What Claude Actually Did to the Riemann Hypothesis

Opening

0:00It'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 sub agents to try and tackle this problem. So between them they they ran like 2400 shell commands um hundreds of Python scripts consumed 31 million output tokens and they autonomously downloaded

Episode 53

0:3954 academic papers from archive. Perhaps Claude like many of us underestimates the rate of AI progress. Hello internet. This is your captain speaking Lester Narre joined as always by my co-host and our resident PhD Krishna Chowdery. 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 Remon hypothesis in this explosion of progress as AI tackles many of these uh open problems and complex areas in mathematics and really understanding what's going on

1:20there. 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. As we've seen an 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.

Has AI reached a mathematical singularity?

1:58>> [music]

2:05>> 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. uh 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 work around and then that was quickly followed by uh Claude's progress on the Remon hypothesis and >> we what we wanted to do today is kind of get a background to understand why this is so important and then also So weed

2:48through the hype versus the sort of this is nothing new >> uh reaction to uh that particular result. >> Yeah, it's it's I think an incredible result. The reman hypothesis is the most important unsolved problem in mathematics. If you ask mathematicians, I think they will agree. If you ask um you know people adjacent to mathematics who have heard about the millennium problems and things like that this is the one right um to highlight that let me give you an example there was a famous interview with Jim Simons who's the famous mathematician behind churn Simons theory but I think everyone else knows him as the creator of the hedge

3:28fund Renaissance Technologies that has the medallion fund which mysteriously just prints money um I think their worst year ever was 20% that was or worst year in the past like 30 years. Usually they do like 100%. Like 40%, 70%. Unbelievable. >> Um he was he was $30 billion rich. He was worth $30 billion when he died. Um and he was asked in this interview, >> if you could trade your wealth for solving the Remon 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 Remon

4:08hypothesis. And then he later on he he like goes back to the scripted, oh, you know, my life has been great. Um, you can't choose how your life ends up. Uh, you know, no regrets. # no regrets. But you could behind it all, you could tell >> it was it was a yes. >> This is a man who was who was like, I I if I could have done that, I >> I would have done it. >> That would have that would have been sick, right? So given the aura around the remon 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

4:50things, 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 Remon 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 300 n 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 remon 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 I particularly there's this interesting dichotomy

5:32between 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 upfront is usually somewhere in the middle. Yes. >> And it's this seems like it's somewhere in the middle. >> 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 Remon hypothesis. I think it's a very cool thing to think about. Um, and two, it's a really nice case study in how AI has progressed from the initial chat bots that we were thinking about, you know, back when chatbt came online and

6:14everybody 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/ a [snorts] little weird. Um, and that is central to this story. 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 >> and the systems that are now being used particularly inside the frontier labs as well as uh for consumers

6:57have a variety of capabilities around the next token prediction piece that make it more than just predicting the next token. >> Exactly. Exactly. And that's that's that's going to be the highlight of this story is 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

Inside Claude’s attempt

7:24of 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 Remon hypothesis. >> It's that Pixar movie Inside Out where you get to see where you get to see inside the the thought process, >> dude. Yeah. And and I think I think it's just really cool like how how gran 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 3 weeks ago for two reasons. Um the first reason this is the Remon hypothesis. Okay, the Jacobian

8:06conjecture people know about it. I had kind of heard about it. The Remon hypothesis I have been hearing about it since even before I knew anything about complex numbers. It's like Millennium Prize and all this other kind of stuff, right? Um two, the person who initiated this, the human being behind it was not a mathematician. Okay, for the Jacobian conjecture one, um, Alpog was a Princeton and Harvard trained mathematician. This time we've got, um, Jared, we've got, uh, 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

8:46mathematicians. Okay, he kind of just prompted the new version of Claude that's unreleased. Anthropic is very clear to say it's unreleased. [laughter] 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 Remon hypothesis?" And then when Claude was like, "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." >> A little bit of do of positive encouragement >> to Claude to an AI >> to an AI. Uh actually what this is okay >> and it's kind of weird that that kind of

9:27worked. So we're going to get into some of that as well. Okay. So first let's go over the reman hypothesis. What is it and why are people obsessed about it? Okay. I could dedicate several 2hour long videos to this thing. Um 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 Eliasstein. If you go through this book, you got to get to chapter seven to really understand Remon Hypothesis. I think you could skip the chapter on fora analysis because that doesn't really tie into this stuff. But everything else you're going to really need to understand in order to

10:08believe some of the magic that I'm about to tell you, but I'm going to go through kind of a sparknotes version of how you would begin to understand why the Remon hypothesis is so important to solve. Okay, I'm going to begin with the basil problem. This was first proposed by Petro Mangoli in the 1600s.

Euler and the Basel problem

10:29The 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 + 1 over 2^2 + 1 over 3 2 + 1 over 4^2 all the way up to infinity. This is a infinite series. So 1 + 1 over4 + 1 over 9 + 1 over6 all the way. What does that equal? This was proposed in the 1600s. It's named the basil problem because the people who were really working on it were Leonard Oiler and the Bernoli family. The Bernoli family is the very famous family that is behind like the Bernoli numbers um the Bernoli principle which is how a lot of people mistakenly say airplanes fly.

11:11It's not that simple. I mean it has something to do with it but it's not that simple. Um, in any case, the Bernoli family tried and they failed. This is a family of mathematicians. Leonard Oiler tried. Honestly, might have been his first time that he just like solved it. Okay? And he said, "This whole thing is equal to p<unk>^2 over 6." Okay, so 1 + 1 over 4 + 1 over 9 + 1 over 16 all the way down. That's going to equal<unk> 2 over 6. The way he did it was kind of hacky. Um, and like it was only proven to be completely correct by wire strauss like a 100 years later. >> It was a little handwavy. >> It was a little hand wavy, but he got the right answer. Like so many things that he did. Okay, cuz his intuition was

11:53just like insane, right? Okay, fine. So that's impressive, right? But that's not all his main thing that Oiler did for the basil problem. So the basil problem is only 1 plus 1 over 4 plus 1 over 9 all the way over, right? um Oiler 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 one over squares but one over cubes >> or one over uh stuff to the fourth power things [clears throat] like that. So he he creates this function called the zeta

12:33function. And crucially he shows that that sum of one over all of the squares and things like that of the natural numbers is equal to a product over the primes. >> Oh. Okay. So 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. >> Very good. Yeah. The generalization was I take one 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

13:16the zeta function is so important in mathematics. It has to do with primes. Okay? And everywhere you'll look, you'll see that the remon 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? Yeah. Okay. So that's what we're going to do right now. Um here's where we're going to start. We're going to start with the zeta function. Let's consider instead of um squares. So in this case, the zeta function is for any arbitrary s. >> Yes. >> But the s is usually two. That'll give you the basil problem of 1 + 4 1 over 4 1 + 1 9th + 1/16th. Instead, let's just consider s= 1. Okay, this is called a

13:57harmonic series. It's just 1 plus all of the fractions added up. This thing crucially does not converge. Okay, this thing goes to infinity. >> So there's the harmonic series which is when the power is one. There's the basil problem when the power is two and then >> and then the zeta is for any power greater than one. Okay, that's the zeta series. Mhm. Okay. So, we're going to consider the harmonic series, which is when the power is one. 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

