. But a lot of times when, um, you know, deep networks or AI can do things that they might be able to find trapdoors or backdoors to solving the problem that use a lot of statistical relationships in the data that a human doesn't. So to me, you know, it- it's important to not, you know, draw this boundary, you know, very sharply between what you can do and what AI can't do because you're gonna be wrong. Um, but that doesn't mean that I don't believe there will emerge some kind of boundary, if only for the fact that, um, I think we over-anthropomorphize the way that they solve problems. And when I'm looking through the OpenAI solutions, they just don't look like a human has solved those problems. So to me, there still seems to be a very big qualitative difference, and that's an opportunity really for collaboration and centaur math more than it is, you know, dread about, uh, the AI overloads. But, you know, who knows? Could be wrong next week.

   22. LFLex Fridman33:13

       What's your intuition about how much of, uh, this is AI just tying together a bunch of thousands of, uh, disparate papers and pulling at the threads, sorry for bringing back the metaphor, but the threads of the different papers and then just basically articulating the proofs that were already there in the different papers and you're making minor connections? Or is it actually developing new mathematical ideas? Have you seen glimmers of whatever that genius alpha, uh, zero move in, in the game of Go?

   23. AMAnkur Moitra33:47

       Yeah, move 37.

   24. LFLex Fridman33:48

       Move 37.

   25. AMAnkur Moitra33:49

       So I haven't read all 700 of their papers. The ones that I have looked at, you know, some people speak about this dismissively like, "Oh, it was in the convex hull." You do the moves that we know as, as humans, and you just do them really well and do a really good tree search. I wouldn't call it the convex hull. I, I think Tim Gowers had a really good way of, uh, talking about it instead as the semigroup generated by all of these things. Yes, it's the same notion of putting together known moves, but the space of known moves, what it can accomplish is sort of much, much bigger than, uh, this kind of derogatory convex hull. So with, in terms of the quasi-RH, uh, the moves that it took are all things that were known to human beings, and, uh, there's a very long cascade of them that is kind of shocking when you look at it and say, "Oh my God, that's so beautiful. How did you think to-- How did you..." I'm anthropomorphizing it, of course. "How did you think to do this move after that move and after that move?" And yeah, that, that does it.

   26. LFLex Fridman34:53

       Could you articulate, uh, to the more general public the intuition, the line between semigroup and convex hull, the derogatory convex hull, and, uh, quasi-RH, of course, is the quasi-Riemann hypothesis that one of the big proofs.

   27. AMAnkur Moitra35:06

       Yeah. Let me try to do it like this. Uh, you have-- think of like a, a balloon. So the inside of the balloon is its convex hull.

   28. LFLex Fridman35:14

       Mm-hmm.

   29. AMAnkur Moitra35:15

       And so we have like all of these results that, all these ideas that mathematicians have thought up, and then anything inside of that is some linear combination of it.

   30. LFLex Fridman35:23

       Mm-hmm.
5. 1:05:56 – 1:29:49

   Riemann hypothesis

   1. LFLex Fridman1:05:56

      So Alex, [chuckles] you're a number theorist. Um, maybe it's fair to say the central problem in your field is r- the Riemann hypothesis, and, uh, you showed a lot of excitement, uh, on the aforementioned, uh, Twitter, X, about the quasi-RH, um, the quasi-Riemann hypothesis proof. Can you explain that? Can you explain what the Riemann hypothesis is, and can you explain the nature of your excitement about it?

   2. AKAlex Kontorovich1:06:25

      Sure. There's sort of a, a one-minute version of the Riemann hypothesis-

   3. LFLex Fridman1:06:28

      Yeah

   4. AKAlex Kontorovich1:06:28

      ... or a five-minute version or a thirty-hour version.

   5. LFLex Fridman1:06:32

      [laughs] Yeah.

   6. AKAlex Kontorovich1:06:32

      I don't know how far you wanna, uh, you wanna go.

   7. LFLex Fridman1:06:34

      Let's start pulling at that string.

   8. AKAlex Kontorovich1:06:36

      So we know what the prime numbers are. People are interested in the distribution of the prime numbers, and what Riemann discovered is that there's a function, this is totally crazy, that there's a function that takes complex numbers as inputs and has complex numbers as outputs, and that function, uh, governs the behavior of the prime numbers. It's called the Riemann zeta function, and by the zeros of the zeta function, we mean complex inputs where the output is the number zero in the complex plane, so the, the origin in the complex plane. And the question is, where do these-- where are these complex zeros? What inputs, what complex number inputs give you a complex number output of zero when you apply this zeta function? Why in the world should that-- wha- complex numbers? What are you talking about? We have, uh, these discrete numbers that aren't divisible by anything else sitting on, in the natural numbers on the number line, and now you wanna talk about functions from a complex n- space to a, to a complex space. Um, that was the, the, the great breakthrough. The great breakthrough was to figure out how to do signal processing on the prime numbers. Uh, if you think about, you know, the reason people can hear you, your podcast is that my voice box is producing pressure waves of sound. That's the natural space. Then that goes into the microphone. The microphone analyzes it, does the Fourier analysis on it, converts it into amplitudes and frequencies, and it's just a bunch of numbers, stores those numbers or sends those numbers, whatever. The speaker then re- takes those numbers and reconstructs back out, uh, what sounds to people like my voice. The same thing is going on. That's what the z- the zeta function does for the prime numbers. It takes the prime numbers, you can think of the prime numbers as particles of air molecules wiggling around, uh, and it does a signal processing analysis on it that can recover the, the same prime numbers back. And the frequencies and the amplitudes correspond to the zeros of the zeta function. The zeta function has that information in it. And the Riemann hypothesis, so the, the amplitudes of all of these, you can call them Riemann, uh, waves, and the superposition of those waves recovers the distribution of the primes, and all of those amplitudes are supposed to be exactly the same. Like, the orchestra that plays the, the, the music of the prime numbers has every single instrument sounding at exactly the same amplitude. That's, that's what the Riemann hypothesis predicts.

   9. LFLex Fridman1:09:09

      So can you-- Actually, uh, we know what the prime numbers are, but maybe, uh, going into the space of philosophy and the romantic notions of aesthetics, um, why are they so interesting and important and central and beautiful at the core of mathematics? And, uh, connected to why the Riemann hypothesis is exciting is why is it so difficult to figure out if and what the pattern is in the prime numbers?

   10. AKAlex Kontorovich1:09:37

       Let me start with your second question first. Why something is difficult, uh, I only know how to answer once I know what the answer is.

   11. LFLex Fridman1:09:44

       [laughs] Once it's made simple.

   12. AKAlex Kontorovich1:09:45

       Right? Maybe it's not difficult.

   13. LFLex Fridman1:09:46

       Okay. [laughs]

   14. AKAlex Kontorovich1:09:47

       Maybe it's not as difficult as we thought it was. I did not think I would see a proof of the quasi-Riemann hypothesis in my lifetime. So the fact that I saw it on Tuesday, uh, was extraordinarily shocking.

