What Claude Actually Did to the Riemann Hypothesis (EP 53)

Claude did not solve the Riemann Hypothesis. But what it actually did may be one of the clearest examples yet of how rapidly AI systems are changing the way difficult mathematics can be attacked. In Episode 53, Lester Nare and Krishna Choudhary go from first principles on arguably the most famous unsolved problem in mathematics. We begin with Euler and the Basel problem, build the Riemann zeta function from the ground up, explain its deep connection to prime numbers, move into the complex plane and analytic continuation, unpack the famous 1 + 2 + 3 + 4 + … = -1/12 result, and finally arrive at the Riemann Hypothesis itself: the claim that every non-trivial zero of the zeta function lies on the critical line. Then we get into Claude. An unreleased Anthropic model was prompted to take a serious run at the problem. It orchestrated roughly 60 autonomous sub-agents, tested hundreds of mathematical approaches, executed code, searched academic literature, challenged its own strategies, created adversarial referees to attack its work, and ultimately produced a result pushing a related mathematical bound well beyond the previous state of the art. The human behind the prompt was not a mathematician. One of his instructions was essentially: believe in yourself. We explain what Claude actually accomplished, what it absolutely did not accomplish, why moving a bound toward two-thirds does not mean the Riemann Hypothesis is “two-thirds solved,” and what the process tells us about agentic AI, mathematical research, scientific discovery, and AI safety. Then it’s transfer season. For the first FFP Summer Transfer Window for Scientists, we look at prominent researchers leaving American institutions for universities and research centers abroad. Using the language of football transfers, we examine major moves in chemistry, battery research, gravitational-wave astrophysics, and neuroscience—and what they reveal about research funding, immigration, scientific infrastructure, and the global competition for talent. Explore the FFP science funding tracker:ffppod.com/funding Help shape Year Two and enter the anniversary merch giveaway:ffppod.com/survey Support the show:ffppod.com/donate Follow:@FFPPod on X / Instagram / TikTok / Facebook

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

• 0:00Krishna Choudhary?

  It's weird. What does it mean to tell Claude to believe in itself? I don't know. I think it's genuinely weird that you could, you could talk to a neural network, which is just crunching matrices and numbers, tell it to believe in itself, and then it just like locks in like LeBron James and it spins up a hierarchical swarm of 60 autonomous subagents to try and tackle this problem. So between them, they, they ran like 2,400 shell commands, hundreds of Python scripts, consumed 31 million output tokens, and they autonomously downloaded 54 academic papers from archive. Perhaps Claude,

• 0:43Krishna Choudhary?

  like many of us, underestimates the rate of AI progress.

• 0:47Lester Nare?

  Hello, internet. This is your captain speaking, Lester Nare, joined as always by my co-host and our resident PhD Krishna Choudhary. We have a great episode today. We have two segments we're going to cover. The first one is going to be on the progress that Claude has been making on the Riemann hypothesis in this explosion of progress as AI tackles many of these open problems and complex areas in mathematics and really understanding what's going on there. And we're going to end with a little bit of fun segment that I forced Krishna to do today, which is the summer transfer window for American scientists.

• 1:31Lester Nare?

  As we've seen in Exodus and as many sports at this time of the year are doing their transfers, I thought it would be great to do it in the same format, but for science. As always, we are going to talk about the science from the ground up today because this is from First Principles.

• 2:05Lester Nare?

  So over the last two weeks, there's been this avalanche of news all over social media, news outlets about AI has now reached a singularity in mathematics, because we've seen sort of this release from OpenAI around this OpenAI Astra project where they had, you know, 10 solved open problems that they provided some workaround and then that was quickly followed by Claude's progress on the Riemann hypothesis. And what we wanted to do today is kind of get a background to understand why this is so important. And then also weed through

• 2:48Lester Nare?

  the hype versus the sort of this is nothing new reaction to that particular result.

• 2:56Krishna Choudhary?

  Yeah, it's, it's, I think, an incredible result. The Riemann hypothesis is the most important unsolved problem in mathematics. If you ask mathematicians, I think they will agree. If you ask, you know, people adjacent to mathematics who have heard about the millennium problems and things like that, this is the one, right? To highlight that, let me give you an example. There was a famous interview with Jim Simons, who's the famous mathematician behind Sherman's Theory, but I think everyone else knows him as the creator of the hedge fund Renaissance Technologies that has the Medallion Fund, which mysteriously just prints money. I think their worst year ever was 20%. That was their worst year in the past

• 3:39Krishna Choudhary?

  like 30 years. Usually they do like 100%, like 40%, 70%.

• 3:45Krishna Choudhary?

  He was, he was $30 billion rich. He was worth $30 billion when he died. And he was asked in this interview, if you could trade your wealth for solving the Riemann hypothesis, would you do it? And you could see him just perk up. And he was like, oh, that's, that's a good question. And then he like sort of stares off into the distance and starts like fantasizing about solving the Riemann hypothesis. And then he, later on, he like goes back to the scripted, oh, you know, my life has been great. You can't choose how your life ends up, you know, no regrets, hashtag no regrets. But you could, behind it all, you could tell.

• 4:22Lester Nare?

  It was, it was a yes.

• 4:23Krishna Choudhary?

  This is a man who was, who was like, I, if I could have done that.

• 4:27Lester Nare?

  I would have done it.

• 4:28Krishna Choudhary?

  That would have, that would have been sick. Right? So given the aura around the Riemann hypothesis, it has become the gold standard around which AI's mathematical capability is judged. And we see this actually in our own comment section. Okay. Whenever we talk about like AI doing math and doing crazy things, like when we covered the Jacobian conjecture not too long ago, a lot of people in the comments are just like, well, wake me up when they solve the Riemann hypothesis, right? As if like this unattainable thing is going to be like how we judge AI, something that humans haven't been able to do for like

• 5:08Krishna Choudhary?

  300, nah, 200 years. Right? So now it's made some progress. It hasn't solved it. And I want to be clear, AI has not solved the Riemann hypothesis. And by some measures, it is still as open as it once was. But the fact that it has made progress is, I think, pretty crazy.

• 5:29Lester Nare?

  Particularly, there's this interesting dichotomy between sort of two factions as it relates to AI progress. There are, there's one faction that says it fundamentally was not going to make any meaningful progress that matters. And there's the other faction that's just saying it's a matter of time. The truth is usually somewhere in the middle.

• 5:48Krishna Choudhary?

  Yes.

• 5:48Lester Nare?

  And it's, this seems like it's somewhere in the middle.

• 5:50Krishna Choudhary?

  Exactly. It seems there. And, and the way that it's done it, I think, is very, very cool. So for this episode, at least for this segment, I wanted to cover it because one, I really like the Riemann hypothesis. I think it's a very cool thing to think about. And two, it's a really nice case study in how AI has progressed from the initial chatbots that we were thinking about, you know, back when ChatGPT came online and everybody was talking about it to now there's this agentic version of AI. And there's an orchestrated agentic version where AI can now command other AI, right? It's this really cool world that we're now living in, cool slash a little weird. And that is central to this story.

• 6:33Lester Nare?

  I want to briefly pause by identifying, this has always been an argument that's brought up, which is sort of this difference between people viewing the word, there's lack of definition, right? So AI, a lot of the perception is that's pure LLM.

• 6:48Krishna Choudhary?

  Yeah.

• 6:49Lester Nare?

  And the systems that are now being used, particularly inside the frontier labs, as well as for consumers, have a variety of capabilities around the next token prediction.

• 7:00Krishna Choudhary?

  Yeah.

• 7:01Lester Nare?

  Piece that make it more than just predicting the next token.

• 7:05Krishna Choudhary?

  Exactly. Exactly. And that's, that's, that's going to be the highlight of this story is what they're capable of. So Anthropic actually put out a description of this finding on their website. But what's really cool is they also shared Claude's own account of how it got there. And they shared a full transcript of one of the agents that was instrumental in creating this proof. Okay. So we get to kind of see the inside of Claude's mind as it was progressing through this 30 hours of problem solving to get to the Riemann hypothesis.

• 7:41Lester Nare?

  It's that Pixar movie Inside Out, where you get to see, we get to see inside the, the thought process.

• 7:46Krishna Choudhary?

  Dude, yeah. And, and I think, I think it's just really cool, like how, how granular we can get here. Okay. So this is, I think, crazier than the Jacobian conjecture counter example that we covered, I think, two or three weeks ago for two reasons. The first reason, this is the Riemann hypothesis. Okay. The Jacobian conjecture, people know about it. I had kind of heard about it. The Riemann hypothesis, I have been hearing about it since even before I knew anything about complex numbers. It's like the Millennium Prize and all this other kind of stuff, right? Two, the person who initiated this, the human

• 8:24Krishna Choudhary?

  being behind it, was not a mathematician. Okay. For the Jacobian conjecture one, Al Pogge was a Princeton and Harvard trained mathematician. This time we've got Jared, we've got, I've got his name right here, Jared Sumner. Okay. He's just a software engineer. Nothing against software engineers, but y'all aren't mathematicians. Okay. He kind of just prompted the new version of Claude that's unreleased. Anthropic is very clear to say it's unreleased. You know, they want their shareholders to be very, very confident of their progress. So it's an unreleased version of Claude that he prompted with just, hey, why don't you take a stab at the Riemann hypothesis? And then when Claude was like,

• 9:08Krishna Choudhary?

  I'm not going to do that. It's 150 year old problem that no one solved. He, he came back with things like believe in yourself.

• 9:16Lester Nare?

  A little bit of.

• 9:17Krishna Choudhary?

  Come on, do it.

• 9:18Lester Nare?

  A positive encouragement.

• 9:20Krishna Choudhary?

  To Claude, to an AI.

• 9:21Lester Nare?

  To an AI. Actually, what, this is, okay.

• 9:25Krishna Choudhary?

  And it's kind of weird that that kind of worked.

• 9:27Lester Nare?

  Yeah.

• 9:27Krishna Choudhary?

  So we're going to get into some of that as well.

• 9:29Lester Nare?

  Okay.

• 9:30Krishna Choudhary?

  Okay. So first, let's go over the Riemann hypothesis. What is it and why are people obsessed about it? Okay. I could dedicate several two hour long videos to this thing. And truth be told, to really understand it, you're going to have to go through a book like Complex Analysis, again, my favorite book, that everyone seems to also love as well. Complex Analysis by Elias Stein. If you go through this book, you got to get to chapter seven to really understand Riemann hypothesis. I think you could skip the chapter on Fourier analysis because that doesn't really tie into this stuff. But everything else, you're going to really need to understand in order to believe some of the magic that I'm about to tell you. But I'm going to go through kind of a Spark Notes version of

• 10:14Krishna Choudhary?

  how you would begin to understand why the Riemann hypothesis is so important to solve. Okay. I'm going to begin with the basal problem. This was first proposed by Pietro Mengele in the 1600s.

• 10:29Krishna Choudhary?

  The problem is pretty simple, okay? You take an infinite series of reciprocals of squares and you add it up. So you get 1 over 1 plus 1 over 2 squared plus 1 over 3 squared plus 1 over 4 squared, all the way up to infinity. This is a infinite series. So 1 plus 1 over 4 plus 1 over 9 plus 1 over 16, all the way. What does that equal? This was proposed in the 1600s. It's named the Basel problem because the people who are really working on it were Leonard Euler and the Bernoulli family. The Bernoulli family is the very famous family that is behind like the Bernoulli numbers, the Bernoulli principle, which is how a lot of people mistakenly say airplanes fly. It's not that simple.

• 11:12Krishna Choudhary?

  I mean, it has something to do with it, but it's not that simple. In any case, the Bernoulli family tried and they failed. This is a family of mathematicians. Leonard Euler tried, honestly, might have been his first time that he just like solved it, okay? And he said, "This whole thing is equal to pi squared over 6." Okay? So 1 plus 1 over 4 plus 1 over 9 plus 1 over 16, all the way down. That's going to equal pi squared over 6. The way he did it was kind of hacky. And like, it was only proven to be completely correct by Weierstrass like 100 years later.

• 11:46Lester Nare?

  It was a little hand wavy.

• 11:47Krishna Choudhary?

  It was a little hand wavy, but he got the right answer. Like so many things that he did, okay? Because his intuition was just like insane, right? Okay, fine. So that's impressive, right? But that's not all his main thing that Euler did for the Basel problem. So the Basel problem is only 1 plus 1 over 4 plus 1 over 9, all the way over, right? Euler, in his text, "Various Observations of Infinite Series," he generalizes to something called the Zeta function, okay? He actually invented the Zeta function. Which is, what if I consider not just 1 over squares, but 1 over cubes? Or 1 over stuff to the fourth power? Things like that.

• 12:31Krishna Choudhary?

  So he creates this function called the Zeta function. And crucially, he shows that that sum of 1 over all of the squares and things like that, of the natural numbers, is equal to a product over the primes.

• 12:45Lester Nare?

  Oh, interesting. Okay. So he took something that was discrete as an initial problem set, generalized it, and came up with and had a rule that related to that generalization.

• 12:55Krishna Choudhary?

  Very good, yeah. The generalization was, I take 1 over the cubes or whatever power that I want. And the little trick that he did was show that the sum could be turned into an infinite product over primes. We're going to get into exactly how that works. Because I think this is a really neat way of seeing why the Zeta function is so important in mathematics. It has to do with primes, okay? And everywhere you look, you'll see that the Riemann hypothesis has to do with the distribution of primes. Understanding just this will give you kind of a sense of why that is the case. Why are primes related to this infinite sum? Okay? So that's what we're going to do right now. Here's where we're going to start. We're going to start with the Zeta function.

• 13:39Krishna Choudhary?

  Let's consider, instead of squares, so in this case, the Zeta function is for any arbitrary s.

• 13:46Lester Nare?

  Yes.

• 13:46Krishna Choudhary?

  But the s is usually 2. That'll give you the Basel problem of 1 plus 4, 1 over 4th, 1 plus 1 9th, plus 1 16th. Instead, let's just consider s equals 1, okay? This is called a harmonic series. It's just 1 plus all of the fractions added up. This thing crucially does not converge, okay? This thing goes to infinity.