14:37+ 1/2, but not not instead of 1/3, we say plus 1/4 >> plus [clears throat] 1/8. So powers of two, the same power over and over again. Okay, this is not now you're not changing the number. >> Yes, >> you're now change you're you're keeping the number the same, but the power is going up. >> The uh the variable moves from being in the power spot to being in the the >> exponent in the exponent [clears throat] spot into the >> Yeah, exactly. Exactly. So 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 halfing each term and then you're adding it up. So you get a one, that's a whole pizza, let's say. Then a half, that's half a pizza plus a quarter of a pizza

15:18plus an eighth plus a 16th plus a 32th or whatever it's called. >> 32. [laughter] >> As you add it all up, you're going to get two. You can see, right, the one is the first, and then as you add a half plus a fourth plus an eighth and so on and so forth, you're going to get a whole number and the whole thing is going to equal two. the way, and this is something that you um 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, cuz you're multiplying a half every time you add a new thing. Um, and you take the

How Euler connected the zeta function to primes

15:51first number. So, the first number in this case is 1. You put that on the top 1 / 1 minus the common ratio. So, in this case, it would be 1 / 1 - 1/2 to get you two. 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 Oiler's genius. He said, consider the geometric series for primes. Okay. We had just seen a geometric series for two. A geometric series for three would look like 1 + 1/3 + 1 9th + 127th because those are the powers of three. >> Mhm. >> We don't do four because four is not a prime. >> Mhm. We um we look at the geometric

16:32series for five >> which is 1 plus 1/5 plus 125th plus 1 125th [clears throat] and then there'll be 625 so on and so forth. All of those things are going to equal 1 / 1 - common ratio. So it's going to be 1 over 1 - 1 over2 over there. That's going to be 1 over 1 - 1/3. So that'll give you 2/3 actually over there. It's it's right or three halves sorry. And so this is the geometric series for powers of primes. >> Okay. [clears throat] >> Mhm. >> Oiler 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,

17:14so 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 um a multiplication of those individual Yes. representations, right? Those individual answers. And now you're already starting to see that secondhand side of Oilers's product formula. >> Yeah, I was going to say that it's the same. Yeah, there's the through line. >> Yeah, there's the through line. The 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. >> Mhm. >> Right. And so now we're seeing the right hand side. >> Yes.

17:55>> Now that's fine. Here's the real genius of Oiler. Why is he even doing this in the first place? Okay, this seems like a lot of stuff to do. >> Mhm. >> Without the punch line. Here is the punch line. 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 + 1/2 + 1/4 the powers of 1/2 the powers of 1/3 the powers of 1/5 you'll get the power of 17th later right? If I multiply those out and I do the foil method, if you remember like um you have to distribute each term has to multiply to each term, right? So if I if I multiply that out, the one can be

18:35multiplied to all the other ones to get a one, right? Then the 1/2 from the first one can be multiplied to all the other ones to get a 1/2. >> Mhm. >> Right. Um similarly, the 1/4 can be multiplied to all the other ones to get a 1/4. >> The 1/3 will be there, the 1/5 will be there. I could also get a 1/6th because the 1/2 times the 1/3 multiplied by all the other ones is going to get me the 1/6. I see >> the 1/8 is going to come from the the 1/2, right? The 1 nth is going to come from the 1/3 and then the 1112th can come from 1/4 multiplied by 1/3. Every single number is going to be represented in that infinite sum

19:15>> because primes make up the numbers. >> Right. >> Right. 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? Right? >> 17th is missing because I haven't included that in this. Right? Mhm. >> Similarly, 11 1th is missing, but that's a prime. That'll be another that'll be another one on its own. Um 113th is

19:56missing, but also 114th is missing because 114th would be the 1 17th multiplied by the 1/2 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 >> because they're just further down. >> Yeah, there's further down. I haven't included it. Right? If I added the if I added the 17th thing, then the 114th would show up and the 17th would show up, but the 111th would not show up, right? Because that would require another. >> 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. >> Yes. And you can do this infinitely. And the key is you will get every single fraction ever. >> That's incredible. Okay. >> Right. >> Yeah. Yeah. Yeah. Yeah. >> This was Oiler's genius. >> Yeah. That

20:36>> and that's how he's showing this is called Oiler's product formula. And it's kind of crazy that like he set out to just solve the basil problem. >> And then and then he's like, "Oh, by the way, I also noticed this. [laughter] >> Just just have this one on the >> Yeah. Yeah. So this here you've already seen now the zeta function is on the left. That's the sum >> of the one over all the natural numbers, right? >> He has tied this now to a product over primes. Mhm. >> And now you can imagine if I do if I want to do um one the zeta function to the second power all I have to do is put the primes to the second power. >> Right. Right. Yes. Yes. >> And that's [clears throat] why the s is a common um exponent

21:18>> in in both across both. >> Oh that's so elegant. >> Right. And and this already shows you how the primes are so intertwined right >> with the zeta function. Right. They are one and the same. And so part of what we're building here is the connection between those two. >> Yes. I'm trying to I'm trying to justify why there's so much hype >> about this. Okay. >> It's because the remon zeta function has to do with primes >> in a very integral way. It's actually encoding the same information. >> Right. >> On one side there's a a product of primes. On the other side there's a sum over every natural number. >> This is really good. >> It's kind of cool. >> This is this is really good.

21:58>> Yeah. And this was this was in 1730s. >> Okay. >> No Wi-Fi then. >> No. Um now 1792

Gauss and the distribution of prime numbers

22:08there's a teenage Carl Friedrich Gaus. >> Gaus. >> Gaus is another amazing mathematician. Um perhaps second or third only to Oiler and maybe Remon actually. Um he starts counting primes when he was a teenager cuz I guess that's what you do when you're a full autist in Germany. 17. >> Yeah. So, so he he starts counting the number of primes by hand. >> Unbelievable. >> 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 if I if I go up to like um a 10,000, right?

22:48>> The probability that numbers around 10 10,000 obviously is not prime, but like any given number around there is prime is 1 divided by the log of 10,000. log base e >> 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 / log n okay there in the blue is you see a pi of n that's the prime counting function n / log n is in red and you can see that n over log n kind of undershoots he also realizes that the >> um 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

23:28the time. If you go to 10 the 10 the 35 then prime numbers go above the the log integral. But like he he actually thought that perhaps the integral function is like always above. >> Mhm. >> But um he didn't know that 10 the 10 the 35 couldn't get that. >> Look [laughter] if you see if you do it by hand and you see a pretty good you're like it's probably you know I get it. So this becomes now the prime number theorem. Okay, which is that the number of primes before a certain level. The number of primes before a certain number is related to n / log n.