   15. LFLex Fridman1:09:59

       What did you do the moment you saw it, like, in, in, um, 30 seconds after, like, what was your-- Do you remember, like, what-

   16. AKAlex Kontorovich1:10:07

       Yeah, I know where I w-- I, I remember where I was-

   17. LFLex Fridman1:10:09

       [laughs]

   18. AKAlex Kontorovich1:10:09

       ... the second I saw the drop. So we were at this, uh, National Museum of Mathematics Gala-

   19. LFLex Fridman1:10:14

       Yeah

   20. AKAlex Kontorovich1:10:14

       ... in New York City. And so I had, you know, a whiskey in one hand, and, uh, I kept refreshing. Uh, I was talking to people, socializing, but I also kept refreshing because I knew it was dropping around six o'clock Eastern Time.

   21. LFLex Fridman1:10:26

       Mm-hmm.

   22. AKAlex Kontorovich1:10:27

       And, uh, and then the drop hit and, in fact, I was talking to, uh, there was a, a Quanta reporter there, uh, and we were just chatting, "Oh, yeah, they're gonna drop these, you know, 300, 400," we didn't know how many, 700 it turns out, um, problems. And I was like, "Oh, wow, that's really cool. It dropped." So I'm looking at my phone and, like, the second thing I see is quasi-RH, and I'm like, "What do you mean quasi-RH? You don't mean, like, for, for what?" And I, and then I look and it's, it's the zeta function, and it's all Dirichlet functions, so they've solved no zero zeros. And, uh, and I'm like, that, that can't, that can't be what's in this drop. That just-- I, I could not compute-

   23. LFLex Fridman1:11:09

       Mm-hmm

   24. AKAlex Kontorovich1:11:09

       ... in that moment what, what I was seeing.

   25. LFLex Fridman1:11:11

       That speed of progress in that little time, yeah.

   26. AKAlex Kontorovich1:11:14

       It's just, it was so far outside my model of what was possible.

   27. LFLex Fridman1:11:20

       Yeah.

   28. AKAlex Kontorovich1:11:20

       And this is-- It's been a month of things, uh, happening way outside of my model of what was possible. The Fermat's Last Theorem formalization, I was sure we were three years away, at least, from being able to do something like that.

   29. LFLex Fridman1:11:33

       So s- uh, maybe you can elaborate on that. It was in Lean, it was formalized as-- How, how big was the, the Fermat's Last Theorem in Lean?

   30. AKAlex Kontorovich1:11:41

       Like, 13 million lines.
6. 1:29:49 – 1:55:45

   Navier-Stokes

   1. LFLex Fridman1:29:49

      So speaking of which, uh, Steve, you told us a story about soup-

   2. AMAnkur Moitra1:29:54

      [laughs]

   3. LFLex Fridman1:29:54

      ... that made you maybe fall in love with the bigger problem of, uh, fluid dynamics. So, uh, going from math to applied math to engineering, the full thing. So maybe can you talk about Navier-Stokes, these set of equations that explain, uh, fluids, why it's such a big deal, why it's beautiful? Similar as we talked about primes, uh, why is it so hard, and why is it such a big deal that Navier-Stokes, uh, w-was proven?

   4. AMAnkur Moitra1:30:22

      So fundamentally, the Navier-Stokes equations is just a way of stating the conservation of momentum for fluids, for objects that can move kind of freely, like gases and liquids. So we know F equals ma is conservation of momentum, you know, classical Newtonian physics. Um, the same basic principle applies to the gas in this room, to water flowing, and it's true in any volume. So it's true in little volumes, big volumes, kind of imaginary volumes. And if you write down that conservation of momentum in a volume, uh, and then you kind of take the limit as that volume becomes arbitrary, uh, and small, you get these Navier-Stokes equations. It's a differential equation, a partial differential equation, that describes how, uh, the velocity field of a fluid changes in space and time, uh, and also the pressure, um, related to the, the forces acting on that fluid.

   5. LFLex Fridman1:31:18

      Can you give intuition about why fluid, given that F equals ma is a pretty simple equation, why fluids are such complicated systems?

   6. AMAnkur Moitra1:31:28

      F equals ma is simple to write down, but it can get complicated really, really quickly. So if I have, like, a single pendulum, it's really easy to write down the equations for that single pendulum. If I put, you know, another pendulum on the end of it, now I have a more complicated equation, and it gives chaos.

   7. SBSteven Brunton1:31:45

      Fluids is, you know, in, in a sense, if I, if I try to model the air in this room, I could chop the room up into a billion little boxes and write down a little differential equation for how the momentum works in all of those billion little boxes, but now I have a really complicated system of equations. And so it's simple to write down, it's simple to conceptually understand what the equation is saying and representing, uh, but actually solving it and understanding how the fluid actually moves in any kind of practical sense is very, very complicated.

   8. LFLex Fridman1:32:18

      I read something like, you know, like one, one space where, uh, fluid dynamics is important is reducing drag for when you're flying, for airplanes. Uh, I read something like, uh, airlines could save like, mm, something like tens of billions of dollars by reducing the drag by one percent. So why is it so hard? So that's one, and drag is maybe easier than something like turbulence. So like, why is, why is these things that are worth so much money so difficult?

   9. SBSteven Brunton1:32:47

      So the good news is, is that if you can improve our ability to predict turbulence, to simulate it even a little bit better, faster, uh, more accurately, there are, you know, dozens of fields where you can have a huge impact, both economical, uh, ecological, human health and, and safety, with a relatively small improvement. So drag reduction on airplanes, uh, better artificial hearts, you know, more efficient wind turbines and gas turbines and, you know, you name it. The, the world is-- We live and work in a fluid and so do all of our machines. The inverse design problem, I, I know it's actually very closely related to this kind of like math problem. I know what I wanna prove. I don't know how to get there. Uh, I know I want a wing that has high lift and low drag, but I don't know how changing the geometry, changing the shape of that wing will affect this whole cascade of smaller and smaller and smaller vortices that, you know, this turbulent fluid that generates over the wing that eventually, uh, results in its lift and drag. I don't know how to inverse design efficiently. Um, part of it's because the equations are expensive to simulate or to test. You know, wind tunnels are expensive. You need supercomputers to simulate these things. This is actually kind of related to pure math. We'll, we'll get back to this, I think.

   10. LFLex Fridman1:34:09

       Mm-hmm.

   11. SBSteven Brunton1:34:10

       But, um, there's this really nice Feynman quote that, you know, we know how to write down these very general equations, Maxwell's equations, Navier-Stokes equations, Schrodinger's equations, but we currently don't know from the equation what phenomena is possible. You know, if you'd never visited Earth, could you have possibly predicted the sunsets and the tides and all the phenomena just from like a partial differential equation? Like, we're not even close to there, and there's some very fundamental math questions about, you know, if I can-- I can write down a, a differential equation. I can't tell you what's inside of it. Um, we still have to simulate, we still have to test, we still have to do this very, uh, elaborate engineering process to understand how these equations manifest drag on an airplane wing or turbulence inside of a, an artificial heart.

   12. AKAlex Kontorovich1:34:59

       Sorry, can I just ask a question on that? I've always wanted-- So like, in the case of Einstein's theory of general relativity, he writes down these equations. A few years later, Schwarzschild finds a blowup, and they're like, "Oh, come on. So what's, what's wrong with these equations that there's a blowup? You know, if you get enough matter into a small enough space, then like light can't escape it. This-- We don't have any physical experience for this. This has got to be a mistake in the equations," until we discover black holes.

   13. SBSteven Brunton1:35:25

       Right.