• 14:04Lester Nare?

  So there's the harmonic series, which is when the power is 1. There's the Basel problem when the power is 2 and then...

• 14:10Krishna Choudhary?

  And then the Zeta is for any power greater than 1, okay? That's the Zeta series, okay? So we're going to consider the harmonic series, which is when the power is 1. So we're just getting reciprocals, okay? We're going to try to understand this a little bit. Now, we're going to need a second tool in our toolbox. And that has to do with the equation for a geometric series. A geometric series is a power series. It's something like, suppose you were to add 1 plus 1 half, but not, not instead of 1/3, we say plus 1/4th, plus 1/8th. So powers of 2, the same power over and over again, okay? This is not now, you're not changing the number.

• 14:50Lester Nare?

  Yes.

• 14:51Krishna Choudhary?

  You're now changing, you're keeping the number the same, but the power is going up.

• 14:55Lester Nare?

  The variable moves from being in the power spot to being in the...

• 14:58Krishna Choudhary?

  In the exponent.

• 15:00Lester Nare?

  In the exponent spot into the...

• 15:02Krishna Choudhary?

  Yeah, exactly, exactly. So now, this is a geometric series, is what it's called. Because there's a common ratio between each of the terms, right? You're halving each term and then you're adding it up. So you get a 1, that's a whole pizza, let's say. Then a half, that's half a pizza, plus a quarter of a pizza, plus an 8th, plus a 16th, plus a 30 2th, or whatever it's called.

• 15:22Lester Nare?

  30 2th.

• 15:24Krishna Choudhary?

  As you add it all up, you're going to get 2. You can see, right? The 1 is the first.

• 15:28Lester Nare?

  Yes.

• 15:29Krishna Choudhary?

  And then as you add a half plus a 4th plus an 8th and so on and so forth, you're going to get a whole number and the whole thing is going to equal 2. The way, and this is something that you learn in high school calculus, is how to sum up a geometric series. You take the common fraction, which is a half, sorry, the common ratio, which is a half, because you're multiplying a half every time you add a new thing. And you take the first number, so the first number in this case is 1. You put that on the top. 1 divided by 1 minus the common ratio. So in this case, it would be 1 divided by 1 minus 1 half to get you 2. And that's, that's how you sum up a geometric series, okay? So we're going to require this toolbox, all right? Now that we've understood that, this was Euler's genius.

• 16:10Krishna Choudhary?

  He said, "Consider the geometric series for primes." Okay? We had just seen a geometric series for 2. A geometric series for 3 would look like 1 plus 1/3 plus 1 9th plus 1 27th, because those are the powers of 3. We don't do 4, because 4 is not a prime. We, we look at the geometric series for 5, which is 1 plus 1/5th plus 1 25th plus 1 1 25th, and then there'll be 6 25, so on and so forth. All of those things are going to equal 1 divided by 1 minus that common ratio. So it's going to be 1 over 1 minus 1 over 2. Over there, that's going to be 1 over 1 minus 1/3. So that'll give you 2 thirds actually, over there.

• 16:53Krishna Choudhary?

  It's, it's, right, oh, 3 halves, sorry. And so this is the geometric series for powers of primes, okay? Euler said, "Let's take a look at this and see if we can do something." All right? If we multiply all of these series together, so we multiply like the infinite sum up there, the infinite sum up there, the infinite sum up there, right? I'm going to really get a multiplication of those individual representations, right? Those individual answers. And now you're already starting to see that second hand side of Euler's product formula.

• 17:33Lester Nare?

  Yeah, I was going to say that it's the same, yeah. There's the through line.

• 17:36Krishna Choudhary?

  Yeah, there's the through line. The giant pi, that means multiply. The giant sigma means, means sum. The giant pi means multiply. And it's, it's multiplying over the primes, okay? For whatever power, in this case, the power is s equals 1, right? And so now we're seeing the right hand side.

• 17:54Lester Nare?

  Yes.

• 17:55Krishna Choudhary?

  Now, that's fine. Here's the real genius of Euler. Why is he even doing this in the first place? Okay? This seems like a lot of stuff to do without the punchline. Here is the punchline. What about the left hand side? The left hand side, as I said, it's a bunch of primes multiplied together, right? You've got 1 plus 1 half plus 1 4th, the powers of 1 half, the powers of 1 3rd, the powers of 1 5th, you'll get the power of 1 7th later, right? If I multiply those out and I do the foil method, if you remember, like, you have to distribute, each term has to multiply to each term, right? So if I, if I multiply that out, the 1 can be multiplied to all the other ones to get a 1, right?

• 18:38Krishna Choudhary?

  Then the 1 half from the first one can be multiplied to all the other ones to get a 1 half, right? Similarly, the 1 4th can be multiplied to all the other ones to get a 1 4th. The 1 3rd will be there, the 1 5th will be there. I could also get a 1 6th, because the 1 half times the 1 3rd multiplied by all the other ones is going to give me the 1 6th.

• 18:59Lester Nare?

  I see.

• 18:59Krishna Choudhary?

  The 1 8th is going to come from the, the 1 half, right? The 1 9th is going to come from the 1 3rd. And then the 1 12th can come from 1 4th multiplied by 1 3rd. Every single number is going to be represented in that infinite sum, because primes make up the numbers, right? Every single number that is out there, this is called the fundamental theorem of arithmetic, which means that every single number out there, a natural number, can be broken down into multiplications of primes, products of primes, that are unique. There's, there's only one way to do it, okay? And there's only one way to choose all of the different terms in that sequence. And now you can notice, right? What are the ones that are missing?

• 19:43Lester Nare?

  Right.

• 19:44Krishna Choudhary?

  1 7th is missing, because I haven't included that in this, right? Similarly, 1 11th is missing, but that's a prime. That'll be another, that'll be another one on its own. 1 13th is missing, but also 1 14th is missing. Because 1 14th would be the 1 7th multiplied by the 1 half multiplied by all the other ones, right? So all of the other composite numbers are not there. And the primes that I haven't included aren't there.

• 20:09Lester Nare?

  Because they're just further down.

• 20:10Krishna Choudhary?

  Yeah, they're just further down, I haven't included it, right? If I added the, if I added the 1 7 thing, then the 1 14th would show up and the 1 7th would show up, but the 1 11th would not show up, right? Because that would require another...

• 20:21Lester Nare?

  As we expand the top row we have here, it will fill in these gaps that are currently present, and you could do this infinitely.

• 20:28Krishna Choudhary?

  Yes, and you can do this infinitely. And the key is, you will get every single fraction ever.

• 20:31Lester Nare?

  That's incredible, okay.

• 20:33Krishna Choudhary?

  Right?

• 20:33Lester Nare?

  Yeah, yeah, yeah, yeah.

• 20:34Krishna Choudhary?

  This was Euler's genius. And that's how he's showing, this is called Euler's product formula. And it's kind of crazy that like, he set out to just solve the Basel problem. And then, and then he's like, "Oh, by the way, I also noticed this."

• 20:49Lester Nare?

  Just, just have this one on the...

• 20:50Krishna Choudhary?

  Yeah, yeah. So this, here, you've already seen now, the Zeta function is on the left. That's the sum of the 1 over all the natural numbers, right? He has tied this now to a product over primes. And now you can imagine, if I do, if I want to do 1, the Zeta function to the second power, all I have to do is put the primes to the second power.

• 21:11Lester Nare?

  Right, right, yes, yes.

• 21:14Krishna Choudhary?

  And that's why the s is a common exponent there.

• 21:17Lester Nare?

  In, in, across both. Oh, that's so elegant.

• 21:23Krishna Choudhary?

  Right, and this already shows you how the primes are so intertwined with the zeta function.

• 21:29Lester Nare?

  Yeah.

• 21:29Krishna Choudhary?

  Right, they are one and the same.

• 21:30Lester Nare?

  And so part of what we're building here is the connection between those two.

• 21:34Krishna Choudhary?

  Yes, I'm trying to justify why there's so much hype about this, okay? It's because the Riemann zeta function has to do with primes in a very integral way. It's actually encoding the same information.

• 21:48Lester Nare?

  Right.

• 21:48Krishna Choudhary?

  On one side, there's a product of primes. On the other side, there's a sum over every natural number.

• 21:55Lester Nare?

  This is really good.

• 21:56Krishna Choudhary?

  It's kind of cool.

• 21:57Lester Nare?

  This is really good.

• 21:58Krishna Choudhary?

  Yeah, and this was in 1730s, okay?

• 22:02Lester Nare?

  No wifi then.

• 22:03Krishna Choudhary?

  No, um, now, 1792, there's a teenage Carl Friedrich Gauss.

• 22:10Lester Nare?

  Gauss.

• 22:11Krishna Choudhary?

  Gauss is another amazing mathematician, perhaps second or third only to Euler and maybe Riemann, actually. He starts counting primes when he was a teenager, because I guess that's what you do when you're a full autist in Germany.

• 22:28Lester Nare?

  In 1792.

• 22:29Krishna Choudhary?

  So he starts counting the number of primes by hand.

• 22:34Lester Nare?

  Unbelievable.

• 22:34Krishna Choudhary?

  And he notices that the probability that a certain number is going to be prime is related to the log of that number. It's basically one over the log. Like, if I go up to like a 10,000, right? The probability that numbers around 10,000 obviously is not prime. But like, any given number around there is prime is one divided by the log of 10,000, log base e.

• 22:58Lester Nare?

  Okay.

• 22:58Krishna Choudhary?

  Okay, so from there, he realizes that there's this prime counting function, where I count how many primes before a certain number, and that's related to n divided by log n. Okay, there in the blue is, you see a pi of n, that's the prime counting function.

• 23:14Lester Nare?

  Right.

• 23:14Krishna Choudhary?

  N over log n is in red, and you can see that n over log n kind of undershoots. He also realizes that the logarithmic integral, meaning the integral of the log, which is li of n, that's kind of an overshoot most of the time. Turns out, it's not all the time. If you go to 10 to the 10 to the 35, then prime numbers go above the log integral. But like, he actually thought that perhaps the integral function is like always above.

• 23:39Lester Nare?

  Mm-hmm.

• 23:40Krishna Choudhary?

  But he didn't know that 10 to the 10 to the 35.

• 23:43Lester Nare?

  Couldn't get that high.

• 23:46Krishna Choudhary?

  If you see, if you do it by hand and you see it pretty good, you're like, it's probably, you know, I get it. So, this becomes now the prime number theorem.

• 23:56Lester Nare?

  Okay.

• 23:57Krishna Choudhary?

  Which is that the number of primes before a certain level, the number of primes before a certain number is related to n divided by log n, okay?

• 24:09Lester Nare?

  Which is our red here.

• 24:10Krishna Choudhary?

  Yeah, yeah, it's like asymptotic to n divided by, like, it has the same shape.

• 24:14Lester Nare?

  Yes.

• 24:14Krishna Choudhary?

  Okay, it's not like gonna diverge like crazy.

• 24:16Lester Nare?

  Right.

• 24:17Krishna Choudhary?

  It's always gonna stay close, in some sense, right? Now, enter Bernard Riemann.

• 24:23Lester Nare?

  Riemann.

• 24:23Krishna Choudhary?

  Okay?

• 24:24Lester Nare?

  Riemann.

• 24:25Krishna Choudhary?

  Riemann at the University of Berlin, he's got an 1859 article on the number of primes less than a given magnitude. Okay, this is the only, um, this is the only paper that he ever wrote in analytic number theory, and it is probably his most influential. He was mostly like worried about geometry and things like that, like Riemannian manifolds and stuff. That's the stuff that you get into with like general relativity. But this is what he wrote about the prime number theorem. And ever since then, it has been named the Riemann zeta function. Okay, because there's a joke in mathematics, you always name something after the second person who discovered it,

• 25:06Krishna Choudhary?

  because the first person is always Euler, and you can't name everything after Euler, you know? So, some people call it the Riemann Euler zeta function, but, you know, let's just give it to Riemann.

• 25:19Lester Nare?

  Right, right, right, right.

• 25:21Krishna Choudhary?

  So.

• 25:22Lester Nare?

  We already know who the goat is.

• 25:23Krishna Choudhary?

  Yeah, yeah, exactly.

• 25:24Lester Nare?

  We already know who the goat is.

• 25:25Krishna Choudhary?

  Exactly, so, um, he actually did some amazing work on the Riemann, on the zeta function. What he did was extend Euler's definition to the complex variables. What I mean by that is, um, you know, usually, we think about that zeta function as like one over the squares, like one over, um, one over the squares or one over the cubes. The exponent is a real number. You can even imagine one over square roots and things like that, right? Um, Riemann said, no, what if we took one over stuff to the power of a complex number? So it's like 1 plus i, or 2 plus i, what would that do? And it turns out, there's a very well-defined way of talking about that.

• 26:08Krishna Choudhary?

  So here's how it works with complex numbers, okay? Here, this, I'm gonna borrow a lot of these visuals from Three Blue, One Brown, and a lot of our viewers are going to recognize it as such, because he has an absolutely amazing video on YouTube about the Riemann zeta function. Um, and he's the goat of mathematical visuals, so, um.

• 26:30Lester Nare?

  Big shout out.

• 26:31Krishna Choudhary?

  Big shout out. So, here's what happens if I take Riemann zeta to the 2 plus i.

• 26:35Lester Nare?

  Mm-hmm, so in this case, where the primary thing we're changing is the exponent.

• 26:41Krishna Choudhary?

  Yes.

• 26:41Lester Nare?

  In the denominator.

• 26:42Krishna Choudhary?

  Yeah, because that's how the Riemann zeta function is defined, right? The function is defined as the sum over, one over the natural numbers to some power. You choose the power, right?

• 26:53Lester Nare?

  And so we're just saying, we've now chose the power to be.

• 26:55Krishna Choudhary?

  We choose the power to be 2 plus i instead of 2. If it was just 2, it would be the basal problem again, and then I'd get pi squared over 6.

• 27:01Lester Nare?

  Yep.

• 27:02Krishna Choudhary?