24:08Okay. >> Which is our red here. >> Yeah. Yeah. It's like asmtoic to n / like it has the same shape. Yes. >> Okay. It's not like going to diverge like crazy, right? It's always going to stay close in some sense. Right. Now enter Bernard Reman. >> Reman. [clears throat] >> Okay. >> Reman. >> Reman at the University of Berlin. He's got a 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 we're worried about geometry and things like that. Um like Remanian manifolds and stuff. That's the stuff that you get into with like general relativity. But this is what he wrote

24:50about um the prime number theorem and ever since then it has been named the reman zeta function function. Okay. Because there's a there's a joke in mathematics you always name something after the second person who discovered it because the first person is always Oiler and you can't name everything after Oiler you know. So some people call it the remon oiler zeta function but let's just give it to Right. Right. Right. Right. >> So >> we already know who the goat is. >> Yeah. Yeah. Exactly. >> We already know who the goat is. >> Exactly. So um he actually did some amazing work on the remon on the zeta

25:31function. What he did was extend Oilers'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 the exponent is a real number you can even imagine one over square roots and things like that right um

Riemann takes the zeta function into the complex plane

25:55remon said no what if we took one over stuff to the power of a complex number so it's like 1 + i or 2 plus i. What what would that do? And it turns out there's a very well- definfined way of talking about that. So here's how it works with complex numbers. Okay, here this I'm going to borrow a lot of um 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 remon zeta function. Um and he's an abs he's the goat of mathematical visuals. So um >> big shout out >> big shout out. So here's here's what happens if I take remon zeta to the 2 plus i.

26:36>> So in this case we're the primary thing we're changing is uh the exponent. Yes. In the denominator. >> Yeah. Because that's how the remon zeta this function is defined. >> Right. The function is defined as the sum over one over the natural numbers to some power. You choose the power. >> Yeah. Right. And so we're just saying we've now chose the power. >> Yeah. We chose we choose the power to be 2 plus i instead of two. If it was just two, it would be the basil problem again. and then I'd get square. But because it's 2 + 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 + 1/4 + 1 9th and I'd go a little bit little bit closer to p<unk> square over 6 and I'd just be moving on the

27:17number 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 two on the x, one on the y. 2 + 2 i would mean two on the x, two on the y. So that's that's how complex numbers are defined. 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 one. So that's why you go one first. But then two, 1 / 2 to the^ of 2 plus i. That's going to curve a little bit downward. And it's going to curve a little bit downward for for the three. And it's going to spiral to that point. Mhm. >> So the remon zeta function takes as

27:58input 2 plus i >> and maps it to that point down there >> which is like 1.15 minus 0.44 i. Okay, that's how it works in the complex plane. >> Yes. >> Now turns out as long as the real part is greater than one, your sum is going to be fine. And we've got an animation here courtesy of three blue one brown. Again, >> as I change the input, the spiral is going to change, but it's going to it's going to be well defined. It's going to end up somewhere. >> Okay. So, here I'm taking my input, which is the yellow dot, and the output of the remon zeta function is where that spiral ends up. Okay. Perfectly well-

28:39definfined function. [clears throat] Really nice. Okay. >> Mhm. >> Um, let's see what this function does to the grid lines. Okay. Here we've seen individual points getting mapped. Now, let's see what this function does to the grid lines. as long as the power the real part is greater than one. This is how the the the grid lines >> Mhm. >> map. >> Okay. Now, why do we need the powers to be greater than one for the real? That's because, you know, imagine if I did um the the power equals zero, right? Then I just have 1 + 1 + 1 + 1 that's going to diverge to infinity. [clears throat] >> Does it's not well defined at least in the series representation. But as if my powers are greater than one, then each

29:20fraction gets smaller and smaller and so my spiral actually converges somewhere. That's the idea. >> Okay, got it. >> 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? >> Right. >> Right. It seems like I could just draw some lines, [laughter] >> right? I could just continue the function right? >> 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. >> We're and we're and specifically you're

30:01referencing to the to the left of of that line of the y ais. >> Yeah. Yeah. Exactly. There's a little part on the on the right of the y- axis too that that thing is not Right. It's like about like half. >> It's a little there. >> Yeah. But like but like my point is >> the parts where we can use the series. >> Yes. >> Give me this really nice functional map. >> Right. >> Okay.

Analytic continuation

30:26>> The idea is can't we just extend what we're already seeing? >> Exactly. That's what Reman said. He's like, can 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 zero, it's not going to make sense cuz I'm just adding 1111. 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. [laughter] That's what Remon'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 one over a power

31:08plus 2 over a power plus 3 over a power instead I make up another function that behaves the same way on the right hand side and it looks like that two uh 2 to the power * pi to the power minus one time s of something time the gamma function of something time the remon zeta of the flip hand flip left 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 1 - s, multiply by all this other stuff, and I'll be fine. >> It's just it it'll be fine. >> Okay, it'll be fine because and this is the magic of complex analysis. >> It turns out there's a very good reason,

31:49right, >> why complex functions do this. Okay, >> I can just flip it on the other side of the board. >> 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 >> okay >> 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 So if you found the way that's it >> that's it. Yeah >> that's it. You have found the

32:29continuation of that function. There's only one way to do it. >> Okay. And so this isn't like >> it it is imaginary. I mean it's in complex variable. So there's an imaginary unit in it. Right. But the imag there's only one way to imagine >> it in this imaginary space >> 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. There's a there the logical consistency only has one answer. >> Exactly. And so that is the extension that is the analytic continuation of a meamorphic function outside of its domain where it's like nicely well behaved. Now it's everywhere. Okay. >> Mhm. >> That's the key thing that Remon

33:11discovers. Okay. Side note,

Why 1 + 2 + 3 + 4 + … = -1/12

33:15um there was a lot of hype over the um the sum 1 plus 2 plus 3+ 4 + 5 equaling -112th. Now you can finally kind of understand why that is the case. If I plug in negative 1 to the remon zeta function, right? Then that's going to be 1 + 1 over 2. But 21 means I just flip. So I get 1 + 2 plus 1 over 31. So I just flip. Y >> right. So that's just 1 + 2 + 3 + 4. Right? It doesn't make sense on its own. But if you use the remon zeta function, you analytically continue it outside of its domain to the negative numbers and then you plug in1 to that functional thing that I showed you earlier with the

33:55signs and the gamma function and everything. You plug in negative1 to that, you're going to get112. >> 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 -112. Okay. To show you another reason why this is the case and why Shrinasa Rammanujan who's a big hero of mine is on this is um Rammanujan 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

34:37big 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 1 plus 2 plus 3+ 4 dot dot dot equals -112. And he said, I know you'll probably think of putting me into an insane assignment. [laughter] But look, I have techniques that assign numbers to divergent series and I got -112. Hardy looked at that and he was like, hold on. This guy makes no mention of complex analysis and imaginary numbers, but he still figured out a way >> to get to -112th. And this this was a known thing at the because the Hardy is after remon right. So there you can just

35:17plug in negative 1 and you know that Rammenujan had no idea about remon 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 number112th. Just goes to show you how closed math is. it. He was able to in it from a totally separate pathway, but it still converged at the same the same destination regardless of his lack of not going down the road already traveled. >> Yeah. Yeah. He went a complete but 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

35:57and say define what this is and it can't be infinity. The only correct answer is -112th. Okay. >> So fascinating. >> 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 -112th, right? And the remon zeta of negative1 is 1112th, 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 >> to like deal with their infinities. >> Yes. >> They're like a12th. It's like okay, >> the number of dimensions is the denominator. [laughter] >> They probably have some party trick where they where they do that. Um I I just thought that's that's a really cool

36:37like little aside. >> No, no, that's and that's I think it's because it's it part of what we're doing is we're connecting all of these component parts, you know, ma mathematics being almost this closed system, >> right? That has has these bounded box and and and a level of like the logic logical consistency that exists through it even if you come from different disciplines within that subject matter. Um and seeing how those different disciplines address the same question. >> Yeah. >> Um or try to approach the same questions. It's really fascinating. >> Yeah. I mean it's it's really cool. I mean the basil problem is another example of that. Let me just give you another little anecdote from my personal

