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What Claude Actually Did to the Riemann Hypothesis (EP 53) · From First Principles

0:00
It's weird.
0:01
What does it mean to tell Claude to believe in itself?
0:06
I don't know.
0:07
I think it's genuinely weird that you could talk to a neural network, which is just crunching matrices and numbers, tell it to believe in itself, and then it just locks in like LeBron James, and it spins up a hierarchical swarm of 60 autonomous sub-agents to try and tackle this problem.
0:27
So between them, they ran like 2,400 shell commands, hundreds of Python scripts, consumed 31 million output tokens, and they autonomously downloaded 54 academic papers from Archive.
0:41
Perhaps Claude, like many of us, underestimates the rate of AI progress.
0:48
Hello, Internet.
0:48
This is your captain speaking, Lester Nare, joined as always by my co-host and our resident PhD, Krishna Chowdhury.
0:58
We have a great episode today.
1:00
We have two segments.
1:01
We're going to cover the first one.
1:04
is going to be on the progress that Claude has been making on the Riemann hypothesis in this explosion of progress as AI tackles many of these open problems and complex areas in mathematics and really understanding what's going on there.
1:21
And we're going to end with a little bit of fun.
1:23
segment that I forced Krishna to do today, which is the summer transfer window for American scientists.
1:31
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.
1:43
As always, we are going to talk about the science from the ground up today because this is from First Principles.
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.
2:18
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 online.
2:29
some workaround, and then that was quickly followed by Claude's progress on the Riemann hypothesis.
2:37
And what we wanted to do today is kind of get a background to understand why this is so important, and then also weed through the hype versus the sort of this is nothing new reaction to that particular result.
2:56
Yeah.
2:56
Yeah, it's, I think, an incredible result.
3:00
The Riemann hypothesis is the most important unsolved problem in mathematics.
3:05
If you ask mathematicians, I think they will agree.
3:08
If you ask, you know, people adjacent to mathematics who have heard about the millennium problems and things like that,
3:15
This is the one, right?
3:17
To highlight that, let me give you an example.
3:19
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 fund Renaissance Technologies that has the medallion fund, which mysteriously just prints money.
3:34
I think their worst year ever was 20%.
3:37
That was their worst year in the past 30 years.
3:40
Usually they do like 100%.
3:41
Like 40%, 70%.
3:43
Unbelievable.
3:46
He was $30 billion rich.
3:48
Yeah.
3:49
He was worth $30 billion when he died.
3:51
And he was asked in this interview...
3:54
if you could trade your wealth for solving the Riemann hypothesis, would you do it?
3:59
And you could see him just perk up and he was like, oh, that's a good question.
4:04
And then he sort of stares off into the distance and starts fantasizing about solving the Riemann hypothesis.
4:09
And then later on he goes back to the scripted, oh, my life has been great.
4:16
You can't choose how your life ends up.
4:18
No regrets.
4:19
Hashtag no regrets.
4:20
But you could –
4:21
Behind it all, you could tell.
4:22
It was a yes.
4:23
This is a man who was like, if I could have done that, that would have been sick.
4:30
Right?
4:30
So given the aura around the Riemann hypothesis, it has become the gold standard around which AI's mathematical capability is judged.
4:43
And we see this actually in our own comments section.
4:46
Whenever we talk about AI doing math and doing crazy things, like when we covered the Jacobian conjecture not too long ago, a lot of people in the comments are just like, well, wake me up when they solve the Riemann hypothesis.
5:00
As if this unattainable thing is going to be how we judge AI, something that humans haven't been able to do for like 300, nah, 200 years.
5:14
It's made some progress.
5:15
It hasn't solved it.
5:16
And I want to be clear.
5:17
AI has not solved the Riemann hypothesis.
5:20
And by some measures, it is still as open as it once was.
5:24
But the fact that it has made progress is, I think, pretty crazy.
5:29
And particularly there's this interesting dichotomy between sort of two factions as it relates to AI progress.
5:36
There are there is one faction that says it fundamentally was not going to make any meaningful progress.
5:42
It matters.
5:43
And there's the other faction that's just saying it's a matter of time.
5:46
Mm hmm.
5:46
the truth is usually somewhere in the middle.
5:48
Yes.
5:48
And this seems like it's somewhere in the middle.
5:50
Exactly.
5:51
And the way that it's done it, I think, is very, very cool.
5:54
So for this episode, at least for this segment, I wanted to cover it because, one, I really like the Riemann hypothesis.
6:01
I think it's a very cool thing to think about.
6:04
And two, it's a really nice case study in how AI has progressed from the initial chatbots that we were thinking about, you know, back when ChatGPT came online and everybody...
6:15
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?
6:25
It's this really cool world that we're now living in, cool slash a little weird.
6:31
And that is central to this story.
6:33
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.
6:42
There's lack of definition, right?
6:43
So AI, a lot of the perception is that's pure LLM.
6:48
Yeah.
6:49
And the systems that are now being used, particularly inside the frontier labs, as well as for consumers, are...
6:57
have a variety of capabilities around the next token prediction piece that make it more than just predicting the next token.
7:05
Exactly.
7:05
Exactly.
7:06
And that's going to be the highlight of this story is what they're capable of.
7:10
So Anthropic actually put out a description of this finding on their website.
7:15
But what's really cool is they also shared Claude's own account of how it got there.
7:22
And they shared a full transcript of one of the agents that was instrumental in creating this proof.
7:29
Okay?
7:30
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…
7:39
to the Riemann hypothesis.
7:41
It's that Pixar movie Inside Out where you get to see inside the thought process.
7:46
Dude, yeah.
7:47
And I think it's just really cool how granular we can get here.
7:53
Okay?
7:53
So this is, I think, crazier than the Jacobian conjecture counterexample that we covered, I think, two or three weeks ago for two reasons.
8:02
The first reason, this is the Riemann hypothesis.
8:04
okay the jacobian conjecture people know about it i had kind of heard about it the riemann hypothesis i have been hearing about it since even before i knew anything about complex numbers it's like 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
8:32
Alpogay was a Princeton and Harvard trained mathematician.
8:35
This time we've got Jared
8:39
We've got his name right here.
8:41
Jared Sumner.
8:42
Okay.
8:43
He's just a software engineer.
8:44
Nothing against software engineers, but y'all aren't mathematicians.
8:47
Okay.
8:48
He kind of just prompted the new version of Claude that's unreleased.
8:52
Anthropic is very clear to say it's unreleased.
8:54
You know, they want their shareholders to be very, very confident of their progress.
8:59
So it's an unreleased version of Claude that he prompted Claude.
9:03
With just, hey, why don't you take a stab at the Riemann hypothesis?
9:07
And then when Claude was like, I'm not going to do that.
9:09
It's a 150-year-old problem that no one solved.
9:12
He came back with things like, believe in yourself.
9:16
A little bit of positive encouragement.
9:20
To Claude.
9:21
To an AI.
9:21
To an AI.
9:23
Actually, this is, okay.
9:25
And it's kind of weird that that kind of worked.
9:27
Yeah.
9:28
So we're going to get into some of that as well.
9:29
Okay.
9:30
Okay.
9:30
So first, let's go over the Riemann hypothesis.
9:33
What is it and why are people obsessed about it?
9:36
Right.
9:36
Okay.
9:37
I could dedicate several two-hour long videos to this thing.
9:41
And truth be told, to really understand it, you're going to have to go through
9:45
A book like Complex Analysis, again, my favorite book that everyone seems to also love as well.
9:52
Complex Analysis by Elias Stein.
9:54
If you go through this book, you've got to get to chapter 7 to really understand Riemann hypothesis.
9:59
I think you could skip the chapter on Fourier analysis because that doesn't really tie into this stuff.