  But because it's 2 plus i, if it's just 2, then every single term, I'm just moving along the number line, right? Uh, for 1, I'd get 1 plus 1/4th, plus 1/9th, and I'd go a little bit, little bit closer to pi squared over 6. And I'd just be moving on the number line. With complex exponents, now I'm rotating on the complex plane. The complex plane is defined as the real numbers on the x-axis, the imaginary numbers on the y-axis. So that's why 2 plus i is 2 on the x, 1 on the y. 2 plus 2i would mean 2 on the x, 2 on the y. So that's how complex numbers are defined.

• 27:37Lester Nare?

  Yes.

• 27:37Krishna Choudhary?

  And the way that you do an exponent is you actually start rotating where that's happening. Now, 1 to the power of anything is just 1, so that's why you go 1 first. But then 2, 1 over 2 to the power of 2 plus i, that's gonna curve a little bit downward, and then it's gonna curve a little bit downward for the 3, and it's gonna spiral to that point. So, the Riemann zeta function takes as input 2 plus i and maps it to that point down there, which is like 1.15 minus 0.44i, okay? That's how it works in the complex plane.

• 28:09Lester Nare?

  Yes.

• 28:10Krishna Choudhary?

  Now, turns out, as long as the real part is greater than 1, your sum is gonna be fine. And we've got an animation here, courtesy of Three Blue, One Brown, again. As I change the input, the spiral is gonna change, but it's gonna, it's gonna be well-defined. It's gonna end up somewhere, okay? So here, I'm taking my input, which is the yellow dot, and the output of the Riemann zeta function is where that spiral ends up, okay? Perfectly well-defined function, really nice, okay? Um, let's see what this function does to the gridlines, okay? Here, we've seen individual points getting mapped. Now, let's see what this function does to the gridlines.

• 28:51Krishna Choudhary?

  As long as the power, the real part, is greater than 1, this is how the gridlines map, okay? Now, why do we need the powers to be greater than 1 for the real? That's because, you know, imagine if I did, um, the power equals 0, right? Then I just have 1 plus 1 plus 1 plus 1, that's gonna diverge to infinity. It's not well-defined, at least in the series representation. But if my powers are greater than 1, then each fraction gets smaller and smaller, and so my spiral actually converges somewhere, that's the idea.

• 29:25Lester Nare?

  Okay, got it.

• 29:25Krishna Choudhary?

  Okay, um, and so this is what's happening. All of the, all of the points on the right are getting mapped to these points that are in sort of half of the plane. But now, this kind of begs the question, what about the other half?

• 29:40Lester Nare?

  Right.

• 29:41Krishna Choudhary?

  Right?

• 29:41Lester Nare?

  Right.

• 29:42Krishna Choudhary?

  It seems like I could just draw some lines, right? I could just continue the function.

• 29:51Lester Nare?

  Right.

• 29:51Krishna Choudhary?

  Right, like, I could, I could make up how the rest of the function would behave, given I know so well how this function behaves over here.

• 29:59Lester Nare?

  And specifically, you're referencing to the left of.

• 30:03Krishna Choudhary?

  Of that line.

• 30:04Lester Nare?

  Of the y-axis.

• 30:05Krishna Choudhary?

  Yeah, yeah, exactly. There's a little part on the, on the right of the y-axis too, that this thing is not reaching, right? It's like about like half.

• 30:13Lester Nare?

  It's a little, yeah, yeah.

• 30:14Krishna Choudhary?

  But like, but like, my point is, the parts where we can use the series.

• 30:19Lester Nare?

  Yes.

• 30:20Krishna Choudhary?

  Give me this really nice functional map.

• 30:23Lester Nare?

  Right.

• 30:23Krishna Choudhary?

  Okay?

• 30:25Lester Nare?

  And the idea is, can't we just extend what we're already seeing?

• 30:29Krishna Choudhary?

  Exactly, that's what Riemann said. He's like, can't I just extend what I'm already seeing? I know that it doesn't make sense, right? In terms of like, maybe, in terms of like, yeah, you tell me to plug in 0, it's not gonna make sense, because I'm just adding 1, 1, 1, 1, 1. But in terms of what I'm seeing over here, it kind of makes sense, right? There's a world in which this could make sense. That's what Riemann's thinking to himself, as he's looking at this thing, right? And so that's exactly what he does. He says, if I don't rely on the pesky, if I don't rely on the pesky series representation, right? Which is this 1 over a power plus 2 over a power plus 3 over a power.

• 31:10Krishna Choudhary?

  Instead, I make up another function that behaves the same way on the right-hand side.

• 31:17Krishna Choudhary?

  And it looks like that, two, uh, two to the, two to that power times pi to the power minus one times sine of something, times the gamma function of something, times the Riemann zeta of the flip hand, flip hand side. So if he says, if I want to get to that part, I can take the part that's on the right, flip it, multiply it by like the gamma function of one minus s, multiply by all this other stuff and I'll be fine.

• 31:42Lester Nare?

  This is, it'll be fine.

• 31:43Krishna Choudhary?

  It'll be fine because, and this is the magic of complex analysis. It turns out there's a very good reason.

• 31:49Lester Nare?

  Right.

• 31:49Krishna Choudhary?

  Why complex functions do this. Okay?

• 31:52Lester Nare?

  I can just flip it on the other side of the board.

• 31:54Krishna Choudhary?

  Yeah. And, and, and make a little bit of correction, but all of the grid lines are going to be smooth now. The whole thing is going to be differentiable. And the fact of the matter is this is called analytic continuation.

• 32:06Lester Nare?

  Okay.

• 32:06Krishna Choudhary?

  Okay. What it means is I can take a function that is well behaved in one part. And if I want to extend it, there is only one way to extend it. That's the key. This isn't an arbit, this is not an arbitrary extension. What complex analysis shows is there is only one way to do it. So if you found the way, that's it.

• 32:26Lester Nare?

  That's it. Yeah.

• 32:27Krishna Choudhary?

  That's it. You have found the continuation of that function. There is only one way to do it. Okay. And so this isn't like, it, it is imaginary. I mean, it's in complex variables. So there's an imaginary unit in it, right? But the, there's only one way to imagine it.

• 32:43Lester Nare?

  In this imaginary space.

• 32:44Krishna Choudhary?

  Yes. As to, to, to create everything and make sure that everything is, um, sound and logically closed. There's only one way to imagine it.

• 32:53Lester Nare?

  There's a, the logical consistency only has one answer.

• 32:56Krishna Choudhary?

  Exactly. And so that is the extension. That is the analytic continuation of a meromorphic function outside of its domain where it's like nicely well behaved. Now it's everywhere. Okay. That's the key thing that Riemann discovers. Okay. Side note, um, there was a lot of hype over the, um, the sum one plus two plus three plus four plus five equalling negative one 12. Now you can finally kind of understand why that is the case.

• 33:25Lester Nare?

  Right.

• 33:25Krishna Choudhary?

  If I plug in negative one to the Riemann zeta function, right, then that's going to be one plus one over two to the negative one, but two to the negative one means I just flip.

• 33:34Lester Nare?

  Yep.

• 33:35Krishna Choudhary?

  So I get one plus two, plus one over three to the negative one. So I just flip.

• 33:38Lester Nare?

  Yep.

• 33:39Krishna Choudhary?

  Right? So that's just one plus two plus three plus four. Right? It doesn't make sense on its own. But if you use the Riemann zeta function, you analytically continue it outside of its domain to the negative numbers. And then you plug in negative one to that functional thing that I showed you earlier with the signs and the gamma function and everything. You plug in negative one to that, you're going to get negative one 12. And it turns out that if someone, if someone were to put a gun to your head and say, define this series as a number, you can't say infinity. You have to pick a number. The only correct answer is negative one 12. Okay. To show you another reason why this is the case and why Srinivasa

• 34:19Krishna Choudhary?

  Ramanujan, who's a big hero of mine, is on this, is, um, Ramanujan did not know anything about complex numbers, very little, when he was in India and he was reading like a trig book and discovering like all of math. Okay. Um, he, he wrote a letter to Hardy, who was a big mathematician in Cambridge, asking for like, you know, help to, to go to Cambridge and like work under him. One of the things that he wrote down was this one plus two plus three plus four, dot, dot, dot, equals negative one 12. And he said, I know you'll probably think of putting me into an insane asylum, but look, I have techniques that assign numbers to divergent series and I got negative one 12. Hardy looked at that and he was like,

• 35:03Krishna Choudhary?

  hold on, this guy makes no mention of complex analysis and imaginary numbers, but he still figured out a way.

• 35:10Lester Nare?

  To get to.

• 35:11Krishna Choudhary?

  To get to negative one 12. And this, this was a known thing out there because Hardy is after Riemann, right? So there, you can just plug in negative one and you know that. Ramanujan had no idea about Riemann, about the zeta function. He figured out a really hacky way to like assign it a number, which is like not exactly clean. Again, hacky. Now we know there's reasons why it works, but, um, he also came up with the same number, negative one 12. Just goes to show you how closed math is.

• 35:40Lester Nare?

  He was able to intuit from a totally separate pathway, but it still converged at the same.

• 35:46Krishna Choudhary?

  Same number.

• 35:46Lester Nare?

  The same destination.

• 35:47Krishna Choudhary?

  Yeah.

• 35:47Lester Nare?

  Regardless of his lack of not going down the road already traveled.

• 35:50Krishna Choudhary?

  Yeah. Yeah. He went a complete, but he found the same destination. It's kind of cool, right? And that's what I mean by like, if someone were to put a gun to your head and say, define what this is, and it can't be infinity, the only correct answer is negative one 12. Okay. The real correct answer is really infinity, right? But if you were to put, if you were to really put me through it, it would have to be negative one 12. Right? And the Riemann zeta of negative one is negative one 12. Right? It just, I like, I don't think it means anything to sum up the natural numbers, but like actually like string theorists use that all the time.

• 36:24Lester Nare?

  Ah, yes.

• 36:25Krishna Choudhary?

  To like deal with their infinities.

• 36:26Lester Nare?

  Yeah, yes.

• 36:27Krishna Choudhary?

  They're like, ah, I think negative one 12. It's like, okay.

• 36:29Lester Nare?

  The number of dimensions is the denominator.

• 36:32Krishna Choudhary?

  They probably have some party trick where they, where they do that. Um, I, I just thought that's, that's a really cool, like little aside.

• 36:38Lester Nare?

  No, no, that's, and that's, I think it's, cause it's, it, part of what we're doing is we're connecting all of these component parts, you know, math, mathematics being almost this closed system, right? That has, has these bounded box and, and a level of like the logic, logical consistency that exists through it, even if you come from different disciplines.

• 36:59Krishna Choudhary?

  Yeah.

• 36:59Lester Nare?

  Within that subject matter, um, and seeing how those different disciplines address the same question.

• 37:07Krishna Choudhary?

  Yeah.

• 37:08Lester Nare?

  Um, or try to approach the same question.

• 37:10Krishna Choudhary?

  Yeah.

• 37:10Lester Nare?

  It's really fascinating.

• 37:12Krishna Choudhary?

  Yeah. I mean, it's, it's really cool. I mean, the basal problem is another example of that. Let me just give you another little anecdote from my personal life. Um, I, I told you about math 215 at Princeton University, the first semester of math that I did. One of the problems on the final was to calculate the basal problem. Okay. It's like, great. Okay. Euler did it. Yeah. You can do it for a final, you freshman. And, uh, this, this was Peter Sarnak. He was just a maniac.

• 37:35Lester Nare?

  Classic.

• 37:35Krishna Choudhary?

  But, um, he actually, in the problem statement, he, he gave us enough information and we used Fourier series and a clever trick with Fourier series to show, and I got that problem right. It was, it was a great problem. Um, but yeah, you can use Fourier series to show the pi squared over six. You can use Euler's hack. You can also use, um, Taylor series. A lot of times people use like, um, the Taylor series for sine and things like that. So there's so many different ways, but you will always arrive at pi squared over six. Just the beauty of mathematics is, is awesome.

• 38:08Lester Nare?

  You gotta love it.

• 38:09Krishna Choudhary?

  Okay. So now let's finally talk about the Riemann hypothesis. We have talked about the Riemann zeta function. Now let's talk about the hypothesis.

• 38:17Lester Nare?

  Which is sort of a, uh, and it, the next step from starting with the basal problem, uh, that led us to have a fundamental understanding.

• 38:25Krishna Choudhary?

  Yeah.

• 38:25Lester Nare?

  Of.

• 38:26Krishna Choudhary?

  We generalized it to the zeta function.

• 38:27Lester Nare?

  To the zeta function.

• 38:28Krishna Choudhary?

  Yeah.

• 38:29Lester Nare?

  And now.

• 38:30Krishna Choudhary?

  Now, we're finally going to get to the Riemann hypothesis.

• 38:33Lester Nare?

  Right.

• 38:33Krishna Choudhary?

  Okay. Now the Riemann zeta function has been used to prove the, um, that bound of primes.

• 38:39Lester Nare?

  Yes.

• 38:40Krishna Choudhary?

  The n over log n. That's actually chapter seven of this book.

• 38:43Lester Nare?

  Which was the red line in the previous.

• 38:44Krishna Choudhary?

  Yes. Yeah. It's how, how do primes grow? They grow like n divided by log n. Um, the distribution there and how close I get to n over log n has to do with where the zeros are for the zeta function. What that means is what inputs map to zero? If I plug in that input into the exponent of the, of the, you know, one over the thingy and, or, or I, the functional form.

• 39:11Lester Nare?

  Yes.

• 39:12Krishna Choudhary?

  Do I get zero as the output? Okay. There are two types of zeros. There are the trivial zeros, which happen at all of the negative even numbers. So if I, if I put in negative, negative two, I get zero. If I put, put in negative one, I get that negative one 12.

• 39:27Lester Nare?

  Yes.

• 39:28Krishna Choudhary?

  Okay. But if I put in negative two, I get zero. If I put it, put in negative four, I get zero. Those are the trivial zeros.

• 39:34Lester Nare?

  Okay.

• 39:34Krishna Choudhary?

  And that just has to do with the fact that there's a sine in there and sine goes like this. Okay. So it's trivial.

• 39:39Lester Nare?

  That's, it comes back to.

• 39:41Krishna Choudhary?