37:18life. 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 bezel problem. Okay. It's like great. Okay. Oiler did it. Yeah. You can do it for a final you freshman. And uh this this was Peter Sarnac. He was just a maniac. But um he actually in the problem statement he he gave us enough information and we [snorts] used Forier series >> and a clever trick with Forier series to show and I got that problem right. It was a it was a great problem. Um but yeah, you can use Forier series to show the pi squared over 6. You can use Oilers's hack. You can also use um tailaylor series. A lot of times people use like um the tailaylor series for

38:00sign and things like that. So there's so many different ways but you will always arrive at pi^ square over 6. Just the beauty of mathematics is is love. You got to love it. >> Okay. So now let's finally talk about the reman hypothesis. We [clears throat] have talked about the reman zeta function. >> Now let's talk about the hypothesis >> which is sort of a uh an the next step from starting with the basil problem >> uh that let us have a fundamental understanding of of >> we generalized it to the zeta function >> to the zeta [clears throat] function. >> Yeah. And now >> now we're finally going to get to the remon hypothesis. Okay. Now the remon

What is the Riemann Hypothesis?

38:34zeta function has been used to prove the um that bound of primes the n over log n. That's actually chapter 7 of this book >> which was the red line in the previous. >> Yes, it's how do primes grow? They grow like n / 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. And what that means is what inputs map to zero if I plug in that input into the exponent [clears throat] >> of the of the you know one over the thingy and or or I the functional form. >> Yes. >> Do I get zero as the output? Okay. There

39:14are 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 -2 I get zero. If I put put in -1, I get that -112th. Yes. Okay. But if I put in -2, I get zero. If I put put in4, I get zero. Those are the trivial zeros. And that just has to do with the fact that there's a sign in there. And sign goes like this. >> Okay. So it's trivial. >> That's it comes back to >> 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 at >> at a cons at a consistent interval. >> Yeah. Yeah. And that's that's okay. Fine. >> Fine. >> Now there are the nontrivial zeros. Oh, my favorite. Non-trivial.

39:55>> Yes. And they are non-trivial for a reason because we don't know where they are. >> And that is the remon hypothesis. >> Where are they? >> Non-trivial because everything else is well defined at this point. >> Yeah. >> Except for the non-trivial zeros. >> The non-trivial zeros. We know that there's a bunch of more zeros, but we don't know where they are. Okay. Remon proved that they are inside of that strip. He said they're somewhere in here between 0 and one on the real axis. That's called the critical strip. So Remon proved that and then in his in in his paper he writes probably [laughter] they're on the critical line. They're on the one half right in the middle of that strip. >> Okay. >> Okay. That's what he writes. He's like

40:36probably >> 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. >> There's there's a search space that we believe these non-trivial zeros exist in. >> Yeah. >> And within that context there is a discrete line >> right in the middle. >> Right in the middle which is the sort of uh most likely place the hypothesis of within this search space >> this is where you should look. >> This is where you should look. And if we can prove that they all lie only on that line

41:16>> and nowhere else. >> They're not in [clears throat] like some fudged like part around it. Every single non-trivial zero is on that line. That is the remon hypothesis. >> We want to resolve it similarly to how cleanly the trivial zeros resolve or the -112th resolve. >> Yeah. Yeah. >> Where it's always true. >> Yes. Yes. It's always true that every single zero is going to be on that line. Okay. Okay. That that has been the quest. >> Yes. >> Okay. >> Okay. >> Oh, it seems so easy. >> Yeah. [laughter] Um and if we know why would we care, right? Well, if we know where the zeros are for the remon hypothesis, >> Mhm. >> then we could prescribe an exact form of

Why the non-trivial zeros matter

41:58the prime number counting theorem. >> Okay. And this next animation shows that. So the the jagged line, that's the prime number counting theorem. >> And that has to be jagged, right? Because at some point it's going to increment by one. At two [clears throat] it goes up by one. At three it goes up by one. At five it goes up by one. >> So every all the vertical increases are that >> are just like, oh, here's another prime. Here's another prime. Um 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

42:39form of that line. >> So this is the idea that the this initial plot of of the um of our where these primes are going to be as we get higher. >> It's fuzzy. >> Yeah. >> Because we don't know where the non-trivial zeros are. >> Yes. And if we knew where the non-trivial zeros are, it would res it would resolve the resolution would effectively be we could then do >> we could predict every >> that seems like a pretty big deal. >> Yeah, we could we could predict exactly the prime number um function, the counting function. >> That's okay. >> Right. >> Yeah. >> And the primes are everything. >> Yeah. So I get now why this is so important. >> Yes. It it's basically like uh it it's

43:20sort of like um 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. >> You'd still have to like you know sum up to all of the infinity of zeros but at least I got a procedure >> to do so and it just becomes a process. >> Yeah. And if I like however accurate I need to be I need to just find as that many that many zeros >> and with the amount of compute we have nowadays. >> 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 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. >> Okay. [laughter] Yeah. And that's when

44:01people say we're trying to solve the remon hypothesis. >> No, when when people are saying they're they're trying to solve the remon hypothesis, they're just trying to show that every zero is here. It doesn't have to be a constructive proof. You could just show that there's no other zeros anywhere else. >> Okay? Right? You know what I mean? 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. >> So there's sort of two levels of let's success is not the right word but but but goals to reach. One is just proving that everything's on the critical line. >> Yeah. And that's the remon hypothesis >> and it can't be anywhere else. >> Um >> if you do that you've won the millennium problem >> like which and it's >> Simons would have would have given us wealth for that. the idea that the the

44:42construction the constructive is is well beyond even where we can imagine given that we haven't even solved the we haven't even be able to solve the problem of of it's where they are not >> yeah I mean it could be that the constructive proof is the way to prove it right >> okay >> you know there there there's there's history in mathematics that show that um 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 koshi sequences of rational numbers. Right? So there's a like both are like the remon hypothesis is such a black box >> that we don't even know >> which is going to be easier >> which is angles the way to go.

45:24>> That's something that Terrence Ta has said is in some of his lectures right if we want to prove like the whole thing >> we it seems we don't even have the toolkits to understand which way to go. >> There's a fork in the road and it's unclear which one is closer to the destination. >> Yeah. Yeah. And there could be multiple forks [laughter] >> right? It's like we we see like two I guess >> right but there could be if the fog goes away there could be like [laughter] a hundred over here like we don't know. >> No but this that's so that's really interesting um in terms of understanding the value the fundamental value of solving this. Yeah >> uh as as a problem. >> Yeah it's like I mean it would give us an understanding of the atoms of numbers right like it would give us an

46:04understanding of the periodic table of numbers. The periodic table of numbers consists of the primes, right? Everything else is made out of them. >> And this would tell us inherently like how that is structured. >> I mean, I think that is just beautiful in itself, right? >> Would it be would it be like um the photograph the the it's sort of like what makes up uh the um Rosalyn Franklin uh X-ray crystalallography photo? It would have that kind of similar like in a different field, that level of fundamental. >> Yes. Yeah. Yeah. Like if we like yeah if you figured out the structure of DNA that's insane right this would be like yeah that kind of [laughter] yeah I mean it would be insane so some other

46:44stuff right if the remon 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. Mhm. >> So let's just assume it's true. And then there's so much other mathematics that falls through because of that. >> And so the the surrounding surface area kind of points to yes, it is true even though we can't yet. >> Yeah. >> Uh either by saying that the zeros are nowhere else or constructively proving it that it's true. >> Yeah. Yeah. I mean it's it it's so it

47:24would be huge, right? >> Yeah. Yeah.