10:04
But everything else you're going to really need to understand in order to believe some of the magic that I'm about to tell you.
10:11
But I'm going to go through kind of a Sparknotes version of...
10:13
of how you would begin to understand why the Riemann hypothesis is so important to solve.
10:19
Okay?
10:20
I'm going to begin with the basal problem.
10:23
This was first proposed by Pietro Mengoli in the 1600s.
10:29
The problem is pretty simple.
10:30
Okay?
10:31
You take an infinite series...
10:34
of reciprocals of squares, and you add it up.
10:37
So you get 1 over 1 plus 1 over 2 squared plus 1 over 3 squared plus 1 over 4 squared all the way up to infinity.
10:44
This is an infinite series.
10:47
So 1 plus 1 over 4 plus 1 over 9 plus 1 over 16 all the way.
10:50
What does that equal?
10:52
This was proposed in the 1600s.
10:54
It's named the Basel problem because the people who were really working on it were Leonard Euler and the Bernoulli family.
11:00
The Bernoulli family is the very famous family that is behind the Bernoulli numbers, the Bernoulli principle, which is how a lot of people mistakenly say airplanes fly.
11:11
It's not that simple.
11:12
I mean, it has something to do with it, but it's not that simple.
11:15
In any case, the Bernoulli family tried and they failed.
11:18
This is a family of mathematicians.
11:20
Leonard Euler...
11:21
tried honestly might have been his first time that he just like solved it okay and he said this whole thing is equal to pi squared over six okay so one plus one over four plus one over nine plus one over 16 all the way down that's going to equal pi squared over six the way he did it was kind of hacky um and
11:41
Like, it was only proven to be completely correct by Weierstrauss like 100 years later.
11:46
It was a little hand-wavy.
11:47
It was a little hand-wavy, but he got the right answer.
11:50
Like, so many things that he did.
11:51
Okay?
11:52
Because his intuition was just like insane.
11:54
Right?
11:55
Okay, fine.
11:55
So that's impressive, right?
11:57
But that's not his main thing that Euler did for...
12:02
The basal problem.
12:03
So the basal problem is only 1 plus 1 over 4 plus 1 over 9 all the way over, right?
12:09
Euler, in his text, Various Observations of Infinite Series, he generalizes to something called the zeta function.
12:16
Okay?
12:17
He actually invented the zeta function, which is, what if I consider not just 1 over...
12:25
squares but 1 over cubes or 1 over stuff to the fourth power, things like that.
12:31
So he creates this function called the zeta function and crucially he shows that that sum of 1 over all of the squares and things like that of the natural numbers is equal to a product
12:44
Over the primes.
12:46
Oh, interesting.
12:46
Okay.
12:47
So we took something that was discreet as an initial problem set, generalized it, and came up with it and had a rule that related to that generalization.
12:55
Very good.
12:55
Yeah.
12:56
The generalization was I take one over the cubes or whatever power that I want, and the
13:03
Little trick that he did was show that the sum could be turned into an infinite product over primes.
13:09
We're going to get into exactly how that works.
13:12
Because I think this is a really neat way of seeing why the zeta function is so important in mathematics.
13:19
It has to do with primes.
13:21
And everywhere you look, you'll see that the Riemann hypothesis has to do with the distribution of primes.
13:26
Understanding just this will give you kind of a sense of why that is the case.
13:30
Why are primes related to this infinite product?
13:34
So that's what we're going to do right now.
13:37
Here's where we're going to start.
13:38
We're going to start with the zeta function.
13:39
Let's consider instead of squares.
13:43
So in this case, the zeta function is for any arbitrary S. Yes.
13:46
But the S is usually two.
13:48
That'll give you the basal problem of one plus one over fourth, one plus one ninth plus one sixteenth.
13:53
Instead, let's just consider S equals one.
13:56
Okay.
13:56
This is called a harmonic series.
13:58
It's just one plus all of the fractions added up.
14:00
Yep.
14:01
This thing crucially does not converge.
14:03
Okay?
14:03
This thing goes to infinity.
14:04
So there's the harmonic series, which is when the power is one.
14:07
There's the basal problem when the power is two.
14:09
And then the zeta is for any power greater than one.
14:13
Okay?
14:13
That's the zeta series.
14:15
Okay?
14:16
So we're going to consider the harmonic series, which is when the power is one.
14:19
So we're just getting reciprocals.
14:21
Okay?
14:22
We're going to try to understand this a little bit.
14:24
Now, we're going to need a second tool in our toolbox, and that has to do with the equation for a geometric series.
14:31
A geometric series is a power series.
14:34
It's something like, suppose you were to add 1 plus 1 half, but not instead of 1 third, we say plus 1 fourth, plus 1 eighth.
14:43
So powers of 2, the same power over and over again.
14:47
This is not now...
14:48
You're not changing the number.
14:50
Yes.
14:51
You're keeping the number the same, but the power is going up.
14:55
The variable moves from being in the power spot to being in the exponent.
15:00
In the exponent spot into the power.
15:03
Yeah, exactly.
15:04
So now this is a geometric series is what it's called.
15:07
Because there's a common ratio between each of the terms, right?
15:10
You're halving each term.
15:12
and then you're adding it up.
15:13
So you get a one, that's a whole pizza, let's say.
15:16
Then a half, that's half a pizza, plus a quarter of a pizza, plus an eighth, plus a 16th, plus a 32th, or whatever it's called.
15:23
32th.
15:24
As you add it all up, you're going to get two.
15:26
You can see, right?
15:27
The one is the first.
15:29
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.
15:35
The way, and this is something that you learn in high school calculus, is how to sum up a geometric series
15:41
You take the common fraction...
15:43
which is a half, sorry, the common ratio, which is a half, because you're multiplying a half every time you add a new thing.
15:50
And you take the first number.
15:52
So the first number in this case is 1.
15:54
You put that on the top.
15:55
1 divided by 1 minus the common ratio.
15:57
So in this case, it would be 1 divided by 1 minus 1.5 to get you 2.
16:01
And that's how you sum up a geometric series.
16:04
Okay?
16:04
So we're going to require this toolbox.
16:06
All right?
16:07
Now that we've understood that, this was Euler's genius.
16:10
He said, consider...
16:12
The geometric series for primes.
16:16
Okay?
16:16
We had just seen a geometric series for two.
16:19
A geometric series for three would look like one plus one-third plus one-ninth plus one-twenty-seventh because those are the powers of three.
16:27
We don't do four because four is not a prime.
16:31
We look at the geometric series for five, which is one plus one-fifth plus one-twenty-fifth plus one-twenty-fifth and then there'll be six-twenty-five, so on and so forth.
16:40
All of those things are going to equal 1 divided by 1 minus that common ratio.
16:45
So it's going to be 1 over 1 minus 1 over 2.
16:47
Over there, that's going to be 1 over 1 minus 1 third.
16:50
So that'll give you 2 thirds, actually, over there.
16:53
It's 3 halves, sorry.
16:56
And so this is the geometric series for...
17:01
powers of primes.
17:04
Okay?
17:05
Euler said, let's take a look at this and see if we can do something.
17:08
Alright?
17:09
If we multiply all of these series together, so we multiply the infinite sum up there, the infinite sum up there, the infinite sum up there, I'm going to really get
17:23
a multiplication of those individual representations.
17:26
Right?
17:27
Those individual answers.
17:28
And now you're already starting to see that second-hand side of Euler's product formula.
17:33
Yeah, I was going to say that it's the same.
17:35
Yeah, there's the through line.
17:36
Yeah, there's the through line.
17:38
The giant pi, that means multiply.
17:40
The giant sigma means sum.
17:43
The giant pi means multiply.
17:44
And it's multiplying over the primes.