   14. AKAlex Kontorovich1:35:25

       And then it turns out that actually the theory perfectly explains, uh, you know, and predicts what, what's supposed to happen. In the case of blowup of Navier-Stokes, are we going to discover something new about water? Or is the fact that at some scale you get to particles, you know, you get to an atomic structure that does not follow the liquid equations anymore, it-- why, why would we have predicted that, uh, there should be solutions to a continuous process that's allowed to be pushed to, to its limits?

   15. SBSteven Brunton1:35:56

       Yeah, I-- It's a great question. Um, I-I'm gonna give you the engineering answer, which is these equations are an exceptionally good model of how fluids work at a certain scale.

   16. AKAlex Kontorovich1:36:06

       Right.

   17. SBSteven Brunton1:36:06

       And, and you're absolutely right. Like, you need some number of individual molecules to be averaged over, uh, in a volume for these to be constitutive equations that are valid. And so if you zoom in small enough, you know, now we're talking about individual, uh, molecules, and Navier-Stokes breaks down. So, uh, the mathematical problem of these equations blowing up is, is interesting from a mathematical standpoint. Uh, by the time they blow up, it stops being a fluid. Um, it's like other-

   18. AKAlex Kontorovich1:36:37

       There's no black holes.

   19. SBSteven Brunton1:36:38

       There's no black holes, like other physics takes over.

   20. AKAlex Kontorovich1:36:41

       Yeah.

   21. SBSteven Brunton1:36:41

       Yeah.

   22. LFLex Fridman1:36:41

       But it's poss-- Wait, I, I think it's interesting from a, a philosophy of science perspective that where equations break down could, and maybe statistically high likelihood, is a window, a door in, in a breakthrough of s- needing some other discovery. Like Newtonian, uh, physics breaking down needs Einstein.

   23. AKAlex Kontorovich1:37:03

       Right. And the grand unification is that-

   24. LFLex Fridman1:37:05

       Yeah

   25. AKAlex Kontorovich1:37:06

       ... we, we don't have gravity at the same scales as we have, you know, interactions of, uh, subatomic particles.

   26. SBSteven Brunton1:37:12

       But, but this idea that, um, we don't know something so fundamental about these equations, there are so many people on Earth right now simulating and testing Navier-Stokes in every industry imaginable. I mean, it's, it's probably one of the most studied and analyzed and used equations in the entire engineering, uh, world, and we don't know this fundamental property of the PDE. Does it blow up in finite time or not? That's like the zeroth order version of the bigger question is, you know, what other kinds of solutions does it admit? Does it, you know, are there sunsets and hurricanes and tornadoes and all the phenomenology that, that could exist? You know, existence and uniqueness is like the The simplest baby step. And that's where it gets interesting to me, is even if there's not a direct tangible use of this mathematical result in an engineering sense, it gets us that one step closer to understanding the nature of these equations and what is possible, which is very fundamentally interesting to us.

   27. AKAlex Kontorovich1:38:14

       Could there be-- could this solution that, that has the blowup be used somehow in a engineering application and, you know, you wanna drive it into this state and get some funny behavior that-

   28. SBSteven Brunton1:38:24

       Yeah

   29. AKAlex Kontorovich1:38:24

       ... helps airplanes fly with less drag or something?

   30. SBSteven Brunton1:38:26

       Yeah, I mean, I think in principle, a huge industry, a huge part of what we try to do is manipulate fluid flows, like bend them into our will, right? So if you watch a, a bird fly or an insect fly, they are shaping vortices and carefully funneling, you know, energy into structures for their, you know, objectives of their daily life to, like, perch on a rock or, you know, hunt. And we try to do the same thing. So flow control is, um, is one of the highest challenges. It's very, very challenging. Um, but all of our science fiction technology would be kind of accelerated if we had better ability to control these flows. And th-this, this proof is one very interesting construction of this kind of, you know, very specialized flow control in a way.
7. 1:55:45 – 2:04:39

   How nature computes

   1. SBSteven Brunton1:55:45

      very tangled.

   2. AMAnkur Moitra1:55:45

      So one of the things I would wonder, you know, scientifically, I mean, these are really interesting problems, and that was a great explanation. So I'm always curious about what computational problems nature can solve already, right? And there are a lot of examples of this. You know, there's a famous one with like slime mold finding shortest paths and mazes In fact, some of the precursors to Navier-Stokes were, you know, like some of Terry Tao's work on showing that you can have blowup for Navier-Stokes-like equations because they can simulate Turing machines.

   3. SBSteven Brunton1:56:16

      Mm-hmm.

   4. AMAnkur Moitra1:56:17

      So I guess I would also wonder what sorts of computational tasks these computations with vortices colliding can do and how that imposes limits on what sorts of coordinate systems ought to exist.

   5. SBSteven Brunton1:56:33

      Hmm.

   6. AMAnkur Moitra1:56:33

      'Cause, you know, you mentioned PV equals nRT. That's an amazing dimensionality reduction-

   7. SBSteven Brunton1:56:39

      Mm-hmm. Yep

   8. AMAnkur Moitra1:56:39

      ... because you don't keep track of all of these, uh, positions and velocities of particles, but there's just, you know, a small list of numbers that describes a meta state, and that's it. But if I really had a, a tremendous reduction, you know, like that in the case of turbulence, then it would seem to me very difficult to, to compute very interesting things too.

   9. SBSteven Brunton1:57:02

      Could it be like the Game of Life, where you have this very simple, uh, cellular automata system, and it's a universal Turing machine, and you can cook up whatever behavior you want in it?

   10. AMAnkur Moitra1:57:13

       Yeah. That's, that's right. I mean, so Game of Life, a lot of times people think about, you know, what are minimal rules to have all of these exotic behaviors, and it's really shocking how small the rules need to be to get that, right? But I guess, you know, I'm a theoretical computer scientist. One of the biggest exports from our field is really the algorithmic lens.

   11. SBSteven Brunton1:57:31

       Mm-hmm.

   12. AMAnkur Moitra1:57:31

       So it's not just about, you know, computation where you have a well-defined problem and a laptop and you wanna solve it, but it's also about, you know, computation in the natural world, you know, and what that teaches us about, uh, what the rules of nature might be. So, you know, some famous examples of this are things like equilibria in multiplayer games, and Nash equilibrium is a great solution concept. But of course, the fact that it's hard to compute Nash equilibrium, you know, throws into question, is it really a universal mechanism that explains how people play games? There was this amazing quote by Kamal Jain that I really like, which is, "If my laptop can't find it, then neither can the market." And, you know, to me, what's really exciting is the nexus of a lot of these different perspectives. Um, you know, what-- When you were describing it, it sounded to me like there are very interesting computational problems under the hood, and that one can kind of get a dual perspective on these things by thinking about how that computation imposes limits on what a good coordinate system ought to look like.