A century of progress toward the critical line

47:28>> You kind of get it now, right? >> That's a big deal. >> Yeah. So this is why everyone's like reload isn't anything until it really sol well if it solves it then that anyway. >> Yeah. If it solves it then that'd be great. It hasn't solved it and we're going to get into exactly what it did. Okay. So >> um after Remon'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. Um, Reman did a lot of the leg work, but then there was Jacus Hatamard, the guy behind Hatamard Matrices for those in quantum computing. And um, also I don't know how do you say this? Can you can you try? >> Charles Jean Dea Valet Poin.

48:10>> Yeah, there there we go. So th those two guys, they they proved um using Remon's complex analytic methods. And in 1914, the great English mathematician um GH Hardy, who is um shown here, he's most well known for um discovering Rammanujan. >> Mhm. >> 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. Reman said [clears throat] 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. >> Mhm. >> It also says almost nothing about the

48:53Remon hypothesis because it could be that there's many many infinitely more elsewhere on that strip. Right. >> Right. So, it's it's a step in the right direction [laughter] >> I'd say, >> but it's not quite the whole thing. Right. [snorts] >> Um, >> next we've got finally in in 1942 at Lelay Selberg. >> Mhm. he makes a step in the direction of the remon hypothesis. Okay. >> Um he won the Fields Medal later in 1950 and this is a photo of him at the Institute of Advanced Study at Princeton. So he proves that there is a pro there is a positive proportion of

49:34non-trivial zeros on the critical line. What that means is there is some percent of zeros at least at least some percent of non of of zeros on that 1/2 line. >> So the point is there there are these there's this number of non-trivial zeros. >> There's infinitely many >> there's infinitely many because of uh Hardy. >> Mhm. >> And some percentage of those cuz we don't yet know if they're only on the critical line. >> Yeah. Yeah. And so if they're only on the critical line it would be 100%. Right. >> He said the percentage is more than zero. >> Zero. Right. >> I don't know what the number is. >> Right. >> I'm just telling you that the percentage is more than zero. >> There's at least one.

50:15>> Yeah. >> On the critical line. >> Well, no, we know that there this is this is I should I should be be careful here. >> We've already proven that there's an infinite number, >> right? >> 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. >> Total infinities because in the larger search area there's an equal amount of there's similarities. >> This is the stuff I was talking about. >> Infinity everywhere. >> Yeah. [laughter] So so you know what I mean. >> No, that's a that's a good distinction. No, that that's a good distinction. It's kind of weird to cuz we're dealing with

50:56some weird stuff here. 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. >> Okay. >> Okay. >> Which is okay. We're 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. [laughter] Um, so next we have Levenson in 1974. He pushes that number up to 33%. >> Okay, that seems like a big deal, right? So he's like a third >> of the zeros at least. >> At least a third of the non-trivial zero >> zeros are on the critical line >> are on the critical line. >> Infinitely many 34%. >> Yeah. Yeah. It's like it's like for it's a crude way of saying it is for every

51:38for 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 going to take two. For this one I'm going to take two. And if you do that you'll cover everything. >> Understood. that that's how that's how you want to think about it. Yep. 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 Levenson shows. Okay. >> Mhm. >> Um and he does this weird thing with like mullifiers where he basically tries to like blur out the reman zeta function and then work with that blurred out version. I don't know how it works, but that's that's what Wikipedia said. [laughter] Okay. So So and and now we're

52:18getting into stuff that I really don't understand. Right. No. Fair. Fair. >> Um, so, so now I'm going to be regurgitating what I sort of figured out with my readings. Um, full disclosure, >> Levenson puts it up to 1/3. >> Um, other people take his method and start inching that up. >> So, we've got Conre in 1989, he pushes it up to two- fifths. >> Mhm. >> So, 40%. >> This is now uh uh what, 15 years later? >> Yeah. 15 years later. Um, at in 2022, so this is what 30 years later. >> Yeah. Yeah. We're up to 41.7%. >> So over Yeah. So So there was a quick jump uh by 33 to to to There's a quick There's an immediate jump to 30 3 from 0

53:00to 34. >> Yeah. There's like three big jumps. I'd say you go from zero to not zero. >> Right. Yes. Yes. >> Which even though on the graph it's like >> at the bottom >> the fact that it starts >> right. >> Right. >> Yes. >> That's huge. >> Sberg. Yeah. >> And 42. >> 42. Um and then in 74 it goes up to 34 [clears throat] per 7. big jump. >> Then it goes to 40.9. Another pretty big jump. And then it took us 30 years to get to 41.7. >> Just to even get one more%. >> Yeah. >> Not even not even one more. >> Okay. And now Claude comes in in 2026 and it has brought it up to 67.25%. >> Mhm. >> Okay.

Claude pushes the bound

53:39>> The remon hypothesis is at 100. >> Mhm. >> Okay. We're getting closer. >> 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 99. >> Mhm. >> 9999. >> Yeah. >> And it doesn't mean anything. >> Doesn't mean anything for the Remon hypothesis. It doesn't, >> right? Because what really matters is >> is the 100. And um Claude and other mathematicians have acknowledged that the way that Claude did this, it's not going to get you to 100. Okay? There is a limit to how we push this. Okay. >> Okay. We're not going to get to 100 this way, but it's still cool to get close,

54:19right? Okay. Because it makes us understand a bit more about the function in some sense. >> Part of it part 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 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 Mh. >> 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. >> Exactly. Exactly. Yeah. So, now let's get into how Claude actually did it. Yep. >> Okay. I've got a few a bit of understanding of how it did it. Okay. Um [snorts] Claude it again tweets. That's

55:00how it they announced I guess that they were doing this right in what appears to be it's like a matter of like 30 hours. They've um they've pushed that bound to 67.2%.

Montgomery, Dyson, and the physics connection

55:10they they are careful to say it didn't solve it but it made um strides on a related problem. Okay. >> Now [snorts] there are two things that Claude used that are important to know here. >> Okay. >> Okay. The first is a technique developed by Hugh Montgomery in 1973. He was a mathematician and um the legend has it that he was actually talking to Freeman Dyson. >> Freeman >> 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 remon zeta function and he analyzed how they sort of clump 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

55:51looking 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 the zeros of the remon zeta function it's kind of Crazy how like the world works that way. >> Now the catch was that when Montgomery did his original breakthrough, >> it required assuming that the Reman 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 remon zeta function works. So this is something that has to do with the remon zeta

56:32function um that Montgomery sort of proved on the side. Okay. >> The second thing is in 2000 the legendary Fields medalist Enrico Bombiieri he wrote a note a little monograph where the clay mathematics institute was um forming the millennium problems. So he wrote the note for the the remon 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 pre 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

57:13the architecture of these zeta zeros. and he dives into something called quadratic forms and um while explicit quadratic functionals. I don't know what they are, but that's what he mentions in his write up. >> Mhm. >> All right. Now, let's see what Claude does. >> Yeah. Cuz you're saying these are two fundamental the Montgomery and Bombiary no relation to Guyier insights uh were

“Believe in yourself”