17:47
Okay?
17:48
For whatever power.
17:49
In this case, the power is s equals 1.
17:50
Mm-hmm.
17:51
right?
17:52
And so now we're seeing the right-hand side.
17:54
Yes.
17:55
Now, that's fine.
17:57
Here's the real genius of Euler.
17:58
Why is he even doing this in the first place?
18:00
Okay, this seems like a lot of stuff to do without the punchline.
18:05
Here is the punchline.
18:06
What about the left-hand side?
18:09
The left-hand side
18:11
As I said, it's a bunch of primes multiplied together, right?
18:14
You've got 1 plus 1 half plus 1 fourth, the powers of 1 half, the powers of 1 third, the powers of 1 fifth.
18:20
You'll get the power of 1 seventh later, right?
18:23
If I multiply those out and I do the FOIL method, if you remember, like you have to distribute.
18:28
Each term has to multiply to each term, right?
18:32
So if I multiply that out, the 1 can be multiplied to all the other ones to get a 1, right?
18:38
Then the 1 half from the first one can be multiplied to all the other ones to get a 1 half, right?
18:46
Similarly, the 1 fourth can be multiplied to all the other ones to get a 1 fourth.
18:49
Mm-hmm.
18:50
The one-third will be there.
18:51
The one-fifth will be there.
18:52
I could also get a one-sixth because the one-half times the one-third multiplied by all the other ones is going to give me the one-sixth.
18:59
I see.
18:59
The one-eighth is going to come from the one-half, right?
19:02
The one-ninth is going to come from the one-third.
19:04
And then the one-twelfth can come from one-fourth multiplied by one-third.
19:08
Every single number is going to be represented here.
19:12
Right?
19:13
in that infinite sum, because primes make up the numbers.
19:19
Right.
19:19
Right?
19:19
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
19:28
into multiplications of primes, products of primes that are unique.
19:33
There's only one way to do it.
19:35
And there's only one way to choose all of the different terms in that sequence.
19:41
And now you can notice, right?
19:42
What are the ones that are missing?
19:44
One seventh is missing because I haven't included that in this, right?
19:48
Similarly, one eleventh is missing, but that's a prime.
19:51
That'll be another one on its own.
19:52
Okay.
19:55
1 13th is missing, but also 1 14th is missing.
19:59
Because 1 14th would be the 1 7th multiplied by the 1 half multiplied by all the other ones.
20:04
Right?
20:04
So all of the other composite numbers are not there.
20:07
And the primes that I haven't included aren't there.
20:09
Because they're just further down.
20:10
Yeah, they're just further down.
20:11
I haven't included it, right?
20:13
If I added the 1 7th thing, then the 1 14th would show up and the 1 7th would show up.
20:17
But the 1 11th would not show up.
20:19
Right?
20:19
Because that would require another 1.
20:21
As we expand the top row we have here, it will fill in these gaps that are currently present.
20:26
And you could do this infinitely.
20:28
Yes, and you can do this infinitely.
20:29
And the key is you will get every single fraction ever.
20:31
That's incredible.
20:32
Okay.
20:33
Right?
20:33
Yeah, yeah, yeah.
20:34
This was Euler's genius.
20:36
And that's how he's showing this is called Euler's product formula.
20:39
And it's kind of crazy that he set out to just solve the basal problem.
20:45
And then he's like, oh, by the way, I also noticed this.
20:49
Just have this one on the frame.
20:50
Yeah, yeah.
20:51
So here, you've already seen now, the zeta function is on the left.
20:55
That's the sum of the one over all the natural numbers, right?
21:00
He has tied this now to a product over primes.
21:03
And now you can imagine if I want to do one, the zeta function to the second power, all I have to do is put the primes to the second power.
21:11
Right, right.
21:13
Yes, yes.
21:14
And that's why the S is a common exponent.
21:17
Exponent across both.
21:20
Oh, that's quite nice.
21:21
That's so elegant.
21:23
Right?
21:23
And this already shows you how the primes are so intertwined with the zeta function.
21:29
Yeah.
21:29
Right?
21:29
They are one and the same.
21:30
And so part of what we're building here...
21:33
is the connection between those two.
21:34
Yes.
21:35
I'm trying to justify why there's so much hype about this.
21:41
It's because the Riemann zeta function has to do with primes in a very integral way.
21:46
It's actually encoding the same information.
21:49
On one side, there's a product of primes.
21:52
On the other side, there's a sum over every natural number.
21:55
This is really good.
21:56
It's kind of cool.
21:57
This is really good.
21:58
Yeah, and this was in the 1730s.
22:02
No Wi-Fi then.
22:03
No.
22:05
Now, 1792, there's a teenage Carl Friedrich Gauss.
22:11
Gauss.
22:11
Gauss is another amazing mathematician, perhaps second or third only to Euler.
22:17
And maybe Riemann, actually.
22:20
He starts counting primes when he was a teenager.
22:23
Because I guess that's what you do when you're a full autist in Germany.
22:29
In 1792.
22:29
Yeah.
22:30
So he starts counting the number of primes by hand.
22:35
And he notices that the probability that a certain number is going to be prime is related to the log of that number.
22:42
It's basically 1 over the log.
22:44
Like if I go up to like a 10,000, right?
22:49
The probability that numbers around 10,000 obviously is not prime.
22:52
But like any given number around there is prime is 1 divided by the log of 10,000, log base e. Okay.
22:58
Okay.
22:58
Okay.
22:59
So from there, he realizes that there's this prime counting function where I count how many primes before a certain number, and that's related to n divided by log n. Okay.
23:10
There in the blue, you see a pi of n. That's the prime counting function.
23:14
Right.
23:14
n over log n is in red, and you can see that n over log n kind of undershoots.
23:18
He also realizes that the logarithmic integral, meaning the integral of the log, which is li of n,
23:25
That's kind of an overshoot most of the time.
23:28
Turns out it's not all the time.
23:29
If you go to 10 to the 10 to the 35, then prime numbers go above the log integral.
23:35
But he actually thought that perhaps the integral function is always above.
23:40
But he didn't know that 10 to the 10 to the 35... Couldn't get that high.
23:46
If you do it by hand and you see a pretty good... You're like, yeah, it's probably...
23:51
I get it.
23:52
So...
23:54
This becomes now the prime number theorem, which is that the number of primes before a certain level, the number of primes before a certain number is related to n divided by log n. Okay?
24:09
Which is our red here.
24:10
Yeah, yeah.
24:11
It's like asymptotic to N divided by, like it has the same shape.
24:14
Yes.
24:14
Okay, it's not like going to diverge like crazy.
24:16
Right.
24:17
It's always going to stay close in some sense.
24:19
Right?
24:20
Now, enter Bernard Riemann.
24:23
Riemann.
24:23
Okay.
24:24
Riemann.
24:25
Riemann at the University of Berlin.
24:26
He's got an 1859 article on the number of primes less than a given magnitude.
24:32
This is the only paper that he ever wrote in analytic number theory, and it is probably his most influential.
24:40
He was mostly worried about geometry and things like that, like Riemannian manifolds and stuff.
24:46
That's the stuff that you get into with general relativity.
24:49
But this is what he wrote about the prime number theorem.
24:52
And ever since then, it has been named the Riemann zeta function.
24:58
Okay, because there's a joke in mathematics.
25:02
You always name something after the second person who discovered it because the first person is always Euler.
25:08
And you can't name everything after Euler.
25:11
You know?
25:12
So some people call it the Riemann-Euler Zeta function, but let's just give it to Riemann.
25:19
Yeah, right, right, right.
25:22
We already know who the goat is.
25:23
Yeah, yeah, exactly.
25:24
We already know who the goat is.
25:25
Exactly.