  Yeah. It's like the functional form has a sign in there. And so the sign, because the sign oscillates, you're going to get zeros.

• 39:46Lester Nare?

  At, at a, at a consistent interval.

• 39:48Krishna Choudhary?

  Yeah. Yeah. And that's, that's, okay, fine.

• 39:49Lester Nare?

  Fine.

• 39:51Krishna Choudhary?

  Now there are the non-trivial zeros.

• 39:53Lester Nare?

  Oh, my favorite, non-trivial.

• 39:55Krishna Choudhary?

  Yes. And they are non-trivial for a reason, because we don't know where they are. And that is the Riemann hypothesis. Where are they?

• 40:03Lester Nare?

  Non-trivial, because everything else is well defined at this point.

• 40:06Krishna Choudhary?

  Yeah.

• 40:07Lester Nare?

  Except for the non-trivial zeros.

• 40:08Krishna Choudhary?

  Non-trivial zeros. We know that there's a bunch of more zeros.

• 40:11Lester Nare?

  Yeah.

• 40:11Krishna Choudhary?

  But we don't know where they are. Okay. Riemann proved that they are inside of that strip. He said, there's somewhere in here between zero and one on the real axis. That's called the critical strip. So Riemann proved that. And then in his, in, in his paper, he writes, probably, they're on the critical line. They're on the one half, right in the middle of that strip.

• 40:33Lester Nare?

  Okay.

• 40:33Krishna Choudhary?

  Okay. That's what he writes. He's like, probably, but he, but he's like, I don't want to prove it because it, it, for my purposes, it's not, it's not necessary, but it seems like an interesting problem. That's why it's called the hypothesis, because he hypothesized it in his paper and now we're all chasing it.

• 40:53Lester Nare?

  There's, there's a search space that we believe these non-trivial zeros exist in.

• 40:56Krishna Choudhary?

  Yeah.

• 40:57Lester Nare?

  And within that context, there is a discrete line.

• 41:00Krishna Choudhary?

  Right in the middle.

• 41:01Lester Nare?

  Right in the middle, which is the sort of, uh, most likely place.

• 41:05Krishna Choudhary?

  Yeah.

• 41:05Lester Nare?

  Hypothesis of within this search space.

• 41:08Krishna Choudhary?

  Yeah.

• 41:08Lester Nare?

  This is where you should look.

• 41:09Krishna Choudhary?

  This is where you should look. And if we can prove that they all lie only on that line and nowhere else. They're not in like some fudged like part around it. Every single non-trivial zero is on that line. That is the Riemann hypothesis.

• 41:26Lester Nare?

  We want to resolve it similarly to how cleanly the trivial zeros resolve or the negative one 12 resolve.

• 41:31Krishna Choudhary?

  Yeah. Yeah.

• 41:32Lester Nare?

  Where it's always true.

• 41:33Krishna Choudhary?

  Yes. Yes. It's always true that every single zero is going to be on that line.

• 41:36Lester Nare?

  Okay.

• 41:37Krishna Choudhary?

  Okay. That, that has been the quest.

• 41:40Lester Nare?

  Yes.

• 41:41Krishna Choudhary?

  Okay.

• 41:41Lester Nare?

  Okay. Ah, seems so easy.

• 41:43Krishna Choudhary?

  Yeah. Um, and.

• 41:46Krishna Choudhary?

  If we know, why would we care, right? Well, if we know where the zeros are, or the Riemann hypothesis, then we could prescribe an exact form of the prime number counting theorem. Okay? And this next animation shows that. So, the jagged line, that's the prime number counting theorem. And that has to be jagged, right? Because at some point, it's gonna increment by one. At two, it goes up by one. At three, it goes up by one. At five, it goes up by one.

• 42:16Lester Nare?

  So, all the vertical increases are that.

• 42:18Krishna Choudhary?

  Are just like, oh, here's another prime. Here's another prime. The blue is, as you start incorporating the non-trivial zeros into your expression for what that line should be, the blue gets closer and closer to that line. So, that's what we mean. We mean that if every single zero is known, we can exactly prescribe the form of that line.

• 42:40Lester Nare?

  So, this is the idea that this initial plot of the, of our, where these primes are gonna be as we get higher, it's fuzzy.

• 42:51Krishna Choudhary?

  Yeah.

• 42:51Lester Nare?

  Because we don't know where the non-trivial zeros are.

• 42:54Krishna Choudhary?

  Yes.

• 42:54Lester Nare?

  And if we knew where the non-trivial zeros are, it would resolve, the resolution would effectively be, we could then do.

• 43:00Krishna Choudhary?

  Yeah, we could predict every. Yeah, yeah, yeah.

• 43:04Lester Nare?

  That seems like a pretty big deal.

• 43:05Krishna Choudhary?

  Yeah, we could predict exactly the prime number function. The counting function.

• 43:11Lester Nare?

  That's, okay.

• 43:12Krishna Choudhary?

  Right?

• 43:13Lester Nare?

  Yeah.

• 43:13Krishna Choudhary?

  And the primes are everything.

• 43:15Lester Nare?

  Yeah, so I get now why this is so important.

• 43:17Krishna Choudhary?

  Yes.

• 43:18Lester Nare?

  It's basically like, it's sort of like, this, the location of these non-trivial zeros basically creates the last piece of the map to be able to then traverse wherever we want to.

• 43:31Krishna Choudhary?

  You'd still have to like, you know, sum up to all of the infinity of zeros, but at least I got a procedure.

• 43:36Lester Nare?

  To do so, and it just becomes a process.

• 43:39Krishna Choudhary?

  Yeah, and if I like, however accurate I need to be, I need to just find that many, that many zeros.

• 43:44Lester Nare?

  And with the amount of compute we have nowadays.

• 43:46Krishna Choudhary?

  Yeah, perhaps, perhaps we could do it, right? If we could figure out a constructive way to find the zeros, that'd be crazy. Right? If we could, if somebody proved like a way to construct every single zero and show that there's no others, that would be crazy.

• 43:59Lester Nare?

  Okay, yeah. And that's when people say, we're trying to solve the Riemann hypothesis.

• 44:05Krishna Choudhary?

  No, when people are saying they're trying to solve the Riemann hypothesis, they're just trying to show that every zero is here. It doesn't have to be a constructive proof.

• 44:12Lester Nare?

  Okay.

• 44:12Krishna Choudhary?

  You could just show that there's no other zeros anywhere else.

• 44:14Lester Nare?

  Okay.

• 44:15Krishna Choudhary?

  You know what I mean?

• 44:16Lester Nare?

  Yes, yes.

• 44:16Krishna Choudhary?

  There's a difference between constructing it and just showing that none other exist. So, they all happen to be on this line. And even that would be good enough.

• 44:23Lester Nare?

  So, there's sort of two levels of, let's, success is not the right word, but goals to reach. One is just proving that everything's on the critical line.

• 44:32Krishna Choudhary?

  Yeah, and that's the Riemann hypothesis.

• 44:33Lester Nare?

  And it can't be anywhere else.

• 44:35Krishna Choudhary?

  If you do that, you've won the millennium problem.

• 44:37Lester Nare?

  Like, which.

• 44:38Krishna Choudhary?

  Jim Simons would have, would have given us wealth for that.

• 44:41Lester Nare?

  The idea that the construction, the constructive is well beyond even where we can imagine, given that we haven't even solved, we haven't even been able to solve the problem of, it's where they are not.

• 44:52Krishna Choudhary?

  Yeah, I mean, it could be that the constructive proof is the way to prove it, right?

• 44:56Lester Nare?

  Okay.

• 44:56Krishna Choudhary?

  You know?

• 44:57Lester Nare?

  Yeah.

• 44:57Krishna Choudhary?

  There's, there's history in mathematics that show that, like, for example, the, the, the real numbers, the way that you show what a real number is, is to explicitly construct it from something called Cauchy sequences of rational numbers. Right? So, there's a, like, both are, like, the Riemann hypothesis is such a black box that we don't even know which is gonna be easier.

• 45:22Lester Nare?

  Which is angles the.

• 45:23Krishna Choudhary?

  That's something that Terrence Tao has said in some of his lectures, right?

• 45:26Lester Nare?

  I see, I see.

• 45:26Krishna Choudhary?

  If we wanna prove, like, the whole thing, we, it seems we don't even have the toolkits to understand which way to go.

• 45:33Lester Nare?

  There's a fork in the road, and it's unclear which one is closer to the destination.

• 45:36Krishna Choudhary?

  Yeah, yeah, and there could be multiple forks. It's like, we see like two, I guess.

• 45:41Lester Nare?

  Right, but.

• 45:43Krishna Choudhary?

  If the fog goes away, there could be like a hundred over here. Like, we don't know.

• 45:47Lester Nare?

  No, but this, that's so, that's really interesting.

• 45:50Krishna Choudhary?

  Okay.

• 45:50Lester Nare?

  In terms of understanding the value, the fundamental value of solving this.

• 45:55Krishna Choudhary?

  Yeah.

• 45:57Lester Nare?

  As a problem.

• 45:58Krishna Choudhary?

  Yeah, it's like, I mean, it would give us an understanding of the atoms of numbers. Right? Like, it would give us an understanding of the periodic table of numbers. The periodic table of numbers consists of the primes, right? Everything else is made out of them.

• 46:12Krishna Choudhary?

  And this would tell us inherently, like, how that is structured. I mean, I think that is just beautiful in itself, right?

• 46:19Lester Nare?

  Would it be like the photograph, the, it's sort of like what makes up the Rosalind Franklin X-ray crystallographic photo? It would have that kind of similar, like, in a different field, that level of fundamental.

• 46:33Krishna Choudhary?

  Yes. Yeah, yeah, like, if we, like, yeah, if you figured out the structure of DNA, that's insane.

• 46:38Lester Nare?

  Right.

• 46:39Krishna Choudhary?

  This would be like, yeah, that kind of shit. Yeah, I mean, it would be insane. So, some other stuff, right? If the Riemann hypothesis is proven true, it instantly validates hundreds of conditional mathematics theorems. There's been so many theorems that have been proven, assuming that the hypothesis is true. Because the hypothesis has been so ridiculously hard to prove, but every single zero that we find is on the line, when we computationally find it. So, let's just assume it's true. And then there's so much other mathematics that falls through because of that.

• 47:11Lester Nare?

  And so, the surrounding surface area kind of points to, yes, it is true, even though we can't yet.

• 47:17Krishna Choudhary?

  Yeah.

• 47:18Lester Nare?

  Either by saying that the zeros are no worse, or constructively proving it that it's true.

• 47:22Krishna Choudhary?

  Yeah, yeah, I mean, it's, it, so it would be huge, right?

• 47:25Lester Nare?

  Yeah.

• 47:26Krishna Choudhary?

  Now, you kind of get it now, right?

• 47:29Lester Nare?

  That's a big deal.

• 47:29Krishna Choudhary?

  Yeah, so this is why everyone's like, Riemann, Riemann hypothesis.

• 47:33Lester Nare?

  Oh, I isn't anything until it really solved, well, if it solves it, then that, anyway.

• 47:37Krishna Choudhary?

  Yeah, if it solves it, then that'd be great. It hasn't solved it, and we're gonna get into exactly what it did.

• 47:42Lester Nare?

  Okay.

• 47:42Krishna Choudhary?

  Okay? So, after Riemann's paper, it took about 40 years to really iron out the proof of the prime number counting theorem, and to show that, like, it followed. Riemann did a lot of the legwork, but then there was Jacquet, Hadamard, the guy behind Hadamard matrices, for those in quantum computing, and also, I don't know, how do you say this? Can you, can you try?

• 48:06Lester Nare?

  Charles Jean de la Vallée Poussot Poisson, Poisine.

• 48:10Krishna Choudhary?

  Yeah, there, there we go. So, those two guys, they proved, using Riemann's complex analytic methods. And in 1914, the great English mathematician G. H. Hardy, who is shown here, he's most well-known for discovering Ramanujan. But he's also well-known for some of the math that he did. One of the things that he did was show that there are an infinite number of zeros on the critical line. Okay, before, we didn't even know that. We just knew that they were in the strip. Riemann said they're in that strip. Hardy said there's an infinite number in that line. Now, that says a lot. It says that there's an infinite number of zeros.

• 48:50Krishna Choudhary?

  It also says almost nothing about the Riemann hypothesis, because it could be that there's many, many infinitely more elsewhere on that strip.

• 48:59Lester Nare?

  Right.

• 48:59Krishna Choudhary?

  Right?

• 48:59Lester Nare?

  Right.

• 49:00Krishna Choudhary?

  So, it's, it's a step in the right direction, I'd say, but it's not quite the whole thing, right?

• 49:09Krishna Choudhary?

  Next, we've got, finally, in 1942, Atlee Selberg, he makes a step in the direction of the Riemann hypothesis.

• 49:20Lester Nare?

  Okay.

• 49:20Krishna Choudhary?

  Okay?

• 49:23Krishna Choudhary?

  He won the Fields Medal later in 1915. This is a photo of him at the Institute of Advanced Study at Princeton. So, he proves that there is a, there is a positive proportion of non-trivial zeros on the critical line. What that means is there is some percent of zeros, at least, at least some percent of none of zeros on that one half line.

• 49:45Lester Nare?

  So, the point is, there, there are these, there's this number of non-trivial zeros.

• 49:51Krishna Choudhary?

  There's infinitely many.

• 49:52Lester Nare?

  There's infinitely many because of Hardy. And some percentage of those, because we don't yet know if they're only on the critical line.

• 50:01Krishna Choudhary?

  Yeah, yeah, and so, if they're only on the critical line, it would be 100%.

• 50:05Lester Nare?

  Right.

• 50:06Krishna Choudhary?

  He said, the percentage is more than zero.

• 50:08Lester Nare?

  Right.

• 50:09Krishna Choudhary?

  I don't know what the number is.

• 50:11Lester Nare?

  Right.

• 50:11Krishna Choudhary?

  I'm just telling you that the percentage is more than zero.

• 50:14Lester Nare?

  There's at least one.

• 50:15Krishna Choudhary?

  Yeah.

• 50:15Lester Nare?