57:39building blocks >> for what Claude is doing. >> For what Claude is doing. >> 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? Cuz like the the the look the 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, [laughter] you know, >> good great marketing. Great marketing. >> It is great marketing. So anyways, let's take their word for it. He's jogging. He opens up a clawed terminal while jogging >> on his phone. Why don't you watch some Netflix or [laughter] something or hey, listen to this podcast, >> right? Right. That's a great that's a

58:19great >> Anyways, um I guess they're always working out in an anthropic. Okay, so he casually prompted the unreleased model to take a real stab at the Remon hypothesis. Okay, the model exhibited initial skepticism about its own ability and the human said, you know, almost comedic encouragements of like keep going and believe in yourself. >> Very Ted Lasso. >> Yeah. Yeah. Yeah, he Ted lassoed Claude into getting closer to the Remon 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

59:00orchestrator

Claude builds a 60-agent research lab

59:02and it spins up a hierarchical swarm of 60 autonomous sub aents to try and tackle this problem. It built a team of pe of other agents, other versions of itself ostensibly that had one thing to focus on. >> Yeah. >> And then it could coordinate them because they're not trying to do too many things as one inst instantiation of itself. >> 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, right? >> And execute those Python programs. So

59:43between them, they they ran like 2400 shell commands, um hundreds of Python scripts, consumed 31 million output tokens, and they autonomously downloaded 54 academic papers from archive. >> Mhm. The uh preprint server run by Cornell. >> 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. Right. Right. There's like a PI and then there's like postocs 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.

1:00:24>> You have some people doing wrote work, you have some people doing sort of ideiation. You have some people doing uh sort of functional mathematical number crunching validation verification. A and the what's so fascinating about this is that the initial orchestrator from the initial prompt can architect the approach. >> Yeah. >> Uh without having to be dictated to what the right construction of that is. >> That's exactly right. This is another point that um I'm glad that you're highlighting here. The initial prompt was just take a real stab at the Remon hypothesis and believe in yourself.

1:01:06That's it. >> Right? Like this this guy was not a mathematician. All of the mathematics is coming from Claude itself. >> Mhm. >> Which is kind of crazy, >> 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'll we're potentially going to 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 and it will necessarily work. But I do think it's an important um to understand

1:01:48that the construction that a lot of us have about what is quote AI of which LLM'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 creative, but but there are so much domain space in taking orthogonal or correlated areas of a variety of areas of study in science and

1:02:28mathematics >> and just making connections no one else has. And there's so there's I think that people underestimate how much can may be able to be done just making existing connections. >> Exactly. No, that's to totally I mean for example we had just covered Montgomery and Dyson, right? If those two didn't have lunch or whatever. >> Bingo. >> Then what the maybe Montgomery would never have done his while whatever you know thing that he did for the Remon Zeta hypothesis. >> Right. >> So but AI now can just do that. Right. >> Right. It can take all of the all of the papers of Dyson and all of the papers of Montgomery and then um you don't have to have lunch. You could have lunch but you

1:03:09can talk about other stuff. I 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 as the foundational material versus >> taking existing copyrighted artists work or creative work and then giving people the ability to just rip them off. This is a very different domain. It's open science and so just these are our different executions of the technology. >> Yeah. Yeah. Great point. Um so let's get into what this orchestra of agents did.

650 approaches and the E2 breakthrough

1:03:44>> Um out of 650 different mathematical approaches that were tested and discarded. There were two specific sub aents 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 aents 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 aent. >> So from that this is what I've gathered happened. It's kind of crazy. that you can peer into it, right? So here's the first thing. The first thing is the orchestrator. It tasks E2 with

1:04:27investigating something called the U pontrian pontrean index of the condition 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 igen values. Something like 33 to 153 negative IGEN values. And this whole thing is inspired by Bombiieri and his like speech his spiel about um one way to tackle the reman 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 values and things like that. Okay. Turns out the prompt's suggestion of doing this was wrong.

1:05:09>> Mhm. >> Okay. So the E2 sub aent 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 warp sync operator like hyperbolic sign operator in there that's like causing some artifact of negative values. So it goes back to the orchestrator and it says, "Yo, actually what you're telling me to do doesn't make any sense." like um you know you're wrong effectively >> it does not compute >> 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

1:05:50going 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 aent >> right >> okay now this sub agent it it it doesn't just like fail the task and give up. It independently starts executing this pivot. Um and then it finds like that the total number of positive igen values of whatever matrix is bounded by some number. Um it starts to compute that is 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 >> on the non-trivial zeros. on the

1:06:31non-trivial zero. We're now all from 41 whatever 41% to now we're at 50%. >> There was this singular insight by making the pivot to this dual use of inertia that that now when it cuz 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 one 4% 1%. >> Yeah. And so now it's at 50%. Right. Okay. It goes back to the orchestrator and the orchestrator expresses intense skepticism. [laughter] 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 postocs or something came back like I got to 50. Yeah. My first

1:07:12>> 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. [laughter] >> Right.

Claude creates hostile referees

1:07:21>> You just got here. >> Yeah. Yeah. So, um, it notes that like 50% would shatter the record. And it goes back to the human to Jared. And it's like, u, 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. >> Um, those referees, go ahead and critique it. >> Referee A, usually it's referee 2. In this case, I guess the AI just [laughter] doesn't have a doesn't have a a ranking, right? So, here, referee A discovers the technical flaw. And um 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

1:08:01as scathing as the stuff that we get in the emails. Like, you should just quit the job [laughter] and go work in a McDonald's, right? Like that's the stuff that we get in the emails. But I I wonder what referee A said to this sub agent E2. But it discovered a technical flaw regarding some in ill conditioning of the matrix and it corrected it. >> Mhm. [clears throat] >> 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. You know, a nice referee at least would like be like, this might be wrong. This is how you could change your experiment to test for this um >> confound and things like that. another referee independently verified that the calculations don't secretly import the

1:08:43Remon 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%. >> Mhm. >> It goes back to the human prompter to Jared >> and it's like, we got we got to 50%. [snorts] >> Jared then asks, "What would you do next?" And Claude responds, "Push it to

Pushing toward two-thirds

1:09:062/3. So then Jared's like, "All right, push it [laughter] to 2/3." Why did it say 2/3? Now the reason [clears throat] why it said 2/3 is because under the assumption of the remon hypothesis, Hugh Montgomery had reached a 2/3 bound by utilizing the fact that there's like some for whatever reason that he was doing and he had found that 2/3 of the of the zeros had some property. >> Mhm. >> Okay. So Claude starts thinking well maybe the two/3s is and human stuff I can now use I already used Bombiaris right >> now maybe instead of he was using like Claude was using something like the

1:09:47Koshy Schwarz inequality not important what we have to get into >> effectively it said I could push this to 2/3 because I know of Montgomery stuff >> there's a reference point that says there's a chance >> and so in a well- definfined space so let's let's go Let's go after it. Right. So again, the orchestrator confidently instructs this next E2 pairs >> to recover some lost efficiency in the Koshy Schwarz inequality. It's like this tool that we're using. It's basically the triangle inequality. But um 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. >> Mhm. >> 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

1:10:29me 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 remon zeta and it's going to do a bunch of math and I'm going to try to optimize this >> 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 prof profound insight into the non-frivial zeros and it went through and started writing up this this little lema, this fiveline lema that's going to help it get to 2/3. All of a sudden the thing crashes. [laughter] >> Okay. Okay. >> This is also documented in Claude's like