25:26
So he actually did some amazing work on the Riemann, on the Zeta function.
25:31
What he did was extend Euler's definition to the complex variables.
25:36
What I mean by that is, you know, usually...
25:42
we think about that zeta function as like one over the squares, like one over the squares or one over the cubes.
25:49
The exponent is a real number.
25:51
You can even imagine one over square roots and things like that, right?
25:56
Riemann said, no, what if we took one over stuff to the power of a complex number?
26:00
So it's like one plus i or two plus i.
26:03
What would that do?
26:04
And it turns out there's a very well-defined way of talking about that.
26:08
So here's how it works with complex numbers.
26:10
Okay?
26:11
Here, I'm going to borrow a lot of these visuals from 3Blue1Brown.
26:16
And a lot of our viewers are going to recognize it as such, because he has an absolutely amazing video on YouTube about the Riemann zeta function.
26:25
And he's the goat of mathematical visuals.
26:30
Big shout out.
26:31
Big shout out.
26:32
So here's what happens if I take Riemann zeta to the 2 plus i.
26:36
So in this case, the primary thing we're changing is the exponent in the denominator.
26:42
Yeah, because that's how the Riemann zeta function is defined.
26:46
Okay.
26:46
The function is defined as the sum over 1 over the natural numbers to some power.
26:51
You choose the power.
26:52
Yeah.
26:52
Right?
26:53
And so we're just saying we've now chose the power to be 2 plus i instead of 2.
26:57
If it was just 2, it would be the basal problem again.
27:00
And then I'd get pi squared over 6.
27:01
Yep.
27:02
But because it's 2 plus i, if it's just 2, then every single term, I'm just moving along the number line.
27:09
Mm-hmm.
27:09
For 1, I'd get 1 plus 1 fourth plus 1 ninth, and I'd go a little bit closer to pi squared over 6.
27:16
And I'd just be moving on the number line.
27:17
With complex exponents, now I'm rotating on the complex plane.
27:22
The complex plane is defined as the real numbers on the x-axis, the imaginary numbers on the y-axis.
27:27
So that's why 2 plus i is 2 on the x, 1 on the y.
27:31
2 plus 2i would mean 2 on the x, 2 on the y.
27:34
Right.
27:35
So that's how complex numbers are defined.
27:37
And the way that you do an exponent is you actually start rotating where that's happening.
27:41
Now, 1 to the power of anything is just 1.
27:44
So that's why you go 1 first.
27:45
But then 1 over 2 to the power of 2 plus i, that's going to—
27:49
curve a little bit downward, and then it's going to curve a little bit downward for the 3, and it's going to spiral to that point.
27:55
So the Riemann zeta function takes as input 2 plus i and maps it to that point down there, which is like 1.15 minus 0.44i.
28:05
Okay?
28:07
That's how it works in the complex plane.
28:09
Yes.
28:10
Now, turns out, as long as the real part is greater than 1,
28:18
your sum is going to be fine.
28:20
And we've got an animation here, courtesy of 3blue1brown again.
28:24
As I change the input, the spiral is going to change, but it's going to be well-defined.
28:30
It's going to end up somewhere.
28:31
Okay?
28:32
So here I'm taking my input, which is the yellow dot, and the output of the Riemann zeta function is where that spiral ends up.
28:38
Okay?
28:39
Perfectly well-defined function.
28:40
Really nice.
28:41
Okay?
28:41
Okay?
28:43
Let's see what this function does to the grid lines.
28:46
Here we've seen individual points getting mapped.
28:49
Now let's see what this function does to the grid lines.
28:51
As long as the power, the real part, is greater than 1, this is how the grid lines map.
28:59
Now, why do we need the powers to be greater than 1 for the real?
29:02
That's because, imagine if I did the power equals 0.
29:09
then I just have 1 plus 1 plus 1 plus 1.
29:11
That's going to diverge to infinity.
29:13
It's not well defined, at least in the series representation.
29:17
But if my powers are greater than 1, then each fraction gets smaller and smaller, and so my spiral actually converges somewhere.
29:24
That's the idea.
29:25
Okay, got it.
29:26
Okay?
29:27
And so this is what's happening.
29:28
All of the points on the right are getting mapped to these points that are in sort of half of the plane.
29:37
Okay.
29:37
But now this kind of begs the question, what about the other half?
29:40
Right.
29:41
Right?
29:41
Right.
29:42
It seems like I could just draw some lines.
29:47
Right?
29:47
I could just continue the function.
29:51
Right.
29:51
Right?
29:51
Like, I could make up...
29:54
how the rest of the function would behave, given I know so well how this function behaves over here.
29:59
And specifically you're referencing to the left of the origin of the y-axis.
30:05
Yeah, exactly.
30:06
There's a little part on the right of the y-axis too that this thing is not reaching, right?
30:12
It's like about half in there.
30:14
But my point is, the parts where we can use the series give me this really nice functional map.
30:23
Okay?
30:25
And what if the idea is, can't we just extend what we're already seeing?
30:29
Exactly.
30:30
That's what Riman said.
30:31
He's like, can I just extend what I'm already seeing?
30:34
I know that it doesn't make sense, right?
30:37
In terms of like, maybe in terms of like, yeah, you tell me to plug in zero, zero.
30:41
It's not going to make sense because I'm just adding one, one, one, one, one.
30:44
But in terms of what I'm seeing over here, it kind of makes sense, right?
30:49
There's a world in which this could make sense.
30:52
That's what Riemann's thinking to himself as he's looking at this thing, right?
30:57
And so that's exactly what he does.
30:59
He says, if I don't rely on the pesky series representation...
31:06
which is this 1 over a power plus 2 over a power plus 3 over a power.
31:10
Instead, I make up another function that behaves the same way on the right-hand side
31:17
And it looks like that.
31:18
2 to that power times pi to the power minus 1 times sine of something times the gamma function of something times the Riemann zeta of the flip hand side.
31:29
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 minus s, multiply by all this other stuff, and I'll be fine.
31:42
It'll be fine.
31:43
It'll be fine because, and this is the magic of complex analysis.
31:47
It turns out there's a very good reason why complex functions do this.
31:52
I can just flip it on the other side of the board.
31:54
Yeah, and make a little bit of correction.
31:57
But all of the grid lines are going to be smooth now.
32:00
The whole thing is going to be differentiable.
32:02
And the fact of the matter is this is called analytic continuation.
32:06
Okay.
32:06
Okay?
32:07
What it means is I can take a function that is well-behaved in one part,
32:12
And if I want to extend it, there is only one way to extend it.
32:16
That's the key.
32:17
This is not an arbitrary extension.
32:21
What complex analysis shows is there is only one way to do it.
32:24
So if you found the way, that's it.
32:26
That's it.
32:27
That's it.
32:27
You have found the continuation of that function.
32:31
There is only one way to do it.
32:33
Okay?
32:33
And so this isn't like...
32:36
It is imaginary.
32:37
I mean, it's in complex variables.
32:38
So there's an imaginary unit in it, right?
32:40
But there's only one way to imagine it.
32:43
In this imaginary space.
32:44
Yes.
32:45
To create everything and make sure that everything is sound and logically closed and
32:52
There's only one way to imagine.
32:53
There's a, the logical consistency only has one answer.
32:56
Exactly.
32:57
And so that is the extension.
32:59
That is the analytic continuation of a meromorphic function outside of its domain where it's like nicely well-behaved.
33:06
Now it's everywhere.
33:08
Okay.
33:09
That's the key thing that Riemann discovers.
33:12
Okay.
33:13
Side note.
33:13
Okay.
33:15
There was a lot of hype over the sum 1 plus 2 plus 3 plus 4 plus 5 equaling negative 1 twelfth.