   13. SBSteven Brunton1:58:36

       So I think that's a really nice way of framing the problem. Uh, and I would, I would take two extreme examples. So Earth's weather and climate systems are actually just complex. They-- all the scales do matter. The weather in this zip code matters to me. Um, you know, everything matters. All the variables matter. Uh, and, and you're right. Like, if I, if I over-reduce that system to, you know, five numbers, then I lose the descriptive capability of, of what I actually care about. That's, that's one extreme. The other extreme, l-let's think about how, you know, a moth flies or a butterfly or a fruit fly. They are proof by existence that you can interact with an extremely complicated turbulent flow and do very advanced engineering control. They can control their forces and moments in gusts, in rain. They can live very rich lives, and they presumably don't have a PhD in fluid dynamics or know the Navier-Stokes. They're probably not running supercomputer simulations. There might be five or six numbers that actually matter to their daily life of maintaining a leading-edge vortex and, you know, stalling their wings to perch and to land. And those are kind of two extreme examples, and I, I really want to be able to handle both of those. We want to be able to handle the full complexity of truly complex systems, and we want to be able to distill down the absolute bare minimum essentials for fast decisions in those kind of insect extreme cases. And, and biological flyers still dramatically outperform, you know, human-designed, uh, fluid systems in many regimes, so we still have a lot to learn about, about these, uh, these, these simple flow-

   14. AMAnkur Moitra2:00:33

       So maybe we're, like, veering into research now, but, you know-

   15. SBSteven Brunton2:00:36

       Yeah

   16. AMAnkur Moitra2:00:36

       ... to me, it's always amazing how, uh, nature not only solves very hard optimization problems, it seems to find very robust solutions. And I'm always struck by the fact that, you know, when we model it, we tend to model it assuming we have full knowledge, you know, about what the boundary and initial conditions are for things like Navier-Stokes. But when a bird flies, you know, it doesn't know all of that. And to some extent, you know, modeling things with uncertainty is really how you reframe the optimization problem to get very, you know, rich and robust solutions. And it's kind of amazing to me how, um, a lot of everyday examples sort of fit into this. I remember, you know, I'm a big baseball fan, and I remember reading a math paper about baseball that when you've got an outfielder, you know, once the ball has been hit, a lot of stuff is known. You know, the velocity, the trajectory. You would think the outfielder would just take a straight-line path to where they expected the ball to be. But when you observe outfielders, they don't take straight-line paths. They take a slight curve to it. So why is that? You know, at least one way to wrap your head around it is that that's actually a very robust strategy 'cause when you're taking a curved path, you can continually reevaluate-

   17. SBSteven Brunton2:01:48

       Mm-hmm

   18. AMAnkur Moitra2:01:48

       ... the trajectory of the ball and then adjust as needed. So there are very interesting solutions that nature finds, but it's not always obvious what is the right formulation for the optimization problem in the first place, and sometimes modeling it in the case of uncertainty so that you can

   19. LFLex Fridman2:02:07

       Design solutions that are robust to the fact that you don't have full system knowledge seems to be an important, uh, piece of the picture.

   20. SBSteven Brunton2:02:14

       So there's a fun aside related to this. Um, insects in particular don't have the ability to measure their full flow field. Um, they can react to gusts of wind, gusts of wind faster than the information that hits their wings goes to their brain and back to their muscles. So researchers have studied this for, for decades, um, in, in biology. And what they've found is that there are strain-sensitive neurons embedded on the wings, maybe a few dozen to a few hundred of these in very stereotypical locations. So brother fruit flies will have the same locations of these, these strain-sensitive neurons, and those strain-sensitive neurons fire when the wing bends in a certain temporal pattern. So it gets hit by a gust of wind, the wing bends, neurons fire, and a local computation is happening in that shoulder muscle of this insect that goes directly back and corrects robustly based on partial information of a system it doesn't have equations for that is nonlinear and turbulent. So it's like, again, proof by-

   21. LFLex Fridman2:03:21

       Yeah

   22. SBSteven Brunton2:03:21

       ... existence of exactly what you're saying is maybe the robust strategy of actually relying on partial information and uncertainty. They can do m-more than we can [chuckles] at this point, uh, with way less information.

   23. LFLex Fridman2:03:34

       Yeah. I have a couple of, uh, I apologize, pothead thoughts 'cause Alex mentioned black holes. I have, uh, a, a fun thought that maybe the singularity observed with a blowup in, um, Navier-Stokes equations, uh, comes into play when we're, uh, as opposed to, uh, airplane wings on Earth, we're traveling through two connected black holes in a warp drive that we figure out, and then there's the flow c-control of spacetime, and that becomes relevant because now you're in a black hole, hence the singularity. This is the, the-- For people just listening to this, the, the look Alex just gave me, uh, it hurts my soul. Okay. So moving on to the next topic-

   24. AMAnkur Moitra2:04:13

       But the forcing, you'd need the, the black hole vortex to also be forced at exactly the right-

   25. LFLex Fridman2:04:18

       Yeah, the forcing. Where's the force? I mean, at that point-

   26. AMAnkur Moitra2:04:20

       You could try to ride it. Yeah. There's a, there's a way-- You can make something out of this.

   27. LFLex Fridman2:04:24

       There's some non-zero probability that future ass that will listen to this, that guy is the one time he said something that was actually correct. Okay. Uh, if it's okay, a quick bathroom break.

   28. AMAnkur Moitra2:04:37

       Yeah.

   29. SBSteven Brunton2:04:37

       Sure. Sounds good.

   30. LFLex Fridman2:04:38

       Cool.

8. 2:04:39 – 2:19:31

   Unique Games Conjecture and complexity theory

   1. LFLex Fridman2:04:39

      Ankur, uh, one of my favorite topics is theoretical computer science. You're one of the world experts in many of its problems. So, uh, I'd love to ask you about one of the big problems that's been solved in this, uh, OpenAI drop, which is the Unique Games Conjecture. Can you give people an intuition about what this problem is, why is it, why it's a big deal, and what are some aspects of the proof that are interesting?