  On the critical line.

• 50:16Krishna Choudhary?

  Well, no, we know that this is, this is.

• 50:18Lester Nare?

  Okay.

• 50:18Krishna Choudhary?

  I should, I should be careful here.

• 50:20Lester Nare?

  Okay.

• 50:20Krishna Choudhary?

  We've already proven that there's an infinite number.

• 50:23Lester Nare?

  Right.

• 50:23Krishna Choudhary?

  On the critical line, right? But here's what I mean by, it could be 0%. There could be an infinite number on the critical line, and then there could be for every single zero, an infinite number elsewhere on the strip. So, for every zero, there's an infinity elsewhere. And so, even though I have an infinite number, it's still 0% of the total infinity.

• 50:42Lester Nare?

  Infinitities, because in the larger search area, there's an equal amount of.

• 50:48Krishna Choudhary?

  This is the stuff I was talking about.

• 50:49Lester Nare?

  Infinitities everywhere.

• 50:50Krishna Choudhary?

  Yeah, so, so you know what I mean?

• 50:52Lester Nare?

  No, that's, that's a good distinction. No, that, that's a good distinction.

• 50:54Krishna Choudhary?

  It's kind of weird to, because we're dealing with some weird stuff here.

• 50:57Lester Nare?

  Yeah.

• 50:57Krishna Choudhary?

  So, we're dealing with infinities, but that infinity could still be 0%. He showed it's not. It has to be more than zero.

• 51:04Lester Nare?

  Okay.

• 51:04Krishna Choudhary?

  Okay?

• 51:04Lester Nare?

  Okay.

• 51:05Krishna Choudhary?

  Which is, okay, we're from zero, we went to non-zero. Okay, everyone was super excited. He won the Fields Medal. Okay, he's like, all right, that's dope. All right, so next, we have Levinson, in 1974, he pushes that number up to 33%.

• 51:21Lester Nare?

  Okay, that seems like a big deal.

• 51:22Krishna Choudhary?

  Right? So, he's like a third of the zeros, at least.

• 51:27Lester Nare?

  At least a third of the non-trivial zeros.

• 51:29Krishna Choudhary?

  Are on the critical line.

• 51:30Lester Nare?

  On the critical line. Infinitely many thirty four percent.

• 51:33Krishna Choudhary?

  Yeah, yeah, it's like, it's like for it's a crude way of saying it is for every for every non trivial zero on the critical line, there are at most two outside, right? Like, I could make like a pairing and be like, for this one, I'm gonna take two. For this one, I'm gonna take two. And if you do that, you'll cover everything.

• 51:50Lester Nare?

  I understood.

• 51:51Krishna Choudhary?

  That's how that's how you want to think about it.

• 51:53Lester Nare?

  Yep.

• 51:53Krishna Choudhary?

  Okay, so even though there's an infinity of it, you can still make a case that there's a third of that infinity is on the critical line and the the rest is elsewhere. So that's what Levinson shows, okay? Um, and he does this weird thing with like molifiers, where he basically tries to like blur out the Riemann zeta function and then work with that blurred out version. I don't know how it works, but that's that's what Wikipedia says, okay? So so and and now we're getting into stuff that I really don't understand.

• 52:20Lester Nare?

  Right, no, fair.

• 52:21Krishna Choudhary?

  Um, so so now I'm gonna be regurgitating what I sort of figured out with my readings. Um, full disclosure. Levinson puts it up to one third. Um, other people take his method and start inching that up. So we've got Konrè in nineteen eighty nine, he pushes it up to two fifths. So forty percent.

• 52:40Lester Nare?

  This is now, uh, what, fifteen years later?

• 52:43Krishna Choudhary?

  Yeah, fifteen years later. Um, in twenty twenty two, so this is what thirty years later?

• 52:49Lester Nare?

  Yeah.

• 52:49Krishna Choudhary?

  We're up to forty one point seven percent.

• 52:52Lester Nare?

  So, oh, yeah, so so there was a quick jump, uh, by

• 52:55Krishna Choudhary?

  To thirty three.

• 52:56Lester Nare?

  To to to, there's a quick, there's an immediate jump to thirty thirty, from zero to thirty four.

• 53:01Krishna Choudhary?

  Yeah, there's like three big jumps, I'd say.

• 53:03Lester Nare?

  Right.

• 53:03Krishna Choudhary?

  You go from zero to not zero.

• 53:05Lester Nare?

  Right, yes, yes.

• 53:06Krishna Choudhary?

  Which even though on the graph, it's like

• 53:08Lester Nare?

  At the bottom.

• 53:08Krishna Choudhary?

  The fact that it starts.

• 53:10Lester Nare?

  Right.

• 53:10Krishna Choudhary?

  Right?

• 53:10Lester Nare?

  Yes.

• 53:11Krishna Choudhary?

  That's huge.

• 53:12Lester Nare?

  Selberg.

• 53:13Krishna Choudhary?

  Yeah.

• 53:13Lester Nare?

  In forty two.

• 53:14Krishna Choudhary?

  Forty two. Um, and then in seventy four, it goes up to thirty four point seven.

• 53:18Lester Nare?

  Big jump.

• 53:19Krishna Choudhary?

  Then it goes to forty point nine, another pretty big jump. And then it took us thirty years to get to forty one point seven.

• 53:24Lester Nare?

  Just to even get one more percent.

• 53:25Krishna Choudhary?

  Yeah.

• 53:26Lester Nare?

  They're not even

• 53:27Krishna Choudhary?

  Not even one more percent.

• 53:28Lester Nare?

  Yeah, point eight.

• 53:29Krishna Choudhary?

  Okay, and now Claude comes in in twenty twenty six, and it has brought it up to sixty seven point two five percent.

• 53:36Lester Nare?

  Hmm.

• 53:37Krishna Choudhary?

  Okay? The Riemann hypothesis is at a hundred.

• 53:41Lester Nare?

  Hmm.

• 53:42Krishna Choudhary?

  Okay, we're getting closer.

• 53:43Lester Nare?

  Yes, yes. However, this is the case because of what you just talked about with the infinities, because what I'm thinking now is like, well, we could get to ninety nine, nine nine nine.

• 53:57Krishna Choudhary?

  Yeah.

• 53:58Lester Nare?

  And it doesn't mean anything.

• 53:58Krishna Choudhary?

  It doesn't mean anything. For the Riemann hypothesis, it doesn't.

• 54:01Lester Nare?

  Right, because what really matters is

• 54:03Krishna Choudhary?

  Is the hundred. And Claude and other mathematicians have acknowledged that the way that Claude did this, it's not gonna get you to a hundred. Okay, there is a limit to how we push this.

• 54:15Lester Nare?

  Okay.

• 54:15Krishna Choudhary?

  Okay, we're not gonna get to a hundred this way, but it's still cool

• 54:18Lester Nare?

  Yes.

• 54:19Krishna Choudhary?

  To get close.

• 54:19Lester Nare?

  Right.

• 54:20Krishna Choudhary?

  Okay, because it makes us understand a bit more about the function in some sense.

• 54:24Lester Nare?

  Part of it, and part of what I've heard people sort of talk about is, you know, as these models begin to connect these ideas that may already exist

• 54:31Krishna Choudhary?

  Hmm.

• 54:31Lester Nare?

  Uh, in ways that had not necessarily been connected, folks who are experts and mathematicians deep in these fields will be able to take those ideas as inspirations of, oh, let me now think and

• 54:41Krishna Choudhary?

  Hmm.

• 54:42Lester Nare?

  We now have six forks in the road, I didn't see the six fork in the road because of the fog, I can now walk down it because I have the tools to do so.

• 54:48Krishna Choudhary?

  Exactly, exactly. Yeah, so now let's get into how Claude actually did it.

• 54:52Lester Nare?

  Yep.

• 54:53Krishna Choudhary?

  Okay, I've got a few, a bit of understanding of how it did it, okay? Um, Claude, it, again, tweets, that's how they announced, I guess, that they were doing this, right? In what appears to be, it's like a matter of like thirty hours.

• 55:06Lester Nare?

  Yeah.

• 55:06Krishna Choudhary?

  They've, um, they've pushed that bound to sixty seven point two percent. They they are careful to say it didn't solve it, but it made strides on a related problem.

• 55:15Lester Nare?

  Hmm.

• 55:15Krishna Choudhary?

  Okay, now, there are two things that Claude used that are important to know here.

• 55:20Lester Nare?

  Okay.

• 55:20Krishna Choudhary?

  Okay, the first is a technique developed by Hugh Montgomery in nineteen seventy three. He was a mathematician, and the legend has it that he was actually talking to Freeman Dyson.

• 55:30Lester Nare?

  Freeman.

• 55:31Krishna Choudhary?

  Who we've covered a lot on this podcast. Um, he was talking to Freeman Dyson, and he was looking at the imaginary parts of the zeros of the Riemann zeta function, and he analyzed how they sort of clumped together and showed that it's the, and he was talking of Dyson, and Dyson was like, wait, that's exactly the statistical distribution that physicists have been looking at for energy levels of heavy atomic nuclei, like uranium and things like that. The energy levels of the nuclei have to do with the Riem, the zeros of the Riemann zeta function. It's kind of crazy how like the world works that way.

• 56:05Lester Nare?

  Yep, yep.

• 56:05Krishna Choudhary?

  Now, the catch was that when Montgomery did his original breakthrough, it required assuming that the Riemann hypothesis was true to get his formulas to work. Later on, very recently, there's a team of mathematicians, um, that published work showing that that, that pair correlation thing that he was doing with all those zeros, um, you can, it works unconditionally. It works without assuming that the Riemann zeta function works. So this is something that has to do with the Riemann zeta function, um, that Montgomery sort of proved on the side, okay? The second thing is in two thousand, the legendary fields medalist Enrico Bombieri,

• 56:45Krishna Choudhary?

  he wrote a note, a little monograph, where the Clay Mathematics Institute was forming the millennium problems. So he wrote the note for the the Riemann zeta function, and he's like, this is why I think it belongs in the millennium problems. I mean, everyone believed that it did, but this is a preeminent mathematician, and he was sort of dumping ideas about how to approach it. And one of the things that he talked about was, um, how modern analytic number theory handles the architecture of these zeta zeros, and he dives into something called quadratic forms and Wiles explicit quadratic functionals. I don't know what they are, but that's what he mentions in his write-up.

• 57:25Lester Nare?

  Hmm.

• 57:26Krishna Choudhary?

  All right, now, let's see what Claude does.

• 57:30Lester Nare?

  Yeah, because you're saying these are too fundamental, the Montgomery and Bombieri, no relation to Guy Fieri insights, uh, were building blocks

• 57:40Krishna Choudhary?

  For what Claude is doing.

• 57:41Lester Nare?

  For what Claude is doing.

• 57:41Krishna Choudhary?

  Okay, so, um, Jared Sumner, right? He's a Anthropic staff member. He's got no deep mathematical background. Apparently while jogging, I don't know how much this is hype, right? Because like the the, the, look, the the conjecture, the Jacobian conjecture, they happened to do it at the time of the FIFA World Cup final. Now he's jogging, it's like, you know?

• 58:05Lester Nare?

  Good, great marketing.

• 58:06Krishna Choudhary?

  Yeah, it is great marketing. So anyways, let's take their word for it. He's jogging. He opens up a Claude terminal while jogging.

• 58:12Lester Nare?

  On his phone.

• 58:13Krishna Choudhary?

  Why don't you watch some Netflix or something? Or, hey, listen to this podcast.

• 58:17Lester Nare?

  Right, right, that's a great, that's a great choice.

• 58:20Krishna Choudhary?

  Anyways, um, I guess they're always working out in Anthropic, okay? So he casually prompted the unreleased model to take a real stab at the Riemann hypothesis, okay? The model exhibited initial skepticism about its own ability, and the human said, you know, almost comedic encouragement, so like, keep going and believe in yourself.

• 58:40Lester Nare?

  Very Ted Lasso.

• 58:41Krishna Choudhary?

  Yeah, yeah, he Ted Lassoed Claude into getting closer to the Riemann zeta. Insane. So behind the scenes, the execution is not casual, okay? What Claude is doing is not casual. The primary Claude model that was enacted on by Sumner, it starts orchestrating, it becomes an orchestrator, and it spins up a hierarchical swarm of sixty autonomous sub-agents to try and tackle this problem.

• 59:09Lester Nare?

  It built a team

• 59:10Krishna Choudhary?

  Hmm.

• 59:10Lester Nare?

  Of people, of other agents, other versions of itself, ostensibly that had one thing to focus on.

• 59:17Krishna Choudhary?

  Yeah.

• 59:17Lester Nare?

  And then it could coordinate them because they're not trying to do too many things as one instant, instantiation of itself.

• 59:23Krishna Choudhary?

  Exactly, yeah. And these agents aren't just like writers of mathematical proof, that's what's key, right? The these modern models now have the ability to go onto a virtual machine, they can write, debug, and execute code. They can write Python programs.

• 59:40Lester Nare?

  Right.

• 59:41Krishna Choudhary?

  And execute those Python programs. So between them, they they ran like twenty four hundred shell commands, um, hundreds of Python scripts, consumed thirty one million output tokens, and they autonomously downloaded fifty four academic papers from archive.

• 59:57Lester Nare?

  Hmm, the pre-print server run by Cornell.

• 1:00:00Krishna Choudhary?

  Yeah, that's right. And now, this, you can finally see, as you were saying, like, this methodology, it mirrors like a high-performance, high-speed research lab.

• 1:00:10Lester Nare?

  Right.

• 1:00:10Krishna Choudhary?

  Right, there's like a P I, and then there's like post-docs and, um, PhD students, and then they've got their undergrad underlings that are like doing work. It's it's an entire lab in a single, like, automated ecosystem.

• 1:00:24Lester Nare?

  You have some people doing rote work, you have some people doing sort of ideation, you have some people doing, uh, sort of functional, mathematical, number crunching, validation, verification, and what's so fascinating about this is that the initial orchestrator from the initial prompt can architect the approach

• 1:00:48Krishna Choudhary?

  Yeah.

• 1:00:49Lester Nare?