33:22
Now you can finally kind of understand why that is the case.
33:25
If I plug in negative 1 to the Riemann zeta function, right, then that's going to be 1 plus 1 over 2 to the negative 1.
33:33
But 2 to the negative 1 means I just flip.
33:34
So I get 1 plus 2 plus 1 over 3 to the negative 1.
33:37
So I just flip.
33:39
Right?
33:39
So that's just 1 plus 2 plus 3 plus 4.
33:41
Right?
33:42
It doesn't make sense on its own.
33:44
But if you use the Riemann zeta function, you analytically continue it outside of its domain to the negative numbers, and then you plug in negative 1 to that functional thing that I showed you earlier with the signs and the gamma function and everything.
33:57
You plug in negative 1 to that, you're going to get negative 112.
34:00
And it turns out that if someone were to put a gun to your head and say...
34:06
define this series as a number.
34:09
You can't say infinity.
34:11
You have to pick a number.
34:12
The only correct answer is negative 112.
34:15
Okay?
34:16
To show you another reason why this is the case and why Srinivasa Ramanujan, who's a big hero of mine, is on this, is Ramanujan did not know anything about complex numbers.
34:26
Very little.
34:27
When he was in India.
34:28
And he was reading like a trig book and discovering like all of math.
34:31
Okay?
34:32
Um...
34:34
He wrote a letter to Hardy, who was a big mathematician in Cambridge, asking for help to go to Cambridge and work under him.
34:44
One of the things that he wrote down was this 1 plus 2 plus 3 plus 4 dot dot dot equals negative 112.
34:49
And he said, I know you'll probably think of putting me into an insane asylum, but look, I have techniques.
34:56
that assign numbers to divergent series.
34:59
And I got negative one 12th.
35:01
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 get to negative one 12th.
35:12
And this, this was a known thing out there because Hardy is after Riemann, right?
35:16
So there you can just plug in negative one and you know that.
35:20
Ramanujan had no idea about Riman, about the Zeta function.
35:23
He figured out a really hacky way to assign it a number, which is not exactly clean.
35:30
Again, hacky.
35:31
Now we know there's reasons why it works.
35:34
But...
35:35
He also came up with the same number, negative 112.
35:38
Just goes to show you how closed math is.
35:41
He was able to intuit from a totally separate pathway, but it still converged at the same destination, regardless of his lack of not going down the road already traveled.
35:50
Yeah, yeah.
35:51
He went a complete, but he found the same destination.
35:53
It's kind of cool, right?
35:54
And that's what I mean by like, if someone were to put a gun to your head and say, define what this is, and it can't be infinity,
36:01
The only correct answer is negative 112.
36:02
Okay?
36:03
That's so fascinating.
36:04
The real correct answer is really infinity, right?
36:07
But...
36:08
If you were to really put me through it, it would have to be negative 112, right?
36:13
And the Riemann zeta of negative 1 is negative 112, right?
36:16
It just – like I don't think it means anything to sum up the natural numbers.
36:20
But like actually like string theorists use that all the time to like deal with their infinities.
36:26
They're like, ah, I think negative 112.
36:28
It's like, okay.
36:29
The number of dimensions is the denominator.
36:32
have some party trick where they do that.
36:36
I just thought that's a really cool little aside.
36:38
No, no.
36:39
And that's, I think it's, because it's part of what we're doing is we're connecting all of these component parts, you know, mathematics being almost this closed system, right, that has these bounded box and a level of, like, the logical consistency
36:55
that exists through it, even if you come from different disciplines within that subject matter, and seeing how those different disciplines exist.
37:05
address the same question or try to approach the same questions.
37:10
Yeah.
37:10
It's really fascinating.
37:12
Yeah.
37:12
I mean, it's really cool.
37:13
I mean, the basal problem is another example of that.
37:15
Let me just give you another little anecdote from my personal life.
37:20
I told you about Math 215 at Princeton University, the first semester of math that I did.
37:24
One of the problems on the final was to calculate the basal problem.
37:28
Okay.
37:28
It's like, great.
37:29
Okay.
37:29
Euler did it.
37:30
Yeah.
37:30
You can do it for a final, you freshmen.
37:32
And, uh, this, this was Peter Sarnak.
37:34
He was just a maniac, but, um, he actually, in the problem statement, he, he gave us enough information and we used Fourier series and a clever trick with Fourier series to show,
37:45
and i got that problem right it was a it was a great problem um but yeah you can use fourier series to show the pi squared over six you can use oiler's hack you can also use um taylor series a lot of times people use like um the taylor series for sign and things like that so there's so many different ways but you will always arrive at pi squared over six just the beauty of mathematics is awesome you gotta love it
38:09
Okay, so now let's finally talk about the Riemann hypothesis.
38:13
We have talked about the Riemann zeta function.
38:15
Now let's talk about the hypothesis.
38:17
Which is sort of the next step from starting with the basal problem that let us have a fundamental understanding.
38:25
Yeah, we generalized it to the zeta function.
38:27
To the zeta function.
38:28
Yeah.
38:29
And now... Now, we're finally going to get to the Riemann hypothesis.
38:33
Right.
38:33
Okay?
38:34
Now, the Riemann zeta function has been used to prove that bound of primes.
38:39
Yes.
38:39
The n over log n. That's actually chapter 7 of this book.
38:43
Which was the red line in the previous show.
38:45
Yes, yeah.
38:45
It's how do primes grow?
38:46
They grow like n divided by log n. The distribution there and how close I get to n over log n...
38:53
has to do with where the zeros are for the zeta function.
38:58
What that means is, what inputs map to zero?
39:02
If I plug in that input into the exponent of the, you know, one over the thingy, or the functional form...
39:11
Yes.
39:12
do I get zero as the output?
39:14
Okay, there are two types of zeros.
39:16
There are the trivial zeros, which happen at all of the negative even numbers.
39:20
So if I put in negative two, I get zero.
39:24
If I put in negative one, I get that negative one twelfth.
39:27
Yes.
39:28
Okay, but if I put in negative two, I get zero.
39:30
If I put in negative four, I get zero.
39:32
Those are the trivial zeros.
39:34
Okay.
39:34
And that just has to do with the fact that there's a sign in there, and sign goes like this.
39:38
Okay, so it's trivial.
39:40
It comes back to...
39:41
Yeah, it's like the functional form has a sign in there, and so the sign, because the sign oscillates, you're going to get zeros.
39:46
At a consistent interval.
39:48
Yeah, yeah, and that's, okay, fine.
39:49
Fine.
39:51
Now there are the non-trivial zeros.
39:53
Oh, my favorite, non-trivial.
39:55
Yes, and they are non-trivial for a reason, because we don't know where they are.
39:59
And that is the Riemann hypothesis.
40:02
Where are they?
40:03
Non-trivial, because everything else is well-defined at this point.
40:06
Yeah.
40:07
Except for the non-trivial zeros.
40:08
The non-trivial zeros, we know that there's a bunch of more zeros.
40:11
Yeah.
40:11
But we don't know where they are.
40:12
Okay, Riemann proved that they are inside of that strip.
40:16
He said they're somewhere in here.
40:18
Between zero and one on the real axis.
40:20
That's called a critical strip.
40:22
So Riemann proved that.
40:23
And then in his paper, he writes, probably they're on the critical line.
40:31
They're on the one half, right in the middle of that strip.
40:33
Okay.
40:33
Okay.
40:34
That's what he writes.
40:35
He's like, probably...
40:37
But he's like, I don't want to prove it because for my purposes, it's not necessary.
40:44
But it seems like an interesting problem.
40:46
That's why it's called the hypothesis because he hypothesized it.
40:50
In his paper.
40:51
And now we're all chasing it.