   2. AMAnkur Moitra2:05:03

      Yes, I'd love to. Uh, so let me give you a little bit of a background. So let's start off with, uh, complexity theory 101. So, um, one of the most important problems in, in theoretical computer science is the satisfiability problem. So the thing to keep in your mind is imagine you've got variables X one to XN, and they can take on either true or false values. And what I'm gonna do is I'm gonna create a big formula out of them, and usually, the way we do this is with an and of ors. So I start off with a clause that's, you know, an or, X one or not X three or X seven. So depending on my assignment, that clause might be satisfied as long as at least one of those things is true. And then I take the and of many of these different clauses together. And the question is, if I give you that formula, called a conjunctive normal form formula, you know, can you find a satisfying assignment? So can you find an assignment of X one, X two, up to XN, setting them either to be equal to true or false so that all of those or clauses are on, so that the entire formula evaluates to true. So this is the canonical example of a problem that's easy to check someone else's answer, but it might be hard to find an answer yourself. So if you told me that it's satisfiable, all I would do is I'd ask you, "Okay, give me the satisfying assignment." You tell me what X one is set to, what X two is set to, and I just go clause by clause and check to make sure that you really have satisfied everything. Now, the, you know, the central problem in, in theoretical computer science is the P versus NP problem. So are problems that are easy to check, is it easy to find their solutions to? And the presumption in our field is that no, it's not easy, and that's the basis, uh, you know, of a lot of things like the, you know, cryptography and e-commerce and what have you. Now, really, the starting point for getting to something like the Unique Games Conjecture is something called the PCP theorem. So this was a landmark result proved in the '90s. It actually was such a big deal that it made the front page of The New York Times. And the crux of it is really this idea that it's even robustly hard to fool a verifier. So you start off with a formula, and either it's satisfiable or not satisfiable, right? And what does that mean? If it's satisfiable, you can write down an assignment, and I can check one by one that the entire thing evaluates to true. But if it's not satisfiable, that means someplace you're lying. You write down this purported satisfying assignment, and when I go clause by clause, I'm quickly gonna find one clause that is not true, and that invalidates the whole thing. And I say, "Aha, I caught you. It's not actually a satisfying assignment." So the idea behind things like probabilistically checkable proofs without getting into a lot of the technical definitions, the idea is really gap amplification. So can you start off with a formula that is hard to, uh, decide whether it's satisfiable or not? And can you engineer a new, maybe much bigger formula that's robustly easy to check in the sense that if it's satisfiable, it'll still be satisfiable. You can write down an assignment, and it'll be fine. But if it's not, I should actually be able to just flip a few coins, query random variables, check a random clause, and with good probability find that you're lying. So the point is that with very sublinear inspection, with only constant amount of inspection, I can find that you're cheating. So this was a momentous result in theoretical computer science because it laid the foundation really for what's called hardness of approximation. So a lot of algorithms and complexity theory is what do you do when you're faced with problems that are genuinely hard to solve? And, you know, one of the things we do is we try to approximately solve them. So maybe we can't find an assignment if there is one that satisfies all the clauses, but can we find one that satisfies many of the clauses? How large a fraction of the clauses? Now, even there, we would expect problems to become computationally hard after a certain point. It should be that it's not just that, you know, deciding whether it's completely satisfiable or not completely satisfiable is hard, but maybe even deciding is it 100% satisfiable or is it only 99% satisfiable. The best case over all assignments satisfy at most 99% of the clauses. That too might end up being hard to decide. So these are what probabilistically checkable proofs give us. In places where we knew that the optimization problem was hard, they tell us that it's even hard to approximate. And the Unique Games Conjecture is a strengthening of the ordinary PCP theorem because it has to do with a certain type of label cover instance. It's a bit technical to go into the details, but the point is that if that version of a probabilistically checkable proof were true, it would have this great organizing effect where it would turn out that for a huge swath of problems, we would actually know what is the best approximation algorithm, and it would be something very simple. So when I started off in the field, the Unique Games Conjecture had been posed, you know, a few years before that, and there was still a lot of debate about whether it should be true or false. And, you know, people worked hard to try and give, you know, proofs of it. People worked hard to try and disprove it. There was this amazing time period maybe in the 2010s where there was quite a lot of back and forth and, you know, there would be new results came out that people's opinions would change whether they believed unique games or not. Uh, I like this quip. I forget who said it, so I can't properly attribute it, but someone told me, "You know, maybe, uh, God's gonna decide whether it's true or false based on which side is working harder." [laughs]

   3. LFLex Fridman2:11:14

      [laughs]

   4. AMAnkur Moitra2:11:14

      Because it really seemed to change year to year. And, um, you know, one of the results, one of the headlining results in the OpenAI drop of proofs was that the Unique Games Conjecture is true. Now, I should mention that this is not without spectacular human progress on it. So, uh, one of my colleagues at MIT, one of my colleagues at IAS, they both contributed among other people to a lot of the major strides, so Dor Minzer and Irit Dinur. They proved a weaker version called the two-to-one games theorem, and this really was a, you know, shot heard round the world in theoretical computer science because at that point, I think a lot of people weren't sure what was a path forward for Unique Games Conjecture, and I think their work really gave a viable path forward to something that was in the vicinity of unique games. So it sounded like, you know, maybe humans could get there with another 10, 20 years. But, you know, just, um, when OpenAI released the results, it turns out that the Unique Games Conjecture is actually true. So this is quite remarkable because, uh, you know, at some points I, I, I, I guess I did find myself on the side of the doubters because Unique Games just sounded too nice to be true because, you know, um, a lot of work like Prasad Raghavendra's work, it implies optimal hardness of approximation for so many different types of constraint satisfaction problems that the conclusion just looked to me too strong to be true.

   5. LFLex Fridman2:12:46

      What did you feel when you found out that night?

   6. AMAnkur Moitra2:12:49

      Okay. Well, for starters, there were already rumors, you know, in the weeks leading up that OpenAI had solved the Unique Games Conjecture and that it was a true. And these weren't even just completely arbitrary rumors. One of my colleagues, Dor Minzer, he actually posted a paper where he wrote something in the footnote at the bottom that, you know, "We would have liked to polish this paper more, but because of the rumors that were going around, we feel the need to put out our approach sooner rather than later." So for me, it was in some sense a bit of a relief that the results were finally out there. Now, um, they're not very readable papers. It's not like you can point to them and really quickly identify what's a new key idea. But I'm glad that they're out there because now we can move forward on the next phase, which is digesting those proofs and figuring out where the field moves after that. Now, in terms of how did I feel that night, uh, you know, as Alex mentioned, we, you know, we all had a drink in hand. Um, and then I saw, you know, on my phone the news that, um, that these proofs had dropped, and, uh, you know, people kept up, uh, coming and talking to me at the gala, but it's already hard to socialize a math- as a mathematician-

   7. LFLex Fridman2:14:05

      Yeah. [laughs]

   8. AMAnkur Moitra2:14:05

      ... let alone, you know, when you have math you really want to read.

   9. LFLex Fridman2:14:08

      Yeah.

   10. AMAnkur Moitra2:14:08

       So it was doubly hard. Um, the other funny thing about it was, you know, I was in New York already, so I took it as an opportunity before the gala to visit a lot of people I knew at NYU. And I just happened to be meeting with Subhash Khot, the one who introduced the Unique Games Conjecture. And I just so happened to ask him, you know, just a few hours before the gala, before the OpenAI paper is released, "Subhash, what would you do, you know, if they release a positive proof of unique games? Uh, are you gonna read it?" And he told me, "No." [laughing] Which, you know, it's, it's, um, that's a lot of people's reactions, unfortunately. It's just kind of true that there are these grand challenges we work on, and sometimes when they get solved, uh, you know, sometimes it feels like, you know, someone has stolen your thunder.

   11. LFLex Fridman2:14:58

       Mm-hmm.

   12. AMAnkur Moitra2:14:59

       You know, it's, it's not like people didn't have ideas about how to solve the Unique Games Conjecture, and I bet you those, those blueprints would have panned out if we'd given them more time. I think the other thing I worry is that, um, you know, that, that we lose out on the human effort of trying to reach those solutions to begin with. You know, math is not just about what is the furthest I can reach with the current set of tools I have. Is it a convex hull? Is it a subgroup? I don't know. But, you know, sometimes when it's a human-driven solution, we learn a lot more in the process. So I worry about the fact that there were these great landmark problems, field-defining problems that a lot of people were reaching towards, and that decisive progress was made, you know, and the results all dropped in one night, and that people lose their enthusiasm for going down those paths. And I wonder if we'd given those people that space to go down those paths, what other exciting things we would have discovered. So I think one of the things I mentioned, you know, we were all trying to process it at MoMath and, you know, there were a lot of really nice talks. Uh, not all of them went up on YouTube. Some of them were, you know, about benefactors and awards named after them. And the thing I kept coming back to is this idea that, you know, why do you do math in the first place is to play and learn about the world. And we saw so many manifestations of it that night-