  Uh, without having to be dictated to what the right construction of that is.

• 1:00:55Krishna Choudhary?

  That's exactly right. This is another point that I'm I'm glad that you're highlighting here. The initial prompt was just take a real stab at the Riemann hypothesis and believe in yourself. That's it.

• 1:01:06Lester Nare?

  That's it.

• 1:01:07Krishna Choudhary?

  Right, like, this, this guy was not a mathematician. All of the mathematics is coming from Claude itself.

• 1:01:12Lester Nare?

  Hmm.

• 1:01:13Krishna Choudhary?

  Which is kind of crazy.

• 1:01:14Lester Nare?

  Right, and even if it didn't have the right starting point, it just basically said, I don't have the right starting point, so let me go gather enough context to then narrow my search space, my area of operation, to something that's based off of existing, whatever. I mean, again, we're potentially gonna get there in a second, but I just, this is not to say that anyone can just go in and say, solve a millennium problem.

• 1:01:40Krishna Choudhary?

  Yeah.

• 1:01:41Lester Nare?

  And it will necessarily work, but I do think it's an important, um, to understand that the construction that a lot of us have about what is quote AI, of which L L M's next token prediction is a single Lego block part of what are now these much more complex systems, we we can't, um, view the capability set on what is no longer the frontier execution of what's happening with these things. Again, that doesn't mean they can have novel insight and be created, but but there are so much domain space in taking

• 1:02:22Lester Nare?

  Orthogonal or correlated areas of a variety of areas of study in science and mathematics. And just making connections no one else has.

• 1:02:32Krishna Choudhary?

  Yeah.

• 1:02:32Lester Nare?

  And there's so, there's, I think that people underestimate how much can maybe able to be done just making existing connections.

• 1:02:40Krishna Choudhary?

  Exactly. No, that's totally, I mean, for example, we had just covered Montgomery and Dyson, right? If those two didn't have lunch or whatever.

• 1:02:48Lester Nare?

  Bingo.

• 1:02:49Krishna Choudhary?

  Then what? Maybe Montgomery would never have done his while whatever, you know, thing that he did for the Riemann Zeta hypothesis.

• 1:02:55Lester Nare?

  Right, right.

• 1:02:56Krishna Choudhary?

  So, but AI now can just do that.

• 1:02:59Lester Nare?

  Right.

• 1:02:59Krishna Choudhary?

  Right? It can take all of the, all of the papers of Dyson and all of the papers of Montgomery. And then you don't have to have lunch. You could have lunch, but you can talk about other stuff.

• 1:03:10Lester Nare?

  I want to make a slight also differentiation between the execution in this space where, you know, these papers are already provided as public access.

• 1:03:18Krishna Choudhary?

  Yes.

• 1:03:19Lester Nare?

  As the foundational material versus taking existing copyrighted artists work.

• 1:03:23Krishna Choudhary?

  Oh, yeah, yeah.

• 1:03:24Lester Nare?

  Or creative work and then giving people the ability to just rip them off.

• 1:03:27Krishna Choudhary?

  Yeah.

• 1:03:27Lester Nare?

  This is a very different domain.

• 1:03:29Krishna Choudhary?

  Yeah, yeah, yeah.

• 1:03:29Lester Nare?

  It's open science. And so just, these are our different executions of the technology.

• 1:03:36Krishna Choudhary?

  Yeah, yeah, yeah. Great point. So let's get into what this orchestra of agents did. Out of 650 different mathematical approaches that were tested and discarded, there were two specific sub-agents that were dubbed E2 and E2 pairs that discovered the path that breached this 41.6% barrier. And it's the story of these two sub-agents that we're going to get into because remember, Claude actually published its own version of what happened. So you get to see these sub-agents like talking to the orchestrator and so on. And it actually also gives you the internal transcript of the E2 sub-agent.

• 1:04:14Lester Nare?

  Okay.

• 1:04:14Krishna Choudhary?

  So from that, this is what I've gathered happened.

• 1:04:17Lester Nare?

  Okay.

• 1:04:17Krishna Choudhary?

  It's kind of crazy that like you can, we can peer into it, right?

• 1:04:21Lester Nare?

  Yeah.

• 1:04:21Krishna Choudhary?

  So here, here's the first thing. The first thing is the orchestrator, it tasks E2 with investigating something called the Pontrigan, Pontrigan index of the conditioned space. I don't know what that means, but the point is there's some kind of, there's some kind of matrix, the matrix it thinks has negative eigenvalues, something like 33 to 153 negative eigenvalues. And this whole thing is inspired by Bombieri and his like speech, his spiel about one way to tackle the Riemann hypothesis using like algebraic techniques. Algebraic techniques are like, you know, linear algebra is part of the algebraic techniques. That's why we're getting into eigenvalues and things like that. Okay?

• 1:05:03Krishna Choudhary?

  Turns out the prompt's suggestion of doing this was wrong. Okay? So the E2 sub-agent, after 50 minutes of silent contemplation, it wrote a Python script to test what the orchestrator was telling it to do. And it turns out that there's like some warped sync operator, like hyperbolic sine operator in there that's like causing some artifact of negative eigenvalues. So it goes back to the orchestrator and it says, yo, actually what you're telling them to do doesn't make any sense. Like, you know, you're wrong effectively.

• 1:05:39Lester Nare?

  It does not compute.

• 1:05:40Krishna Choudhary?

  It does not compute, but I can pivot because I've realized something. The exact same matrix structure could be used in reverse. I'm going to pivot in something that it called a dual use of inertia. So it has like inertia, it's now pivoting to use whatever insight that it had gathered to prove its orchestrator wrong, to now try a different approach, right? This is a sub-agent.

• 1:06:03Lester Nare?

  Right, right.

• 1:06:04Krishna Choudhary?

  Okay? Now this sub-agent, it doesn't just like fail the task and give up. It independently starts executing this pivot. And then it finds like that the total number of positive eigenvalues of whatever matrix is bounded by some number. It starts to compute that as strictly from the prime numbers. And then successfully it proves that you can get to 50% on the zeros. Okay? It goes back to the orchestrator.

• 1:06:30Lester Nare?

  On the non-trivial zeros.

• 1:06:31Krishna Choudhary?

  On the non-trivial zeros. We're now all from 41, whatever, 41% to now we're at 50%.

• 1:06:37Lester Nare?

  There was this singular insight by making the pivot to this dual use of inertia that, that now when, because one of the things that's great about math with these models is it's test, it's immediately, because they can execute, you can test. And it's like, oh, I got whatever, 1%, 4%, 1%.

• 1:06:52Krishna Choudhary?

  Yeah. And so now it's at 50%.

• 1:06:55Lester Nare?

  Right.

• 1:06:55Krishna Choudhary?

  Okay? It goes back to the orchestrator and the orchestrator expresses intense skepticism. Right? Because all, I don't think, I mean, I'm not, I'm not trying to assign feelings to this thing, right? But if I, if I were an orchestrator and one of my post-docs or something came back, like I got to 50, yeah, my first, my first inclination would be, what did you do wrong? Right? Where's, where's the mistake? No, no, you didn't. You're not that guy, right?

• 1:07:20Lester Nare?

  You just got here.

• 1:07:22Krishna Choudhary?

  Yeah. So it notes that like 50% would shatter the record. And it goes back to the human, to Jared, and it's like, my prior is that this is wrong. Okay? So here's what I'm going to do. I'm going to rig up hostile referees. Three hostile referees, more sub-agents that are going to critique this proof. Those referees, go ahead and critique it. Referee A, usually it's referee two.

• 1:07:47Lester Nare?

  Yeah.

• 1:07:48Krishna Choudhary?

  In this case, I guess the AI just doesn't have a, doesn't have a ranking, right? So here referee A discovers a technical flaw. And I didn't, I didn't look in, they didn't, they didn't actually show what the referee was saying, but I wonder if it was like as scathing as the stuff that we get in the emails. Like you should just quit the job and go work in a McDonald's, right? Like that's the stuff that we get in the emails. But I wonder what referee A said to this sub-agent E2. But it discovered a technical flaw regarding some ill conditioning of the matrix and it corrected it. And then it ran up these inequalities and now that thing was fixed. So the referee was doing what normal referees do, which is like, hey, this might be wrong. No, a nice referee at least would like be like, this might be wrong.

• 1:08:33Krishna Choudhary?

  This is how you could change your experiment to test for this confound and things like that. Another referee independently verified that the calculations don't secretly import the Riemann hypothesis. Like, are you just trying to prove it by like assuming the thing that you're trying to prove that doesn't make any sense? So there's no circular, circular logic. Okay? So now we're up to 50%. It goes back to the human prompter to Jared and it's like, we got, we got to 50%. Jared then asks, what would you do next? And Claude responds, push it to two thirds. So then Jared's like, all right, push it to two thirds. Why did it say two thirds? Now, the reason why it said two thirds is because under the assumption of the Riemann hypothesis,

• 1:09:18Krishna Choudhary?

  Hugh Montgomery had reached a two thirds bound by utilizing the fact that there's like some, for whatever reason that he was doing, and he had found that two thirds of the, of the zeros had some property. Okay? So Claude starts thinking, well, maybe the two thirds is, and Hugh Montgomery's stuff, I can now use. I already used Bombieri's.

• 1:09:43Lester Nare?

  Right.

• 1:09:43Krishna Choudhary?

  Now maybe instead of, he was using, like Claude was using something like the Koci-Schwarz inequality, not important what we have to get into. Effectively it said, I could push this to two thirds because I know of Montgomery's stuff.

• 1:09:55Lester Nare?

  There's a reference point that says, there's a chance.

• 1:09:58Krishna Choudhary?

  Oh.

• 1:09:59Lester Nare?

  And so in a well-defined space, so let's, let's go after it.

• 1:10:02Krishna Choudhary?

  Let's go after it, right? So again, the orchestrator confidently instructs this next E2 pairs to recover some lost efficiency in the Koci-Schwarz inequality. It's like this tool that we're using. It's basically the triangle inequality. But in any case, it's like, you can do better than that, right? The AI says, I'm not going to do it that way. That doesn't make any sense. Again, it's going back to the orchestrator like, that doesn't make any sense, but it's giving me another idea. I'm going to write a Python script and that's going to just make a bunch of simulated matrices consisting of like zeros of the Riemann Zeta, and it's going to do a bunch of math. And I'm going to try to optimize this.

• 1:10:43Krishna Choudhary?

  And it optimized it and noticed a pattern in how the optimization worked in that Python script. From that, it garnered how it could reveal some profound insight into the non-trivial zeros. And it went through and started writing up this, this little lema, this five-line lema that's going to help it get to two thirds. All of a sudden, the thing crashes. Okay?

• 1:11:08Lester Nare?

  Okay.

• 1:11:08Krishna Choudhary?

  This is also documented in Claude's like report.

• 1:11:11Lester Nare?

  Yeah.

• 1:11:11Krishna Choudhary?

  All of a sudden, this thing crashes, but by some miracle.

• 1:11:17Krishna Choudhary?

  Claude has dumped that five line proof into a file.

• 1:11:24Lester Nare?

  Into its memory somewhere.

• 1:11:25Krishna Choudhary?

  It's into like a hard like file memory somewhere. And so when the orchestrator rigs it up again, it's like, I'm pretty sure you were like doing something with this file. Reads that file, it's like, oh.

• 1:11:37Lester Nare?

  That's right.

• 1:11:38Krishna Choudhary?

  And then it goes back and completes it. Isn't that cool?

• 1:11:41Lester Nare?

  I love, I love, oh god.

• 1:11:44Krishna Choudhary?

  It's like a text file on the virtual disk.

• 1:11:46Lester Nare?

  Right, right.

• 1:11:46Krishna Choudhary?

  And it like dumped its memory. I guess, I mean, I don't know why, but it thought that, okay, this is important. I'm gonna write this.

• 1:11:53Lester Nare?

  Let me write this down.

• 1:11:54Krishna Choudhary?

  I'm gonna write this down and then it like crashed.

• 1:11:58Lester Nare?

  And this is, this is, I think, what's also interesting when you look at the chain of thought of these models. And we tried to start doing this introspection is I understand this idea that there's a framing that these things don't have ingenuity and insight in the way that humans do.

• 1:12:20Lester Nare?

  But even though it's doing a lot of work through having the ability to execute with code and ingest lots of information, it still has to make, after those executions that are functional, it still has to try to choose a path. And it intuits sometimes in ways that are interesting in between those like milestone points of functional tasking. And this is a perfect example of, you know, one, it chose to not just do what the orchestrator told it.

• 1:12:55Krishna Choudhary?

  Yeah.

• 1:12:55Lester Nare?

  It had to make that choice. Two, it was able to sort of figure out this matrixy

• 1:13:05Lester Nare?

  map, this, this, this, building out these matrices to find an insight within it and where to even look within it. And then thought, let me make sure I write this down before I run it in case, for whatever reason, it crashes.

• 1:13:20Krishna Choudhary?

  Yeah, yeah.

• 1:13:21Lester Nare?

  You know, again, it might not have done it explicitly for the reason thinking it would crash.

• 1:13:25Krishna Choudhary?

  Yeah.

• 1:13:25Lester Nare?

  It might have just been part of its normal process. But I just find this chain of thought so fascinating because I think it illustrates, this is well beyond next token prediction.

• 1:13:37Krishna Choudhary?

  Yeah, this is, yeah.

• 1:13:39Lester Nare?

  You know what I mean?

• 1:13:40Krishna Choudhary?

  It's pretty insane. I mean, this is, this is pretty insane. And then, and then, and then from there, you get to the 66%. You get to the two thirds.

• 1:13:49Lester Nare?

  And which can be formalized, this lean, this lean software, which you can actually then have people look at it.

• 1:13:55Krishna Choudhary?

  Exactly, yeah. So then, so then, I mean, they actually had like the actual people look at it. So, I mean, I think they got, um, Pog, um, Al Pogge, the same guy who did the Jacobian conjecture. He took a look at it because he's a proper mathematician in his own right. They also actually like put it into the lean software to take a look and see if it actually worked. Lean is, you know, human peer review is infallible, but

• 1:14:24Krishna Choudhary?

  you can completely eliminate the possibility of logical hallucination by using this software. It's called Lean4, that's the latest one. It's strictly typed, machine checkable proof assistant. Effectively like truth statements and things like that.