40:53
There's a search space that we believe these non-trivial zeros exist in.
40:57
Yeah.
40:57
And within that context, there is a discrete line.
41:00
Right in the middle.
41:01
Right in the middle, which is the sort of most likely place, the hypothesis of within this search space.
41:08
Yeah.
41:08
This is where you should look.
41:09
This is where you should look.
41:10
And if we can prove that they all lie only on that line and nowhere else, they're not in like some fudged like part around it.
41:21
Every single non-trivial zero is on that line.
41:25
That is the Riemann hypothesis.
41:26
We want to resolve it similarly to how cleanly the trivial zeros resolve or the negative 1 12th resolve.
41:31
Yeah, yeah.
41:32
Where it's always true.
41:33
Yes, yes.
41:34
It's always true that every single zero is going to be on that line.
41:37
Okay.
41:37
Okay.
41:38
That has been the quest.
41:41
Yes.
41:41
Okay.
41:41
Okay.
41:42
That seems so easy.
41:43
Yeah.
41:45
And...
41:47
If we know, why would we care, right?
41:48
Well, if we know where the zeros are for the Riemann hypothesis, then we could prescribe an exact form of the prime number counting theorem.
42:01
Okay?
42:01
And this next animation shows that.
42:03
So the jagged line, that's the prime number counting theorem.
42:08
And that has to be jagged, right?
42:09
Because at some point it's going to increment by one.
42:12
At two, it goes up by one.
42:13
At three, it goes up by one.
42:15
At five, it goes up by one.
42:16
So all the vertical increases are that.
42:18
Are just like, oh, here's another prime.
42:20
Here's another prime.
42:22
The blue is as you start incorporating the non-trivial zeros into your expression for what that line should be,
42:30
The blue gets closer and closer to that line.
42:32
So that's what we mean.
42:33
We mean that if every single zero is known, we can exactly prescribe the form of that line.
42:41
So this is the idea that this initial plot of where these primes are going to be as we get higher, it's fuzzy.
42:51
Yeah.
42:51
Because we don't know where the non-trivial zeros are.
42:54
Yes.
42:55
And if we knew where the non-trivial zeros are, it would resolve, the resolution would effectively be we could then do the primes.
43:01
Yeah, we could predict every... Every primes all the way up.
43:03
Yeah, yeah, yeah.
43:04
That seems like a pretty big deal.
43:05
Yeah.
43:06
We could predict exactly the prime number function, the counting function.
43:11
That's...
43:12
Okay.
43:12
Right?
43:13
Yeah.
43:14
And the primes are everything.
43:15
Yeah.
43:15
So I get now why this is so important.
43:18
Yes.
43:18
Yes.
43:19
It's basically like a...
43:20
It's sort of like...
43:24
The location of these non-trivial zeros basically creates the last piece of the map to be able to then traverse wherever we want to.
43:31
You'd still have to sum up to all of the infinity of zeros, but at least I got a procedure.
43:36
To do so.
43:37
And it just becomes a process.
43:39
Yeah, and however accurate I need to be, I need to just find that many zeros.
43:44
And with the amount of compute we have nowadays...
43:46
Yeah, perhaps we could do it, right?
43:48
If we could figure out a constructive way to find the zeros, that'd be crazy, right?
43:52
If somebody proved a way to construct every single zero and show that there's no others, that would be crazy.
43:59
Okay, yeah.
44:01
And that's when people say we're trying to solve...
44:04
The Riemann hypothesis.
44:05
No, when people are saying they're trying to solve the Riemann hypothesis, they're just trying to show that every zero is here.
44:10
It doesn't have to be a constructive proof.
44:12
You could just show that there's no other zeros anywhere else.
44:14
Okay.
44:15
You know what I mean?
44:16
Yes, yes.
44:16
There's a difference between constructing it and just showing that none other exists.
44:20
So they all happen to be on this line.
44:21
And even that would be good enough.
44:24
So there's sort of two levels of success is not the right word, but goals to reach.
44:29
One is just proving that everything's on the critical line.
44:32
Yeah, and that's the Riemann hypothesis.
44:33
And it can't be anywhere else.
44:35
If you do that, you've won the Millennium Problem.
44:37
Jim Simons would have given us wealth for that.
44:41
The idea that the construction, the constructive, is well beyond even where we can imagine, given that we haven't even been able to solve the problem of where they are not.
44:52
Yeah.
44:53
I mean, it could be that the constructive proof is the way to prove it, right?
44:56
Okay.
44:56
You know, there's history in mathematics that show that, like, for example, the real numbers, the way that you show what a real number is, is to explicitly construct it from something called Cauchy sequences of rational numbers.
45:12
Okay.
45:13
Right?
45:13
Right.
45:13
So there's a, like, both are, like, the remote hypothesis is such a black box that we don't even know which is going to be easier.
45:22
Which angles the better way to go.
45:23
That's something that Terence Tao has said in some of his lectures, right?
45:26
I see, I see.
45:26
If we want to prove, like, the whole thing, it seems we don't even have the toolkits to understand which way to go.
45:33
There's a fork in the road and it's unclear which one is closer to the destination.
45:36
Yeah, yeah.
45:37
And there could be multiple forks.
45:39
It's like we see like two, I guess.
45:41
Right.
45:41
But if the fog goes away, there could be like a hundred over here.
45:45
Like we don't know.
45:47
No, but that's really interesting in terms of understanding the value, the fundamental value of solving this.
45:55
Yeah.
45:57
As a problem.
45:58
Yeah, it's like, I mean, it would give us an understanding of the atoms of numbers, right?
46:03
Like it would give us an understanding of the periodic table of numbers.
46:06
The periodic table of numbers consists of the primes, right?
46:09
Everything else is made out of them.
46:12
And this would tell us inherently like how that is structured.
46:16
I mean, I think that is just beautiful in itself, right?
46:19
Would it be like the photograph – it's sort of like what makes up the Rosalind Franklin X-ray crystallography photo.
46:29
It would have that kind of similar like in a different field, that level of fundamental –
46:34
Yeah, yeah.
46:35
Like, if you figured out the structure of DNA, that's insane.
46:38
Right.
46:39
This would be like, yeah, that kind of shit.
46:41
Yeah.
46:41
I mean, it would be insane.
46:43
So, some other stuff, right?
46:44
If the Riemann hypothesis is proven true, it instantly validates hundreds of conditional mathematics theorems.
46:50
There's been so many theorems that have been proven assuming that the hypothesis is true.
46:55
Because the hypothesis has been so ridiculously hard to prove, but every single zero that we find is on the line.
47:02
When we computationally find it...
47:04
So let's just assume it's true.
47:06
And then there's so much other mathematics that falls through because of that.
47:11
And so the surrounding surface area kind of points to, yes, it is true, even though we can't yet.
47:17
Yeah.
47:18
Either by saying that the zeros are no Ross or constructively proving it that it's true.
47:22
Yeah, yeah.
47:23
I mean, it's...
47:24
So it would be huge, right?
47:25
Yeah.
47:26
Yeah.
47:28
You kind of get it now, right?
47:29
That's a big deal.
47:30
Yeah.
47:30
So this is why everyone's like, we want to have a positive.
47:33
Oh, I isn't anything until it really solved... Well, if it solves it, then that...
47:37
Anyway.
47:37
Yeah.
47:38
If it solves it, then that'd be great.
47:40
It hasn't solved it, and we're going to get into exactly what it did.
47:42
Okay.
47:42
Okay?
47:43
So...
47:45
After Riemann's paper, it took about 40 years to really iron out the proof of the prime number counting theorem and to show that it followed.
47:55
Riemann did a lot of the legwork, but then there was Jakis Hadamard, the guy behind Hadamard matrices for those in quantum computing.