   13. LFLex Fridman2:16:22

       Mm-hmm

   14. AMAnkur Moitra2:16:23

       ... you know, because there were awards named after people who, you know, why were they donating to the museum was because they wanted their kids to have a place to play with mathematical ideas and learn about the world. It was also the opening of an Escher exhibit, and what a great visual of, you know, what happens when we play with mathematical concepts, right? And so it felt to me that i- in this moment, it's really important to remember, uh, as we're going through all of this change together, is we don't know what the future holds, what AI and mathematicians' collaboration will look like in the future, what hiring will look like. But the reason we all got into it was because we were very curious. You know, we wanted to understand the world, and mathematicians are very good at abstracting things. You know, they don't just understand one little piece, but they find some abstraction and some set of definitions that turns out to be unreasonably powerful in some other corner of mathematics. So I think it's really important, whatever the future holds, to make sure that we organize our field in a way where we still encourage people to play with ideas. I don't know what kind of reward system will work, but, you know, when I look out and I talk to young people in the field, they have a lot of anxiety. You know, they hear a lot of stories even about people they know, um, you know, running harnesses to, you know, uh, scrape the web for open problems, to dump as many papers as they can to flood archive. I don't know if that's a good strategy. It doesn't seem like it is, but it certainly doesn't feel like fun to me. It doesn't feel like we're playing with ideas and, and learning about the world, and, um, that's sad. I, I just wanna get back to the things I really enjoy. The other thing maybe I, I just wanna add in there too is, like, you know, it's a big identity crisis for a lot of people 'cause, you know, some people view themselves more as problem solvers, and that can feel very comforting, that you can draw a little circle around the types of problems you work on, the types of techniques you use, the types of problems you care about. And that feels very comforting because the rest of the world is messy and you've got blinders on. This is the part that I care about. Now, I think, is the time to remove those blinders because I just don't find that to be a productive way to do it. You know, we love playing and learning about the world around us, and it's important to remember as, you know, a young mathematician, are you smarter than you are curious, or are you more curious than you are smart? It's very important to be more curious than anything else.

   15. LFLex Fridman2:19:01

       And you mentioned fun. There could be another perspective of it, of more about the struggle from my out-- maybe outsider perspective, is some of the most fun in the moment sure feels like struggle. I've talked to a lot of Olympic gold medalists. A lot of them feel something approximating depression after they win the gold medal because it's like you have given everything, and you realize everything you've given is

9. 2:19:31 – 2:33:08

   How mathematicians think

   1. LFLex Fridman2:19:31

      now over.

   2. AMAnkur Moitra2:19:32

      Thirty years ago, there was a documentary of Andrew Wiles, and there's a great scene, I think they even run it right at the very beginning, um, to your gold medalist, you know, what do you do once you've won the thing? He's sitting there at his desk, and he's describing the moment when he realizes what the solution is, not his initial announcement, which was flawed, and then for many months, he was trying to correct it, then he finally saw how to, how to do it. And, um, he's trying to say, you know, "I will never again in my life have a moment as powerful as the moment that I experienced at this desk that day," and he's choking up. He can't get the words out. So that-- Can you get that if you press the button and it, it gave you the solution? Absolutely not.

   3. AKAlex Kontorovich2:20:21

      So exactly to, to both of your points, like the journey, you know, if, if you see a, a giant hill, a giant mountain, and you say, "One day I'm gonna get to the top of that," and you start training for it, and you're sort of doing little by little, whatever, and then a helicopter lands and just drops you at the top.

   4. LFLex Fridman2:20:40

      Can you actually speak to that? The, the interesting thing about Andrew Wiles and a little bit about Grigori Perelman as well is, uh, working alone for a prolonged period of time. What do you make of that? How common is that in mathematics of sitting there and grinding a problem for, in his case, seven years?

   5. AKAlex Kontorovich2:20:59

      Until the 20th century, almost every math paper that had ever been written was written, you know, had one author. I think now the median is two in mathematics. And so we're not a lab science. It, it really is a solitary process. Now, see, the things that I give my graduate students to do are not the things that I'm working on, sort of by design.

   6. LFLex Fridman2:21:20

      Mm-hmm.

   7. AKAlex Kontorovich2:21:20

      I want them to have their own field. I'm tr-- I, I wanna open doors for them. It's, it's different in different fields, right? So when they have a success, it really should be their success, even if I've sort of-- I had an idea that this would go somewhere and they, they executed on it, that's theirs. Um, but in terms of sitting there for s-- Uh, the, the paper that I put out a month ago is something I've been thinking about for 10 years. It's a five-author paper. Over those 10 years, we made incremental progress to the point where we finally, you know, had a, a solution. And that paper was an instance of there's a phenomenon here, and I don't understand what it is. It wasn't, "Here's a problem, solve it." Like, we could not at the start of the project give you the lean, formalized statement that needs a sorry.

   8. LFLex Fridman2:22:06

      Mm.

   9. AKAlex Kontorovich2:22:07

      Like, you discover what the statement is through the process of hacking away at the problem little by little until you see enough of it to, to say, "This is how it comes together."

   10. LFLex Fridman2:22:17

       Actually, that makes me think, can, can each of you describe how your thinking process of how you think through really hard problems? Is there some aspects to it that are interesting, like, um, uh, a lot of coffee in the morning, at night, alone, brooding with a cigar or on a [chuckles] together at a whiteboard with others? Is, is there some interesting, uh, color, texture, insights you can give?

   11. AMAnkur Moitra2:22:45

       Sure. I think, I think a lot through world models in some way. So, you know, I have some mental model of how I think results in some sub-area go, and I think that colors a lot of how I see surrounding areas too. So for me, a lot of my work is really about, you know, building bridges between algorithmic thinking and other problems that usually aren't studied through that same algorithmic lens. So I think when I first started as a advisor, I sort of made the mistake that I gave students questions that were 100% in my area. And the problem then was, you know, when they would come back and they'd have an idea, I could shoot it down very quickly, and I was either right or I was wrong, but either way, no one was having any fun.

   12. LFLex Fridman2:23:32

       [chuckles]

   13. AMAnkur Moitra2:23:33

       And, you know, over time, what I started to do was I would give students questions that were 10, 20 degrees off from my own area, things that I wanted to understand better myself, papers that I thought were important that I just hadn't had time to sit down and read, and then I have the student explain the result to me. And what's fun is we would go through that process together, not, you know, by printing out the paper and going through it line by line, but trying to reverse engineer why do we think this works in the first place? And there's so many details that once you get the feel for it, you kind of have the sense that they ought to work out. But at least in important papers, there are one or maybe even more than that key ideas that really change your perspective, oh, this really might be doable. So what I found was, you know, as soon as we start trying to understand one of these results, we find our own way of looking at things, and then it's not long after we find some other questions that we now think are approachable because we found our own way of, of organizing our thoughts. So that's often the way I do it is the earlier that I can have the student in the driver's seat, not necessarily with a well-defined question-

   14. LFLex Fridman2:24:45

       Mm-hmm

   15. AMAnkur Moitra2:24:45

       ... but with them teaching me something and them being responsible for distilling and digesting and abstracting it down to its essence. You know, reasoning through with them analogies that aren't meant to literally be true, but help us figure out what to expect in neighboring areas. You know, then it feels like they bring a unique perspective, and we're off to the races. So for me, in, in, in a lot of ways, the technology is only gonna enable that kind of work because it means that a much richer set of problems are approachable. You know, for me, when I find experts in related areas that can really explain things in my own language, I hold onto them for dear life because, you know, uh, they, they can speak the gospel. They can tell me how this, this area is organized and explain it in my own language. But now, you know, I have technology that can do that too, and it also feels a bit sad that you lose that human element that was there once before. I think also students have so much anxiety about, you know, what publications are gonna look like that one of the hardest parts is, you know, when I'm thinking about a project, I'm really trying to think about it in a sort of multi-year time horizon because I have some meta vision that I wanna realize, like how can we use learning theory techniques in quantum many body systems? And there are a lot of stepping stones on route to that, but some of the most exciting problems are, you know, many years out. But students are so worried about the incoming AI tidal wave, you know, the future of work, universal basic income. Are any of us gonna have jobs that they're really thinking in three and six-month horizons? And then it's hard to find problems that we both care about just because-

   16. SBSteven Brunton2:26:30

       You know, we're thinking about different worlds

   17. LFLex Fridman2:26:33

       You don't think we should encourage them to continue to think long term? I mean, I understand that-

   18. SBSteven Brunton2:26:37

       Oh, I do, but they don't always listen to you.