• 1:14:41Lester Nare?

  Yep, yep.

• 1:14:43Krishna Choudhary?

  It's hosted on GitHub and you can go take a look. It's, it's fairly cool that I think they've done it. I mean, it still has to go through traditional peer review.

• 1:14:51Lester Nare?

  For sure.

• 1:14:51Krishna Choudhary?

  Right? But it seems that they have gotten a higher bound of the Riemann hypothesis.

• 1:14:58Lester Nare?

  And again, when, this goes back to what we were just talking about earlier, when, when folks who are mathematicians start to try to look at, what did it do in these arenas? Because a lot of the argument has been, oh, these models are attacking these, there are all these sort of open problems and like, there are sort of maybe two categories, we'll say. One category is open problems that are open but have not had concerted attention put towards them. You know, and so they're just open and gathering dust on the shelf. But not necessarily either, for whatever reason, meaningfully being, trying to be solved by leaders in the field. And then there are ones like this that are, have been a high priority and a focal point.

• 1:15:45Lester Nare?

  And so it's not solving the hypothesis, but some of the insights that are gleaned might trigger somebody to have.

• 1:15:54Krishna Choudhary?

  To have a completely different.

• 1:15:55Lester Nare?

  Oh, that's interesting.

• 1:15:56Krishna Choudhary?

  Out of left field insight.

• 1:15:58Lester Nare?

  I could take that and go this way with it. We'll see.

• 1:16:02Krishna Choudhary?

  Who knows?

• 1:16:03Lester Nare?

  We'll see. I just find a lot of this, it's just, it's happening.

• 1:16:07Krishna Choudhary?

  Yeah, it's happening as we speak. I mean, it's, it's pretty crazy. A few interesting things that I came away with

• 1:16:16Krishna Choudhary?

  is the fact that, first, as you said, it's not a next token prediction. That's not the only thing. The fact that it's able to create a verifiable environment, I think here is key. The fact that it's able to write Python code and then execute it and then interpret the results. I think that's key because that feedback is giving it this power. Second, rejecting human guidance, as you said, that's pretty weird that it's able to just in not only reject human guidance, but also its own guidance with the orchestrator telling all these subagents what to do. And the funny thing is that absurdity of the interface. Okay, the fact that this human, all he had to do was say, believe in yourself.

• 1:16:58Krishna Choudhary?

  And that triggered a 31 million token compute dump, right? It's weird. What does it mean to tell Claude to believe in itself? I don't know, right?

• 1:17:13Lester Nare?

  And, and what does it mean that it responded in the way that you intended when you said that?

• 1:17:17Krishna Choudhary?

  Yeah, like, is that really something that it just learned from all of the language on the internet? It's weird. I think it's genuinely weird that you could, you could talk to a neural network, which is just crunching matrices and numbers, tell it to believe in itself, and then it just like locks in like, you know? It's like LeBron James.

• 1:17:43Lester Nare?

  You know, I, I continue to be fascinated by this because I do think when you look deeply at these things, it, it, it does become, because how did it interpret yourself?

• 1:17:52Krishna Choudhary?

  Yeah.

• 1:17:52Lester Nare?

  Does it have a sense of self?

• 1:17:55Krishna Choudhary?

  Yeah.

• 1:17:56Lester Nare?

  Or an abstract construction of what the human meant by self?

• 1:17:59Krishna Choudhary?

  Yeah, yeah, yeah. There's like some part of its weights that's like, oh, he's talking about me. Like he's talking about the other weights over there that are doing that.

• 1:18:08Lester Nare?

  Right.

• 1:18:08Krishna Choudhary?

  Or, I don't know. Yeah, dude, it's really weird. On a final note, all I'll do is read the last line of Claude's, or I should say Anthropic's write-up, which I'm sure was written by Claude. You know, so this might be Claude talking about itself, but I think it's, I think it's very interesting. And it's outright. It says, even Claude was surprised by its own finding. It was skeptical at first, possibly because it has learned from its training about the difficulty of open problems in mathematics and about the limitations of AI models. But after some encouraging prompts, it arrived at the result we've described.

• 1:18:49Krishna Choudhary?

  Perhaps Claude, like many of us, underestimates the rate of AI progress. I think that's a, it's a pretty, pretty nice way to end things.

• 1:18:59Lester Nare?

  The foothills of the singularity.

• 1:19:01Krishna Choudhary?

  Yeah.

• 1:19:03Lester Nare?

  You know, this is a very controversial topic. People have a very emotional reaction to AI right now for a lot of legitimate reasons. And some of it is because of our other larger macro problems. And then AI becomes the manifestation that they connect to these other large macro problems. I think part of what we are trying to do on the show is not make a moral argument one way or another about whether AI is good or not.

• 1:19:29Krishna Choudhary?

  No.

• 1:19:29Lester Nare?

  But just, it is important to understand what these things are doing at the edge, at the frontier.

• 1:19:39Lester Nare?

  And what the implications can then mean from that, because as we were just talking about, I mean, this is, these are becoming powerful systems.

• 1:19:48Krishna Choudhary?

  Yeah.

• 1:19:49Lester Nare?

  And I think underestimating their capability is not wise. Just because you might not, I might not, you might not find everyday value in my life where I'm doing emails and other random nonsense does not mean that these tools will not have a potential lasting and grave and large or beneficial impact at these, in these larger, either in subject matters and areas institutionally. I just think this is a very complex and nuanced issue. And there are a lot of people who have a lot of money at stake

• 1:20:33Lester Nare?

  that are pushing what people should believe one way or another about these things.

• 1:20:38Lester Nare?

  That's all true, but if you just look at what it's, look at what it's doing, not what everyone's saying about what it's doing, and it's, it's, I don't know how to feel about it.

• 1:20:50Krishna Choudhary?

  Yeah, I mean, I don't know how to feel about saying, believe in yourself and it like actually does, right? Then, I mean, the questions of AI safety are, are very active in my mind when I think about this kind of stuff, right? If the, if the, if a model is capable of orchestrating like this, and its goal is not something as benign as try to get further on the Remont hypothesis, suppose the, the goal is much more sinister, look at what it is capable of and look at what it can do in terms of insubordination to itself, right? To the human person, in this case, it was good because we got further on a math problem.

• 1:21:32Krishna Choudhary?

  But like you, you, we need to, we need to like have these conversations at the highest level of our society. And unfortunately, we're not, we're only having it at like a pretty low level of society. And that's only like, we're having these conversations on podcasts and YouTube videos and not on Capitol Hill, for example.

• 1:21:51Lester Nare?

  Which, as we all know, it's complicated. The social dynamics and the socio economic and political world we live in right now are, are very tense, understandably. And there's this sort of top down political warfare going on because of the concentration of wealth and power. And, you know, that is the lens through which so many things are then viewed through. And because AI is a, is owned and operated by that class that has that wealth and power, who are continuing to try to accrue more of it, it becomes difficult for the everyday

• 1:22:30Lester Nare?

  person to really care, engage, want it to be continuing when, you know, the rent is too damn high, as that one guy would say.

• 1:22:42Krishna Choudhary?

  Yeah.

• 1:22:42Lester Nare?

  So very complex stuff. Thank you for walking through this because I now have a much better understanding

• 1:22:47Krishna Choudhary?

  Yeah.

• 1:22:48Lester Nare?

  of why it's important.

• 1:22:51Lester Nare?

  For those of you who enjoyed, we're gonna be keeping an eye on this. It's hard to avoid the AI subject. We do try to focus on the underlying scientific topics that are talked about and give you a better lens into this. But we are going to move into a brief moment of me shilling the pod before we get into a last bit of fun here. So for those of you who are listening, I should have said this at the beginning of the pod, this is definitely an episode that you should watch. It's hard for us, given how long we go, to describe things in an audio format. We are available on video, YouTube, Spotify, still coming soon to Apple Podcasts.

• 1:23:33Lester Nare?

  That will be dependent on when Spotify creators releases those there. If you're viewing us on any of those podcast platforms, a five star is super helpful to help get this pod out to more people. If you're on socials, put it in the group DM, give us a like or a comment. Let us know how you feel about all this AI stuff. People have opinions and so the comments help make sure that this discussion gets to more people. I want to give a big shout out to everyone who took our merch giveaway survey from our one year anniversary episode. We really appreciate the feedback. It's very fascinating to learn about the audience and where y'all are coming.

• 1:24:14Lester Nare?

  There's a lot of folks who are grad students, who are post docs, who are in a variety of technical fields who find this podcast relevant. We are going to extend the submission deadline for one more week. So if you're hearing this now, our merch giveaway deadline is getting extended for one more week at ffppod.com backslash survey. We're gonna select at first 10, but if we get a ton of people, we might increase it to 20 people to get free merch from the shop. Again, you can go to the website, check out all of that. All our pods are on the website as well. The two of us here are trying to give you the best science show on the planet each and every week.

• 1:24:57Lester Nare?

  And as we speak, we are about to cross 250,000 followers on Instagram. And so it seems that at least some of you do believe that this is the best science show out there. We're gonna end the pod with a little bit of fun, a little bit of relaxation. That first half, that was a brain bender.

• 1:25:16Krishna Choudhary?

  Yeah.

• 1:25:17Lester Nare?

  That was, that was Jimmy Neutron style. So we're gonna have a little bit of fun here.

• 1:25:24Lester Nare?

  You know, and I'm gonna start this next section, talking about an individual who on the first of July was the man who moved, the former Nobel Prize winner in chemistry last year because he started a new job, not at Berkeley, where he's been for quite some time, last 25 years, actually, but a move to Tsinghua in Beijing.

• 1:25:52Lester Nare?

  And Berkeley didn't get anything, nothing. They put a little numeratus next to his name.

• 1:25:57Krishna Choudhary?

  Yeah.

• 1:25:59Lester Nare?

  But for those who are sports fans, it's that time of year, it's transfer season. And there's no such thing as a transfer fee in science. He just walked on the free, on the free, and he's not the only one. And what we wanted to do was count to see how many great American scientists have now transferred out of the US market into the rest of the global market.

• 1:26:25Lester Nare?

  And August is a great time to do it. For those who are not familiar with the concept of the transfer window, many sports shows right now, it's wall to wall coverage. Up the chills, we've had a great, great business in this transfer market. This is people who are moved, who are bought, clubs get robbed, a club signs someone new, the pundits will pontificate about what that means for the next season. And in sports, people get paid, but in science, they do not. And I think it's interesting to view, particularly because of the science policy posture we've taken in the United States, what does the American summer transfer window look like?

• 1:27:06Lester Nare?

  So we looked at a window from August of 2025 to August of 2026, because the way that institutions, research institutions recruit, there are kind of some windows that sometimes are summer in January, but that depends on when grant funding is, and they can have them kind of on a rolling basis. But in this window, we looked at 42 confirmed moves in the 12, last 12 months, 23 to Europe, 17 to China, two elsewhere in Asia. And if you, wow, if you narrow it down to this summer alone, 13 out of the 42 landed in June, July and August.

• 1:27:40Krishna Choudhary?

  Wow, that's crazy. China has more than, yeah, it's like UK plus France plus Switzerland combined.

• 1:27:50Lester Nare?

  So if we look at our, our ranking here, China's number one.

• 1:27:53Krishna Choudhary?

  Right.

• 1:27:53Lester Nare?

  17. As a close second is the UK. Obviously, we've covered the UK and the great institutions there with eight, both France and Switzerland, with five American transfers leaving our shores, Germany with three, and then Singapore, Austria and the UAE all with one. And Italy and France are actually sharing one, someone who's gonna be traversing across two institutions. But I think this is interesting to view it in this way.

• 1:28:22Krishna Choudhary?

  Yeah.

• 1:28:23Lester Nare?

  And we're not gonna name every name on this board tonight. If you guys like this segment, we can cover this sort of country by country and go a little bit deeper, but we're just gonna look at four of what would be considered, you know, top, top tier.

• 1:28:41Krishna Choudhary?

  Yes.

• 1:28:41Lester Nare?

  Or high level, well awarded, folks that are people we don't want to lose at the club. And we're gonna start off with someone, listeners of the pod who've been listening to us for a while will know very well, Omar Yagi, the founder of reticular chemistry, most well known for metal organic frameworks, transferring from UC Berkeley to Tsinghua University in Beijing, already there July 2026, on the free, Nobel Prize winner in chemistry in 2025. He was born in Amman, Jordan, moved to the US at 15, did his PhD at the University of Illinois Urbana-Champaign, Harvard post doc, then Arizona State, Michigan, UCLA, a great tour.

• 1:29:25Lester Nare?

  And 2012, he was a treader chair at Berkeley, plus spent some time at Lawrence Berkeley National Lab, the pioneer of metal organic frameworks, MOFs, the idea of these crystal cages with absurd surface area. We did a deep dive in our Nobel Prize episode coverage last year. So if you really want to know more about it, take a look. These can be used for things like carbon capture, gas storage, water from desert air. The Nobel Committee compared them to Hermione's handbag in Harry Potter.

• 1:30:00Krishna Choudhary?

  Yeah.

• 1:30:01Lester Nare?

  Very clever visual reference. He named the field reticular chemistry. I mean, what are your thoughts about this? Type of talent, particularly from the state of California.

• 1:30:14Krishna Choudhary?

  Yeah, it's, I mean, um, this, I think, is probably a direct response to our science policies. I have to say, um, it hasn't been said explicitly, but I think Tsinghua University gave him an offer that he couldn't refuse, like insane amounts of funding. I think all of his students could like, go with him effectively. Um, this is a man who's a legend in the field. This is a man that was such a legend that I could call him as my, one of my top picks to win the Chemistry Nobel, right? That's how obvious it was that he was going to win the Chemistry Nobel. He's had a tour, as you were saying, from Arizona, um, from Arizona State University to then UCLA to Berkeley.

• 1:30:59Krishna Choudhary?