48:02
And also, I don't know how you say this.
48:05
Can you try?
48:06
Charles-Jean de la Vallée poussant poissin.
48:10
Yeah, there we go.
48:11
So those two guys, they proved using Riemann's complex analytic methods.
48:17
And in 1914, the great English mathematician G.H.
48:21
Hardy, who is shown here, he's most well-known for discovering Ramanujan, but he's also well-known for some of the math that he did.
48:33
One of the things that he did was show that there are an infinite number of zeros on the critical line.
48:38
Okay, before we didn't even know that.
48:39
We just knew that they were in the strip.
48:41
Riemann said they're in that strip.
48:43
Hardy said there's an infinite number in that line.
48:45
Now, that says a lot.
48:47
It says that there's an infinite number of zeros.
48:50
It also says almost nothing about the Riemann hypothesis.
48:55
Because it could be that there's many, many infinitely more elsewhere on that strip.
48:59
Right.
48:59
Right?
48:59
Right.
49:00
So it's a step in the right direction, I'd say.
49:04
But it's not quite the whole thing, right?
49:07
Yeah.
49:07
Yeah.
49:09
Next, we've got, finally, in 1942, Atlee Selberg, he makes a step in the direction of the Riemann hypothesis.
49:20
Okay.
49:20
Okay?
49:23
He won the Fields Medal later in 1950, and this is a photo of him at the Institute of Advanced Study at Princeton.
49:28
So he proves that there is a positive proportion of non-trivial zeros on the critical line.
49:36
What that means is there is some percent of zeros, at least some percent of zeros on that one half line.
49:45
So the point is, there's this number of non-trivial zeros.
49:51
There's infinitely many.
49:52
There's infinitely many because of Hardy.
49:56
And some percentage of those, because we don't yet know if they're only on the critical line.
50:01
Yeah, yeah.
50:02
And so if they're only on the critical line, it would be 100%.
50:04
Right.
50:05
He said...
50:07
The percentage is more than zero.
50:09
Right.
50:09
I don't know what the number is.
50:11
Right.
50:11
I'm just telling you that the percentage is more than zero.
50:14
There's at least one on the critical line.
50:16
Well, no.
50:16
We know that this is...
50:18
Okay.
50:19
I should be careful here.
50:20
Okay.
50:20
We've already proven that there's an infinite number.
50:23
Right.
50:23
On the critical line, right?
50:24
Yes.
50:24
But here's what I mean by it could be zero percent.
50:27
There could be an infinite number on the critical line, and then there could be for every single zero an infinite number...
50:34
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 zero percent of the total infinities because in the larger search area yeah there's an equal amount of there's similar this is the stuff i was talking about infinities everywhere yeah so so you know what i mean no that's a that's a good distinction no that that's a good it's kind of weird to because we're dealing with some weird stuff here yeah so we're dealing with infinities but that infinity could still be zero percent he showed it's not it has to be more
51:04
Okay.
51:04
Okay.
51:05
Which is, okay.
51:06
We're from zero.
51:07
We went to non-zero.
51:08
Okay.
51:09
Everyone was super excited.
51:10
He won the Fields Medal.
51:11
Right.
51:11
Okay.
51:11
He's like, all right, that's dope.
51:13
All right.
51:14
So next we have Levinson in 1974.
51:18
He pushes that number up to 33%.
51:21
Okay.
51:21
That seems like a big deal.
51:22
Right?
51:23
So he's like a third of the zeros, at least.
51:27
At least a third of the non-trivial zeros are on the critical line.
51:29
Are on the critical line.
51:30
Okay.
51:31
Infinitely many 34%.
51:33
Yeah, yeah.
51:34
It's like – a crude way of saying it is for every non-trivial zero on the critical line, there are at most two outside.
51:44
Right?
51:44
Like I could make like a pairing and be like for this one, I'm going to take two.
51:47
For this one, I'm going to take two.
51:49
And if you do that, you'll cover everything.
51:50
I understand.
51:51
That's how you want to think about it.
51:53
Yep, that makes sense.
51:53
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 rest is elsewhere.
52:01
So that's what Levinson shows, okay?
52:03
And he does this weird thing with like mollifiers where he basically tries to like blur out the Riemann zeta function and then work with that blurred out version.
52:12
Mm-hmm.
52:12
I don't know how it works, but that's what Wikipedia says.
52:17
Okay?
52:17
So, and now we're getting into stuff that I really don't understand.
52:20
Right.
52:21
No fair.
52:22
So now I'm going to be regurgitating what I sort of figured out with my readings.
52:27
Full disclosure.
52:28
Okay.
52:28
Levinson puts it up to one third.
52:32
Other people take his method and start inching that up.
52:35
So we've got Conray in 1989.
52:37
He pushes it up to two fifths.
52:39
So 40%.
52:40
This is now 15 years later.
52:41
Yeah.
52:41
Yeah.
52:44
15 years later, in 2022, so this is what, 30 years later?
52:49
Yeah.
52:49
We're up to 41.7%.
52:50
So over, yeah.
52:53
So there was a quick jump by... To 33.
52:58
There's an immediate jump from zero to 34.
53:01
Yeah.
53:01
There's like three big jumps, I'd say.
53:03
Right.
53:03
You go from zero to not zero.
53:05
Right.
53:05
Yes.
53:06
Yes.
53:06
Which even though on the graph, it's like... At the bottom.
53:08
The fact that it starts.
53:10
Right.
53:10
Right?
53:10
Yes.
53:10
Yes.
53:11
That's huge.
53:12
Selberg.
53:13
Yeah.
53:13
In 42.
53:14
42.
53:15
And then in 74, it goes up to 34.7.
53:18
Big jump.
53:19
Then it goes to 40.9, another pretty big jump.
53:21
And then it took us 30 years to get to 41.7.
53:24
Just to even get one more percent.
53:25
Yeah.
53:27
Not even.
53:27
Not even one more percent.
53:28
Yeah, 0.8.
53:29
Okay.
53:30
And now Claude comes in in 2026, and it has brought it up to 67.25%.
53:34
Okay.
53:35
Okay.
53:39
The Riemann hypothesis is at 100.
53:42
We're getting closer.
53:43
Yes.
53:44
Yes.
53:45
However...
53:48
this is the case because of what you just talked about with the infinities.
53:51
Because what I'm thinking now is like, well, we could get to 99, 999.
53:56
Yeah.
53:58
And it doesn't mean anything.
53:58
It doesn't mean anything.
53:59
For the Riemann hypothesis, it doesn't.
54:01
Right?
54:02
Because what really matters is... Is the 100.
54:04
And Claude and other mathematicians have acknowledged that the way that Claude did this...
54:10
it's not going to get you to 100.
54:12
Okay?
54:12
There is a limit to how we push this.
54:15
Okay?
54:15
Okay?
54:16
We're not going to get to 100 this way.
54:17
But it's still cool to get close.
54:19
Right.
54:20
Okay?
54:20
Because it makes us understand a bit more about the function in some sense.
54:24
Part of it, and part of what I've heard people sort of talk about is, you know, as these models begin to connect these ideas that may already exist in ways that had not necessarily been connected, folks who are experts and mathematicians deep in these fields
54:40
oh let me now think and 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 claude
54:58
It, again, tweets.
55:00
That's how they announced, I guess, that they were doing this, right?
55:03
In what appears to be, it's like a matter of like 30 hours.
55:06
Yeah.
55:07
They've pushed that bound to 67.2%.
55:09
They are careful to say it didn't solve it, but it made strides on a related problem.
55:15
Okay.
55:16
Now, there are two things that Claude used that are important to know here.
55:20
Okay.
55:20
Okay.
55:21
The first is a technique developed by Hugh Montgomery in 1973.