   19. LFLex Fridman2:26:39

       Right.

   20. SBSteven Brunton2:26:40

       [laughs] I mean, I don't know if I'm right or wrong. You know, maybe AI will solve all of our problems, but it seems like the only ones that are actually worth us working on at this moment in time are the ones we're still gonna care about in a bunch of years. And you should only be working on future-proof problems. If you're working on things that, you know, you're in a rush because you think the next generation of models will definitely solve those problems, then maybe just go on vacation and don't solve that problem.

   21. LFLex Fridman2:27:06

       Yeah. Can you speak to how you, uh, think-- You, you work with a lot of natural systems. What are the interesting insights that could be pulled out of these natural systems?

   22. SBSteven Brunton2:27:17

       So I find a tension between focusing very hard on an engineering problem I'm trying to solve, which can be very singular. You put on blinders, you only think about this problem. And then giving yourself the kind of mental freedom to look around and just observe and be curious. Very often you find the solution when you turn over a rock and when it's not something you're thinking about, you know, with, with the full front of your mind, but in the side of your mind. If, if you're looking at a problem, you know, through your peripheral vision, you have a much better chance of solving it sometimes. And so this is a, a constant balance for me, is, uh, observing all of the fields that I don't work in and all of the nature that I don't understand and all of the things that I'm curious about, but then also having very focused problems that you don't relent, you don't let go. And I-- this must be true for all of us. We-- I've heard from, from both of you, you know, focus problems you've been working on for years and decades that you haven't let go of. But presumably, uh, that is informed by a million other pieces of information that you happen to pick up through interesting conversations and interesting-

   23. LFLex Fridman2:28:31

       Do, do, do you have example? 'Cause it sounds like neighboring, uh, areas is a common theme. Do you have, uh, examples of things you're like, you're-- the curious peripheral vision of your brain went, "Oh, okay, that's an interesting system, might be relevant," and that ended up being so or maybe in the future will?

   24. SBSteven Brunton2:28:48

       So for me, uh, as a fluid dynamicist, biology and its messiness and complexity-

   25. LFLex Fridman2:28:54

       Mm-hmm

   26. SBSteven Brunton2:28:54

       ... and also material science and its messiness and complexity are neighboring in a way that's very fruitful. I learn a lot from-

   27. LFLex Fridman2:29:02

       Mm-hmm

   28. SBSteven Brunton2:29:02

       ... from both of those fields, and a lot of the kind of basic math that we develop to handle complexity does seem to transfer, and sometimes you have an idea just because you see something is possible or, uh, you start to develop something and you can explain something that we didn't understand before. But I think for me, biology and fluids and materials have this really interesting kind of commonality of multi-scale, non-linear, messy, nebulous objectives, things like that.

   29. LFLex Fridman2:29:30

       By the way, on that topic, do you think there's hope that AI can have some significant breakthroughs, uh, in the messy world of biology?

   30. SBSteven Brunton2:29:40

       I certainly hope so, and I think so. There are large efforts in biology to collect very large data sets and to understand complexity in systems. So the connectome, where you have the full wiring diagram of the brains of, uh, of insects, it looks like we will start to have connectomes of other animals, and eventually it'll be probably like genomes where, you know, it is cheaper and more accessible.
10. 2:33:08 – 2:39:16

    Is math invented or discovered?

    1. AKAlex Kontorovich2:33:08

       we know.

    2. LFLex Fridman2:33:09

       Uh, I apologize for the question, but I'm obligated by the said universe to ask the question, but given this topic, uh, do you think math is invented or discovered? Alex.

    3. AKAlex Kontorovich2:33:19

       Yeah, it's the age-old question. Um, I have a very sort of pragmatic answer to this, which is, uh, the questions that we end up asking, it seems to me, are functions of what's going on in the world, what questions we possibly can be asking given our current state of knowledge. The questions we're trying to answer today are not questions that Euclid could have thought about. Uh, math was sort of-- The, the, the things that people got interested in were different in Greece versus India versus China. But then the answers, once you ask a question precisely, the answer is what God decides the answer is. So we don't have-- So we're discovering answers to questions that we've invented.

    4. LFLex Fridman2:33:57

       We're discovering answer. Okay. Steve, you have worked with a lot of natural systems that look at equations. Equations are, I guess, always approximations of the real world. So what's-- Is math real?

    5. AKAlex Kontorovich2:34:09

       [chuckles]

    6. SBSteven Brunton2:34:10

       You gotta say yes.

    7. LFLex Fridman2:34:11

       [laughs]

    8. AKAlex Kontorovich2:34:12

       [laughs]

    9. SBSteven Brunton2:34:13

       We got a lot of pressure here.

    10. AKAlex Kontorovich2:34:14

        Yeah.

    11. LFLex Fridman2:34:14

        How big is the gap between the mathematical world and the real world?

    12. SBSteven Brunton2:34:18

        Surprisingly small. I think, uh, unexpectedly small. I like, I like your answer a lot, actually. I do somehow fundamentally think that math is both created and discovered. We create mathematical formalisms to understand the world around us, and those approximations are unreasonably effective. It is, uh, it is a miracle that there are simple, compact descriptions, F equals ma equals MC squared, that describe everything that we observe at, you know, all of the scales that we seem to observe. We seem to be able to distill something simple, mathematical that describes them. That feels like we are creating those, but that there is that simple, generalizable description underneath everything, and that seems to be something we're also uncovering.

    13. LFLex Fridman2:35:09

        Do you think that's a good guide, you know, kind of searching for the beautiful simplicity for us? Are we, are we just s- like aesthetically pleasing things?

    14. SBSteven Brunton2:35:21

        I think it's a good guide for us because we are fundamentally simple. We can only handle so much in our minds. We can only envision so much. I think simple is essential for humans to interact with the real world. Um, okay, if I wanna get, like, slightly deeper into the question of created ver-versus discovered, numbers are a good example, integers. We all fundamentally feel like, you know, counting numbers are real in a very essential way. I can have one rock and two rocks and three rocks, and that is a number of rocks. It is a real number of rocks. But those are absolutely abstractions that I have created. A rock is not a rock, is not a rock. It's a collection of atoms. It's a different collection of atoms. It's-- They're not, there's not one rock or two rocks. Do numbers even exist? I wrestle with this. I don't know. Like, uh, do we actually believe that integers are a human creation to interact with our symbolic abstraction of what we observe, quantities, and-

    15. AKAlex Kontorovich2:36:28

        I mean, it took humans thousands of years to discover the number zero.

    16. SBSteven Brunton2:36:31

        Right.