1:11:10report. All of a sudden this thing crashes. But by some miracle, Claude has dumped that fiveline proof >> into a a file >> into its memory somewhere. >> Yeah. It's into 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. >> It reads that file. It's like a f It's like oh >> that's right. >> That and and then it goes back and completes it. Isn't that cool? >> I love I love Oh [sighs] god. >> It's like a text file on the virtual disc. >> Right. Right. >> And it like dumped its memory. It I guess I mean I don't know why but it

1:11:50thought that okay this is important. I'm going to write this. >> Let me write this down. >> I'm going to write this down. And then it like crashed. >> Of course. God. And this is this is I think what's also interesting when you look at the chain of thought of these models um and we try to start doing this introspection is >> I understand this idea that there's a framing that these things don't have ingenuity and insight yeah in the way that humans do. Um, but even though it's doing a lot of work through having the ability to execute ex execute with code and ingest lots of information, >> it still has to make after those

1:12:32executions that are functional. >> It still has to try to choose a path >> and it intuitits 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. >> Yeah. >> It had to make that choice. >> Two, it was able to sort of figure out this matricy map, this this this building out these matrices to find an insight within it >> and where to even look within it. and

1:13:14then thought, "Let me make sure I write this down before I run it >> in case for whatever reason it crashes." >> Yeah. >> Yeah. >> You know, again, it might not have done it explicitly for the reason thinking it would crash. It might just been part of it normal process, but um I just find this chain of thought so fascinating because I think it illustrates this is well beyond next token prediction. >> Yeah. This is Yeah. >> You know what I mean? >> 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 2/3

Can the proof be machine-verified?

1:13:49>> and which can be formalized this this lean this lean software which can you can actually then have people look at it. >> Exactly. Yeah. So then so then I mean they actually had like the actual people look at it. So I mean um I think they got um Pog um Al Pogate the the same guy who did >> the uh 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. Um lean is you know human peer review is infallible but um you can completely illumin eliminate the possibility of logical hallucination by

1:14:30using this software. It's called Lean 4. That's the latest one. It's strictly typed machine checkable proof assistant. >> Mhm. effectively like truth statements and things like that. >> Yep. Yep. >> Um 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 >> for sure, >> right? Um but it seems that they have gotten a higher bound of the Remon hypothesis. 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

1:15:11models are attacking these there there are all these um sort of open problems and like there are sort of maybe two categories. will 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 uh 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. Um, and so it's not solving the hypothesis, but some of the insights that are

1:15:52gleaned might trigger somebody to have >> to have a completely different Oh, that's interesting. >> Out of left field insight. >> I could take that >> and go this way with it. >> We'll see. Who knows? We'll see. >> I just find a lot of this it's just it's happening. >> Yeah, it's happening as we speak. I mean, it's it's pretty crazy. a few interesting things that I came away with. >> Um, is the fact that first, as you said,

What this actually tells us about AI

1:16:19it'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. Um, second, rejecting human guidance, as you said. Um, that's pretty weird that it's able to just not only reject human guidance, but also um its own guidance with the orchestrator telling all these sub agents what to do. And um the funny thing is that absurdity of the interface. >> Mhm. >> Okay. The fact that this human all he had to do was say believe in yourself and that triggered a 31 million token

1:17:01compute dump. >> Right? It's weird. What does it mean to tell Claude to believe in itself? I don't know. >> Right. >> Um >> and and what does it mean that it responded in the way that you intended when you said that? >> Yeah. Like is that really something that it just learned from all of the language on the internet? I 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.

1:17:42I I 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? >> Yeah. Does it have an a sense of self? >> Yeah. >> Or an abstract construction of what the human meant by self? >> Yeah. Yeah. Yeah. There's like some part of its that's like, oh, he's talking about me. Like he's talking about the other weights over there that are doing that right? >> Or I don't know. Yeah, dude. It's really weird. Um, on a final note, all I'll do is read the last line of Claude's um, or I should say anthropics write up, which I'm sure was written by Claude. [laughter] Um, you know, so this might

1:18:24be Claude talking about itself, but I think it's I think it's very interesting in its 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 limit about the limitations of AI models. But after some encouraging prompts, it arrived at the result we've described. 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. >> The foothills of the singularity. Yeah,

AI capability, risk, and the hype cycle

1:19:03>> you know, this is a very controversial topic. People are have a very emotional reaction to AI right now for a lot of >> legitimate reasons and some of it is because of other 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. No. But just >> it is important to understand what these things are doing at the edge at the frontier. Um >> and what the implications can then mean from that because as we were just

1:19:44talking about I mean this is these are becoming powerful systems. >> Yeah. And I think underestimating their capability uh is um 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 uh 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 uh in subject matters and areas institutionally.

1:20:25Um, 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 that are pushing what people should believe one way or another about these things. >> That's all true, but if you just look at what it's >> look at what it doing, not what everyone's saying about what it's doing. >> Um, and it's it's I don't know how to feel about it. >> Yeah. I mean I I don't know how to feel about saying believe in yourself and it like actually does [laughter] 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

1:21:06the if a model is capable of orchestrating like this um and its goal is not something as benign as try to get further on the remon hypothesis suppose the the goal is much more sinister look at what it is capable love 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. 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

1:21:46and YouTube videos and not on Capitol Hill for example, >> which as we all know it's complicated. The social dynamics and the socioeconomic 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 um and because AI is a is owned and operated by that class that has that wealth and power who are continuing to

1:22:27try to acrue more of it. It becomes difficult for the everyday person to uh really care engage uh want it to be continuing uh when you know the rent is too damn high as that one guy would say. So very complex stuff. Thank you for walking through this because I now have a much better understanding yeah of why it's important. Uh for those of you who enjoyed uh we're going to 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 uh are talked about and give you a better lens into this. But we are going to move into

1:23:09a brief moment of me shilling the pod before we get into a last bit of fun here. Um so for those of you who are

FFP Nation and the Year Two survey

1:23:16listening, I should have said this at the beginning of the pod. This is definitely an episode that you should watch. Uh 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 Podcast. That will be dependent on when Spotify Creators releases those there. If you're viewing us on any of those podcast platforms, a fivestar 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. Uh people have opinions and so the comments help make sure that this discussion gets to more

1:23:58people. I want to give a big shout out to everyone who took our merch giveaway survey from our one-year anniversary episode. Uh we really appreciate the feedback. It's very fascinating to learn about the audience and where y'all are coming. There's a lot of folks who are grad students who are postocs. uh 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 ffpod.com/servey. We're going to select at first 10, but

1:24:39if 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 of 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. 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 going to end the pod with a little bit of fun, a little bit of relaxation. That first half, that was a brain bender. >> Yeah, >> that was that was Jimmy Neutron style.

1:25:20Uh, so we're going to have a little bit of fun here. Um,

The Summer Transfer Window for Scientists

1:25:24you know, and I'm going to start this next section talking about an individual who on the 1st of July, uh, 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 Singua in Beijing And Berkeley didn't get anything. Nothing. They put a little numeritus next to his name. >> Yeah. >> But for those who are sports fans, it's that time of year. It's transfer season.