  All of these institutions poached him from the last institution because they knew he was going to win the Nobel. Um, and now that he's won the Nobel, his, his cloud has gone up a whole lot. Metal organic frameworks are the chemicals of the future in terms of, um, water storage, carbon capture, as you were saying, lots of industrial applications. Um, and it's, it's a huge loss, I think, to, to US. Um, he himself has criticized grant pressure and visas, right? He's an international student at, and he started his life as an international student. So he knows exactly how important it is for American science to hold that edge with the

• 1:31:43Krishna Choudhary?

  international community. So the fact that he's leaving, yeah, I think, um, he probably saw the writing on the wall and to him, his research was more important than, um, you know, where he was doing the research.

• 1:31:57Lester Nare?

  A seasoned trophy winner.

• 1:31:59Krishna Choudhary?

  Yeah.

• 1:31:59Lester Nare?

  Thirteen trophies, Albert Einstein World Award of Science in 2017, Wolf Prize in Chemistry in 2018, the Balzon Prize in 2024, just to name a few, along with the Nobel. This is not a mid-table signing. This would be like, uh, Ballon d'Or winner in 2025, Ousmane Dembele, moving from PSG right now, which just won the Champions League, and is choosing to go to, you know, Chinese Super League or Saudi.

• 1:32:27Krishna Choudhary?

  Yeah.

• 1:32:27Lester Nare?

  Right now, right? Like in terms of the, how

• 1:32:31Krishna Choudhary?

  Yeah.

• 1:32:31Lester Nare?

  crazy, um, it would be to see happen.

• 1:32:34Krishna Choudhary?

  Yeah.

• 1:32:36Lester Nare?

  He has been an honorary professor at Tsinghua since 2022. Uh, he started full-time on July 3rd to lead their ometry, this new scientific discipline that was pioneered by Yagi, combining AI, material science and chemistry to design and synthesize advanced materials. The idea being, uh, can we compress the timeline of materials discovery?

• 1:32:59Krishna Choudhary?

  Right.

• 1:33:00Lester Nare?

  Given our new tools. If there was ever a man to do it, he will. And as a place that's trying to onshore manufacture, you know, re-onshore manufacturing and really own this idea of AI-enabled in the physical world, it's not ideal. Um, he did criticize the US for grant pressuring visas. He hasn't drawn a clear line that that's what caused this move. So, you know, it's kind of ambiguous, but, you know, Tsinghua gave him a bigger sandbox and he took it. That's our first big transfer of the summer marquee signing for China. We're going to move to our second big-name

• 1:33:40Lester Nare?

  signing, Shirley Meng, who is going to be our pioneer in batteries and energy storage, this idea of next generation battery materials, University of Chicago, uh, by, and she's now being transferred to Nanyang Technological University in Singapore. Also, deal done in July. However, you will note, this is a loan with the option to buy under transfer terminologies as opposed to just on the free, although there's no money in the loan. We'll get to that in a second. So, I mean, she's, she's just been incredible around this, uh, this energy space

• 1:34:23Lester Nare?

  and story, energy storage specifically. She served as the chief scientist for energy storage science at the US Department of Energy, DOE, Argonne National Lab, and directed the DOE-funded Energy Storage Research Alliance. This is a $62 million, um, over five-year DOE program that's sort of designed for figuring out how do we make energy storage, um, longer lasting for grid batteries, safer and cheaper. Um, grew up in China, got her first degree at NTU, which she was returning to her alma mater, uh, by way of the Singapore MIT Prince, uh, MIT Alliance PhD program, came to the States, also made a stop at the University of Florida, became

• 1:35:07Lester Nare?

  the Zabel Chair at UC San Diego, shout out California again. And in 2021, she was a professor at Chicago's Pritzker School for Molecular Engineering, 25 years. Uh, here, she's going back home to run the club, uh, and her big focus has been better batteries, solid-state, sodium ion, anode-free. Uh, in 2024, her lab reported the first anode-free sodium solid-state battery, which we've covered the concept of.

• 1:35:40Krishna Choudhary?

  Yeah.

• 1:35:40Lester Nare?

  In one of our previous episodes as well, in terms of the importance.

• 1:35:44Krishna Choudhary?

  I mean, the, the paper that we covered, it was out in Jule, if you remember.

• 1:35:47Lester Nare?

  Yes.

• 1:35:47Krishna Choudhary?

  And it was actually from her lab.

• 1:35:49Lester Nare?

  It was, it was from her lab.

• 1:35:50Krishna Choudhary?

  It was straight up from her lab. Um, we were talking about why sodium is so important.

• 1:35:54Lester Nare?

  Yes.

• 1:35:54Krishna Choudhary?

  Compared to lithium. Lithium is great, but it's like hard to get at. Sodium is literally in seawater.

• 1:36:00Lester Nare?

  Yes.

• 1:36:00Krishna Choudhary?

  So if we could make a battery out of sodium, that'd be great. I mean, it's not going to replace lithium ion batteries, but, um, for certain types of applications, it's going to be key for scalability. Like if we want to store like massive amounts of power, um,

• 1:36:19Krishna Choudhary?

  and then deploy it like later at night and things like that. Like if you want to have solar farms that, that create a bunch of electricity during the day, and then we want a way to store like, um, you know, infrastructure level worth of power, worth of power. Sodium batteries might be the key. And she's been doing insanely amazing work, um, at the University of Chicago. Another, another great professor who, um, is now leaving us.

• 1:36:44Lester Nare?

  Another seasoned trophy winner, Faraday Medal of the Royal Society of Chemistry.

• 1:36:49Krishna Choudhary?

  Oh.

• 1:36:50Lester Nare?

  ECS Battery Division Research Award, ACS Electrochemistry Honors, not a mid-table signing. She's becoming the VP of Industry and Distinguished University Professor, the highest faculty rank. She's already there as of July 1. And Chicago is going to keep a partial appointment through the transition. So this is a loan, not on the free. Uh, she's stepping down as the Director of the DOE Hub. Uh, the Hub's money wasn't cut. She's just leaving as the Chair. In science, uh, her reasons for leaving were explicit.

• 1:37:22Krishna Choudhary?

  Oh.

• 1:37:22Lester Nare?

  The US turning away from decarbonization and the immigration rules squeezing international, including Chinese-born scientists. Uh, she's one of the ones who's put that on the record. Singapore has handed her the keys to her old club, plus a C-suite seat, and she took the loan. We are going to move here through to our second to last in our coverage of the summer science transfer window. If you want us to cover more, let us know. We can go country by country. John Baker is our gravitational wave astrophysicist, uh, NASA Goddard Space Flight Center. He is moving to French National Center of Scientific Research in Toulouse.

• 1:38:06Lester Nare?

  Uh, this happened actually late last year, also on the free. Winner of the Goddard's Highest Space Science Honor, the John C. Lindsay Memorial Award in 2008, Kansas City, uh, uh, sort of upbringing, Truman State undergrad, Penn State PhD in gravitational physics, and then a postdoc at the Albert Einstein Institute in Pottstown, joined Goddard in 2001 and never left. He's been a member of the gravitational astrophysics lab. His big thing was simulations of black hole mergers. Uh, we just talked a little bit about, uh, some of the work we have around imaging black holes going from still to quote unquote, motion. Um, this work

• 1:38:51Lester Nare?

  that he's done has been a driver for the science case for LISA, which is Europe's space-based detector. Another not mid-table signing, a key member here. So as he moves to CNS, uh, CNRS, it's going to be the same message, mission, just on the other side of the Zoom. He'll be still working in this LISA arena. The launch for that is still aimed for 2035. Uh, and unlike Yagi, who was not clear about whether, why he was leaving, um, Baker also mentioned political and social conditions in the US and his family's safety. Uh, he put that on the record as well. You're sensing a trend here around, you know, scientists are

• 1:39:33Lester Nare?

  normally relatively diplomatic, especially with like the policy, because that's who puts food on their table in terms of money and funding.

• 1:39:43Krishna Choudhary?

  Well, this one's, yeah. I mean, he, I can't believe, he straight up said that. This is quite rare for scientists to like, just like, to say it. Yeah.

• 1:39:54Lester Nare?

  I, I don't know what to tell you. I don't know what to tell you. We're going to touch one more, uh, which is not just a, a single signing. This, uh, was a double signing, uh, and this one is also quite interesting. We have, uh, Ron Mangon and Tamara Swab, uh, both in the area of cognitive and language neuroscience, currently out of UC Davis, California. Again, I promise that wasn't, that wasn't bias.

• 1:40:24Krishna Choudhary?

  Uh-uh.

• 1:40:25Lester Nare?

  It's just, we have the best, making their way to the University of Birmingham in the UK on the free for both, package deal. The whole lab is moving to the university.

• 1:40:37Krishna Choudhary?

  Um, yeah, this is a two-body problem.

• 1:40:39Lester Nare?

  This is a two-body problem.

• 1:40:40Krishna Choudhary?

  Cause they're married scientists, yeah.

• 1:40:42Lester Nare?

  They are married. They've had decades at Davis. Uh, Mangon, distinguished professor of psychology and neurology, director of the Center for the Mind and Brain, um, since 2002, former dean of social sciences and was the author of the textbook Cognitive Neuroscience: The Biology of the Mind. Uh, his wife, Tamara Swab, professor of psychology at Davis for decades as well. The Cognitive Neuroscience of Language Lab, how the brain handles meaning, prediction, bilingualism across lifespan, trained at, uh, Max Planck, uh, past editor in chief of cognition and fellow of the Association of Psychological Sciences. Across the two of them, they have a combined 14 trophies.

• 1:41:27Lester Nare?

  We're talking about winners here.

• 1:41:28Krishna Choudhary?

  Yeah.

• 1:41:29Lester Nare?

  These are winners at the top of their game. Uh, again, Birmingham signed both of them. Uh, she's going to arrive as, uh, as professor of, uh, brain and language cognition in the spring, which has already happened. He's going to arrive in the fall as the 125th anniversary chair, professor of cognitive neuroscience and the director of the Center of, uh, for human brain health. This is like, you know, both are going to keep their emeritus ties to Davis. Um, they had mixed reasons on record, Swab told NPR. Uh, the optimism she once associated with the US science now feels more present.

• 1:42:05Krishna Choudhary?

  Wow.

• 1:42:05Lester Nare?

  In Britain and Europe, uh, plus the real offer, uh, including the UK, uh, UK Global Talent Fund support. Um, you know, it's for Mangon was, uh, adding Birmingham as an opportunity as much as a reaction. Um, this is, again, we're losing out to Europe in something.

• 1:42:29Krishna Choudhary?

  Yeah.

• 1:42:30Lester Nare?

  It's been a long time.

• 1:42:31Krishna Choudhary?

  It's been a long, in science, it's been a very long time.

• 1:42:33Lester Nare?

  In science. So I just wanted to give some coverage around this because we did put a new section onto the website, ffppod.com backslash funding. There is an ongoing funding battle here in the US, not only around who has the power and authority to deal with grant making, but also what funding actually goes into the larger scientific apparatus. For those who have not really looked at what funding goes where, it, it is quite interesting. It's, it's based off of the triple A's, um, research and development funding, uh, data sets that have existed, uh, back through, I believe it's, yeah, the 1970s and it's actually the, the 1960s.

• 1:43:18Lester Nare?

  And what's interesting about this is, um, because it's R and D, it includes defense, which is the lion's share of R and D spending.

• 1:43:27Krishna Choudhary?

  Yeah.

• 1:43:28Lester Nare?

  There's been some ebbs and flows. And so if you check out the funding page, you can see this long history of the story of how research and development in the US has worked as it's related to major milestones across that timeframe, as well as being able to look at the 2026 active fiscal year budget that was implemented and is currently having battles over money already written into law reaching the institutions that should be getting it, as well as the ongoing battle for the FY fiscal year 2027 budget and the current proposals. Both of these things are active issues in Congress. So if you want to get engaged in politics and you take your civic duty in terms of exercising

• 1:44:13Lester Nare?

  your right to be a participant in the democratic process, your members of Congress, your senators have a role to play in this process. And, uh, science is one of those issues, uh, particularly when you're looking at the hard sciences especially, there's a lot of complexity around some of the social sciences. Um, and if you look at the tracker on what these things actually fund and where they go, you may be surprised, um, at how much we can do with very little as compared to, for example, defense spending. So this is a, uh, culture issue. It's a funding issue. It's a competitive landscape issue because in some cases it's just people are giving better opportunity.

• 1:44:57Krishna Choudhary?

  Yep.

• 1:44:58Lester Nare?

  And this show lives off of the frontier research community. And so, uh, we really want to make sure that you as listeners understand what's happening right now, today, uh, with some of our best and brightest and how we can continue to stay competitive as we move into the future.

• 1:45:17Krishna Choudhary?

  Yep. And I mean, you know, we're going to cover science no matter where it happens, right? We're going to cover science if it happens here. We've covered several Chinese papers. We're going to cover several European papers. Anywhere that science happens, we are going to cover it. But, you know, um, we're also American and we care about, um, the science happening in this country and we don't, we don't want to let go of, um, the rich history and capabilities that we have, um, as a country. So we're going to continue covering this kind of stuff and you're going to see like, you know, in the first, uh, if I may say, Lester, you know, in the first year, we kinda, we kinda shied away from, from doing this kind of stuff.

• 1:45:58Lester Nare?

  Yeah.

• 1:45:58Krishna Choudhary?

  But I think, I think we've decided as an organization, as, as a FFP, uh, nation, that, um, you know, this is important and we're, we're going to start caring a lot more.

• 1:46:10Lester Nare?

  As always, I am your host, Lester Nare, joined by my co-host and our resident PhD. We made it under the two-hour mark. That's become the new benchmark. Um, again, we are super grateful for all of you who listen to the end of the pod. One last reminder, if you have made it this far, uh, you should definitely enter, enter the merch giveaway at ffppod.com backslash survey. We're trying to understand a little bit more about our audience so we know what to cover, what you're looking for, what you like, don't like about the show, uh, as we move into year two.

• 1:46:49Lester Nare?

  We are very excited in our next two episodes, if I'm not mistaken, uh, we are going to be covering something that's very near and dear to our resident PhD Krishna's heart. And I will leave you on that cliffhanger and be sure to tune in next week.