    17. AKAlex Kontorovich2:36:32

        And then negative numbers. Negative numbers were actually accepted by mathematicians in the West after complex numbers-

    18. SBSteven Brunton2:36:39

        [chuckles]

    19. AKAlex Kontorovich2:36:39

        ...were accepted as a mechanism for study, which seems insane, but there's a whole-

    20. SBSteven Brunton2:36:44

        Yeah

    21. AKAlex Kontorovich2:36:44

        ...story behind that.

    22. LFLex Fridman2:36:46

        What's your relationship like with infinity? Is it real to you, or is it a nice thing to have as part of proofs?

    23. AKAlex Kontorovich2:36:53

        So much of modern mathematics is, uh, exactly dealing seriously with infinity, right? Uh, one of the first times people really start to hit a wall in their mathematical education is when they hit calculus, and f- there's sort of several reasons for it. One is they probably have a lot of gaps in their understanding in previous, uh, you know, they got a, a eighty-one on geometry and a seventy-five in trigonometry or whatever. Those are pointing to gaps in their understanding that all of which is gonna come together in one-

    24. SBSteven Brunton2:37:22

        Mm-hmm

    25. AKAlex Kontorovich2:37:23

        ...uh, fell swoop, and the Swiss cheese is just, you know, not, they're not able to cross that threshold. That's one issue with calculus. The other issue with calculus is the first time you deal seriously with infinity. You have to add up infinitely many infinitely small things and get a finite number and have that number make sense. Um, human beings, uh, Newton and Leibniz did not understand infinity the way we understand it today. They had an intuition for what it's supposed to represent. Uh, but if you read Newton, I mean, Newton, in his time, was ridiculed for the way he wrote about, you know, the, the ghosts of departed quantities.

    26. SBSteven Brunton2:37:58

        [laughs]

    27. AKAlex Kontorovich2:37:59

        Uh, you know, you, you, you, you have to divide zero by zero-

    28. SBSteven Brunton2:38:02

        Yeah

    29. AKAlex Kontorovich2:38:02

        ...and, and have it make sense, and you don't do it, uh, you don't do it when they're zero. You do it after they become zero. I mean, the, the way that he was trying to write about these things turned into the, the crisis in the nineteenth century when Cauchy and Fourier proved conflicting theorems. Cauchy said you have a sequence of continuous functions, and you take the limit of continuous functions, you're gonna get a continuous function, right? Limits are a continuous process. Continuous functions are continuous. The, the answer has to be continuous. And Fourier says, "I'm gonna add up a bunch of sines and cosines. Doesn't get more continuous than sines and cosines, and I'll get a sawtooth. I'll get whatever you want. I'll get a discontinuous function in the limit." They broke math.

    30. SBSteven Brunton2:38:38

        Mm-hmm.
11. 2:39:16 – 2:49:31

    Can AI solve P vs NP?

    1. LFLex Fridman2:39:16

       Uh, speaking of crises, um, let me get back [laughs] to, uh, P vs NP question Ankur was talking about. Do you think-- How hard of a problem is that, uh, proving? So this is sort of the central question of complexity theory, P equals NP. Do you think AI will be able to prove this? In fact, we should mention that you're, uh, kind of a part of a panel. There was initially a debate about whether LLM based methods, uh, could one day make progress on the hardest problems in mathematics, like PNP or, uh, the Riemann hypothesis. Um, that was a year ago.

    2. AMAnkur Moitra2:39:54

       [laughs]

    3. LFLex Fridman2:39:56

       So what, what is your updated view on this?

    4. AMAnkur Moitra2:39:59

       Well, clearly they can make progress on-

    5. LFLex Fridman2:40:02

       [laughs]

    6. AMAnkur Moitra2:40:02

       ... central questions in the field. Uh, P vs NP, maybe it's just because it's closest to my heart. That seems very difficult to make progress on. Um, you know, it's kind of remarkable the things we don't understand. So that problem I was describing about satisfiability, we don't actually know any super linear lower bounds. You know, we think that it takes exponential time, not that much faster than enumerating all possible assignments and checking them, and yet we don't know anything that sort of is asymptotically more running time than just reading the darn input.

    7. LFLex Fridman2:40:39

       Mm-hmm. And we should actually mention, y-you've talked about hard and easy. What is a more, um, technical way to think about what is hard and what is easy?

    8. AMAnkur Moitra2:40:51

       Right. So this is polynomial time versus, uh, not polynomial time, or, you know, NP, just to be clear, stands for non-deterministic polynomial time. So the question is whether things which you can solve with non-determinism, like guess the assignment and then check it, can also be solved in polynomial time. So the question there is, if I give you a description of the formula, you know, I tell you what are the variables, what clauses they appear in, is there an algorithm that runs in time polynomial in the size of the input and correctly decides whether or not there is a satisfying assignment?

    9. LFLex Fridman2:41:28

       We should mention that the polynomial time or exponential time or super linear time, all this refers to worst case analysis.

    10. AMAnkur Moitra2:41:37

        That's right. That's right.

    11. LFLex Fridman2:41:38

        So can you explain what worst case analysis is and what are its-

    12. AMAnkur Moitra2:41:43

        Pitfalls. [laughs]

    13. LFLex Fridman2:41:45

        Yeah, I should say what are its pitfalls, 'cause I think it's a little bit counterintuitive from people outside the field, like, to, to, like, to understand exactly when we say a certain problem is hard, you're actually talking about worst case.

    14. AMAnkur Moitra2:41:56

        That's right.

    15. LFLex Fridman2:41:57

        It's like analyzing, I don't know, I don't know why that popped into my head, but, like, marriage. It's like analyzing whether I should marry this person or not by saying, "Well, what's the worst thing that could possibly happen?"

    16. AMAnkur Moitra2:42:09

        [laughs]

    17. LFLex Fridman2:42:09

        I think that's, uh, for human relationships, maybe it's the, it's not such a good strategy. Maybe the average case is a better, uh, is a, is a better guide for making decisions. Anyway, I don't know why I brought that up.

    18. AMAnkur Moitra2:42:20

        Okay, I have no advice about who you should marry.

    19. LFLex Fridman2:42:23

        [laughs] This is comp-- Yeah, complexity theory does not give you-- I think there is some-- Isn't there a dating, um, it's like an optimal stopping theory of, uh, the optimal-

    20. AMAnkur Moitra2:42:35

        Secretary problem.

    21. LFLex Fridman2:42:37

        The secretary problem.

    22. AMAnkur Moitra2:42:37

        Yeah.

    23. LFLex Fridman2:42:38

        There's a lot of different names for it.

    24. AMAnkur Moitra2:42:39

        Yeah. [laughs]

    25. LFLex Fridman2:42:39

        But there is a mathematical way. There's an optimal strategy.

    26. AMAnkur Moitra2:42:44

        So first, so what is worst case analysis? So-

    27. LFLex Fridman2:42:46

        Yes, please

    28. AMAnkur Moitra2:42:47

        ... you know, I was describing the SAT problem earlier, and the algorithmic problem you want to solve is, given any formula, you want to decide is it satisfiable or not.

    29. LFLex Fridman2:42:56

        Yeah.

    30. AMAnkur Moitra2:42:57

        Now, what's crucial there is that when I'm asking to solve that algorithmic problem, I'm not telling you anything more about the structure of the formula.

Episode duration: 3:34:26