1:26:04And 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. 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. Uh up the shells, we've had a great great business in this transfer market. Uh this is people who are moved, who are bought, clubs get robbed, a club sign someone new, the pundits will pontificate about what that

1:26:46means 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? 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 and January, but that depends on when grant funding is and they can happen kind of on a rolling basis. Um, but in this window, we looked at 42 confirmed moves in the 12 last 12

1:27:28months. 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. >> Wow, that's crazy. China has more than Yeah, it's like UK plus France plus Switzerland combined. >> So, if we look at our our ranking here, China's number one, >> right? >> 17. Uh, 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

1:28:10Italy and France are actually sharing one uh uh someone who's going to be traversing across two institutions. >> But I think this is interesting to view it in this way. >> Yeah. Um, and we're not going to 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 going to look at um four of what what would be considered, you know, uh, top top tier or high level, well-awwarded uh, folks that are people we don't want to lose at the club. And we're going to start off

1:28:50with someone

Omar Yaghi: Berkeley → Tsinghua

1:28:52listeners 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 Singa University in Beijing, already there July 2026 on the free Nobel Prize winner in chemistry in 2025. He was born in Aman, Jordan, moved to the US at 15. uh did his PhD at the University of Illinois, Urbana Champagne, Harvard postocck, then Arizona State, Michigan, UCLA, a great tour uh and in 2012 he was a treader chair at Berkeley plus spent some time at Lawrence Berkeley National Lab, the

1:29:33pioneer >> of metal organic frameworks, moths, the idea, 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, uh take a look. These can be used for things like carbon capture, gas storage, water from desert air. Uh the Nobel committee compared them to Hermione's handbag in Harry Potter. >> Yeah, >> very clever visual reference. He named the field reticular chemistry. I mean, what are your thoughts about this type of talent particularly from the

1:30:13state of California? >> Yeah. [clears throat] 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 Singua 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

1:30:54saying, from Arizona um from Arizona State University to then UCLA to Berkeley. 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 clout has gone up a whole lot. um 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 in he started his life as an

1:31:35international student. So he knows exactly how important it is for American science to hold that edge with the 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. >> A seasoned trophy winner. >> Yeah. >> 13 trophies. Albert Einstein World Award of Science in 2017, Wolf Prize in Chemistry in 2018, the Balsan Prize in 2024, just to name a few along with the Nobel. This is not a mid-table signing. This would be like uh Balandor winner in

1:32:172025, Usman 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. Yeah. Right now, right? like in terms of the how >> crazy um it would be to see happen. Um >> he has been an honorary professor at Singa since 2022. Uh he started full-time on July 3rd to lead their 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

1:32:58materials discovery 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 manufacturing and really own this idea of AI enabled in the physical world, it's not ideal. Um he did criticize the US for granting 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, Singa 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 bigname signing, Shirley

1:33:41Mang, 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

Shirley Meng: Chicago → Singapore

1:33:54now 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 and story, energy storage specifically. She served as the chief scientist for energy storage science at the US Department of Energy, DOE, uh Argon National Lab, and directed the DOE funded energy storage research

1:34:35alliance. 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 longerlasting for grid batteries safer and cheaper. Um grew up in China, got her first degree at NTU, so 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 the Zable chair at UC San Diego. Shout out California again. And in 2021, she was a professor at Chicago's Pritsker School

1:35:16for Molecular Engineering. 25 years. Uh here she's going back home to run the club. Uh and her big focus has been better, solid state, sodium ion, >> anodefree. uh in 2024 her lab reported the first anodef free uh sodium solidate battery which we've covered the concept of >> yeah in one of our previous episodes as well in terms of the importance >> I mean the the paper that we covered it was out in Juel if you remember and it was actually from her lab >> it was it was from her lab >> it was straight up from her lab um we were talking about why sodium is so important >> compared to lithium lithium is great but

1:35:56it's like >> hard to get at sodium is literally in seawater >> 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 and then deploy it like later at night and things like that. Like if we 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

1:36:38Chicago. another another great professor who um is now leaving us. >> Another seasoned trophy winner, Faraday Medal of the Royal Society of Chemistry. Oh, >> ECS battery division research award, ACS uh 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

1:37:21were explicit. Oh, >> the US turning away from decarbonization and the immigration rules squeezing international, including Chinese-born scientists. Uh, she's one of the ones

John Baker: NASA → France

1:37:32who's put that on the record. Singapore has handed her the keys to her old club plus a seuite 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 uh space flight center. He is moving to French National Center of Scientific Research in Tulus. Uh this happened actually late last year. Also on the free winner of the Goddard's highest space science

1:38:13honor, the John C. Lindsay Memorial Award in 2008, Kansas City uh uh uh sort of upbringing Truman State undergrad Penn State PhD in gravitational physics and then a posttock at the Albert Einstein Institute in Pott joined Gddard 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 that he's done has been a driver for the science case for LISA

1:38:55which is Europe's space-based detector. Another not midtable signing a key member here. So, as he moves to uh CNS uh CNRS, it's going to be the same miss 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 whe um Baker also mentioned political and social conditions in the US and his family safety. Uh he put that on the record as well. You're sensing a trend here around, you know, scientists are normally relatively diplomatic,

1:39:35especially relate to policy because that's who puts food on their table in terms of money and funding. [sighs and gasps] >> Wow. This one's Yeah. I mean, uh, he I can't believe he he straight up said that. This is quite rare for scientists to like just like to say it. Yeah. 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 neuroscience package deal to Birmingham

1:40:01single signing. This uh was a double signing. Uh and this one is also quite interesting. We have uh Ron Mang Gun and Tamara Swab uh both in the area of cognitive and language neuroscience. Currently out [clears throat] of UC Davis, California. Again, I promise that wasn't that wasn't bias. M >> 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. Um >> yeah, this is a two-body problem. >> This is a two problem cuz they're married scientists. Yeah,

1:40:43>> they are married. They've had decades at Davis Mang Gun, 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 Schwab, 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-inchief of cognition and fellow of the Association of Psychological Sciences. Across the

1:41:23two of them, they have a combined 14 trophies. We're talking about winners here. >> Yeah, >> 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

1:42:04more present >> well >> in Britain and Europe uh plus a real offer uh including the UK uh UK global talent fund support um you know it's for man Gunu is uh adding Birmingham as an opportunity as much as a reaction um this is again

The battle over U.S. science funding

1:42:27>> we're losing out to Europe something. >> Yeah, >> it's been a long time. >> It's been a long in science. It's been a very long time [laughter] >> in science. So, I just wanted to give some coverage around this because we did put a new section onto the website ffpod.com/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 tripleas um

1:43:07>> research and development funding uh data sets that have existed uh back through I believe it's u the 1970s and it's actually the the 1960s and what's interesting about this is um because it's R&D it includes defense which is the lion share of R&D spending >> there's been some es 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 time frame as well as being able to look at the 2026 active fiscal year budget that was

1:43:48implemented 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 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

1:44:30especially 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. Um yep >> 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

1:45:12best and brightest and how we can continue to stay competitive as we move into the future. 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 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, if I may say, Lester, you know,

1:45:53in the first year, we kind of we kind of shied away from from doing this kind of stuff. >> Yeah. >> 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 going to start caring a lot more.

Closing

1:46:10As always, I am your host, Lester Narre, joined by my co-host and our resident PhD. We made it under the 2hour 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 ffpod.com/servey. 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, we are very

1:46:50excited. 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 [music] next week.

1:47:18>> [music]