55:25
He was a mathematician.
55:27
And the legend has it that he was actually talking to Freeman Dyson.
55:30
Freeman!
55:31
Who we've covered a lot on this podcast.
55:34
He was talking to Freeman Dyson, and he was looking at the imaginary parts of the zeros of the Riemann zeta function.
55:40
And he analyzed how they sort of clumped together.
55:42
and showed that it's the... And he was talking of Dyson, and Dyson was like, wait, that's exactly the statistical distribution that physicists have been looking at for energy levels of heavy atomic nuclei, like uranium and things like that.
55:57
The energy levels of the nuclei have to do with the zeros of the Riemann-Zeta function.
56:02
It's kind of crazy how, like, the world works that way.
56:05
Now, the catch was that when Montgomery did his original breakthrough...
56:10
It required assuming that the Riemann hypothesis was true to get his formulas to work.
56:15
Later on, very recently, there's a team of mathematicians that published work showing that that
56:21
that pair correlation thing that he was doing with all those zeros, it works unconditionally.
56:26
It works without assuming that the Riemann zeta function works.
56:30
So this is something that has to do with the Riemann zeta function that Montgomery sort of proved on the side.
56:37
The second thing is in 2000, the legendary Fields medalist Enrico Bombieri, he wrote a note, a little monograph,
56:48
where the Clay Mathematics Institute was forming the Millennium Problems.
56:53
So he wrote the note for the Riemann zeta function, and he's like, this is why I think it belongs in the Millennium Problems.
57:00
I mean, everyone believed that it did, but this is a preeminent mathematician, and he was sort of dumping ideas about how to approach it.
57:07
And one of the things that he talked about was how modern analytic number theory handles the architecture of these zeta zeros, and he dives into something called quadratic forms,
57:18
And Wiles explicit quadratic functionals.
57:22
I don't know what they are, but that's what he mentions in his writeup.
57:26
All right.
57:27
Now,
57:28
Let's see what Claude does.
57:30
Yeah, because you're saying these are two fundamental, the Montgomery and Bombieri, no relation to Guy Fieri insights, were building blocks.
57:40
For what Claude is doing.
57:41
For what Claude is doing.
57:41
Okay.
57:42
So Jared Sumner, right?
57:44
He's an Anthropoc staff member.
57:46
He's got no deep mathematical background.
57:49
Apparently while jogging, I don't know how much this is hype.
57:52
Right?
57:53
Because like the, look, the conjecture, the Jacobian conjecture, they happened to do it at the time of the FIFA World Cup final.
58:02
Now he's jogging.
58:03
It's like, you know.
58:05
Great marketing.
58:06
Yeah, it is great marketing.
58:07
So anyways, let's take their word for it.
58:09
He's jogging.
58:10
He opens up a clawed terminal while jogging.
58:12
On his phone.
58:14
Why don't you watch some Netflix or something?
58:16
Or, hey, listen to this podcast.
58:18
Right, right.
58:18
That's a great choice.
58:20
I guess they're always working out in Anthropic, okay?
58:23
So he casually prompted the unreleased model to take a real stab at the Riemann hypothesis, okay?
58:30
The model exhibited initial skepticism about its own ability.
58:35
And the human said, you know, almost comedic encouragements of like, keep going and believe in yourself.
58:40
Very Ted Lasso.
58:42
Yeah.
58:42
Yeah.
58:43
He Ted lassoed Claude into getting closer to the Riemann Zeta.
58:46
Insane.
58:47
So behind the scenes, the execution is not casual.
58:50
Okay.
58:50
What Claude is doing is not casual.
58:52
The primary Claude model that was enacted on by Sumner, it starts orchestrating.
58:59
It becomes an orchestrator.
59:02
And it spins up a hierarchical swarm of 60 autonomous sub-agents to try and tackle this problem.
59:09
It built a team of other agents, other versions of itself.
59:14
ostensibly that had one thing to focus on.
59:17
Yeah.
59:17
And then it could coordinate them because they're not trying to do too many things as one instantiation of itself.
59:23
Exactly, yeah.
59:25
And these agents aren't just like writers of mathematical proof.
59:28
That's what's key, right?
59:30
These modern models now have the ability to go onto a virtual machine...
59:35
They can write, debug, and execute code.
59:39
They can write Python programs and execute those Python programs.
59:43
So between them, they ran like 2,400 shell commands, hundreds of Python scripts, consumed 31 million output tokens,
59:53
And they autonomously downloaded 54 academic papers from Archive.
59:58
The preprint server run by Cornell.
1:00:00
Yeah, that's right.
1:00:01
And now this, you can finally see, as you were saying, like this methodology, it mirrors like a high performance, high speed research lab.
1:00:10
Right.
1:00:10
Right.
1:00:10
There's like a PI and then there's like postdocs and...
1:00:15
PhD students and then they've got their undergrad underlings that are like doing work.
1:00:20
It's an entire lab in a single like automated ecosystem.
1:00:23
Yeah.
1:00:24
You have some people doing rote work.
1:00:25
You have some people doing sort of ideation.
1:00:28
You have some people doing sort of functional mathematical number crunching, validation, verification.
1:00:37
And what's so fascinating about this is that the initial orchestrator from the initial prompt is,
1:00:45
can architect the approach without having to be dictated to what the right construction of that is.
1:00:55
That's exactly right.
1:00:56
This is another point that I'm glad that you're highlighting here.
1:00:59
The initial prompt was just take a real stab at the Riemann hypothesis and believe in yourself.
1:01:06
That's it.
1:01:06
That's it.
1:01:07
Right?
1:01:07
Like, this guy was not a mathematician.
1:01:10
All of the mathematics is coming from Claude itself, which is kind of crazy.
1:01:14
Right.
1:01:15
And even if it didn't have the right starting point, it just basically said, I don't have the right starting point.
1:01:19
So let me go gather enough context.
1:01:21
Yeah.
1:01:22
to then narrow my search space, my area of operation, to something that's based off of existing whatever.
1:01:30
I mean, again, 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 –
1:01:46
to understand that 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.
1:02:03
We can't view the capability set on what is no longer the frontier execution of what's happening with these things.
1:02:13
Again, that doesn't mean they can have novel insight and be created.
1:02:16
But there are so much domain space in taking –
1:02:22
orthogonal or correlated areas of a variety of areas of study in science and mathematics, and just making connections no one else has.
1:02:32
Yeah.
1:02:32
And there's so, there's, I think people underestimate how much can, may be able to be done just making existing connections.
1:02:40
Exactly.
1:02:41
No, that's totally, I mean, for example, we had just covered Montgomery and Dyson, right?
1:02:46
If those two didn't have lunch or whatever...
1:02:48
Bingo.
1:02:49
Then what?
1:02:50
Maybe Montgomery would never have done his while whatever, you know, thing that he did for the Riemann-Zeta hypothesis.
1:02:56
Right, right.
1:02:56
So, but AI now can just do that.
1:02:59
Right.
1:03:00
Right?
1:03:00
It can take all of the...
1:03:02
All of the papers of Dyson and all of the papers of Montgomery.
1:03:05
And then you don't have to have lunch.
1:03:08
You could have lunch, but you can talk about other stuff.
1:03:11
I want to make a slight also differentiation between the execution in this space where, you know, these papers are already provided as public access.
1:03:18
Yes.
1:03:19
As the foundational material versus...
1:03:21
taking existing copyrighted artists work or creative work and then giving people the ability to just rip them off.
1:03:27
This is a very different domain.
1:03:29
It's open science.
1:03:30
And so just these are our different executions of the technology.
1:03:36
Yeah, yeah, yeah.
1:03:37
Great point.
1:03:39
So let's get into...
1:03:41
what this orchestra of agents did.
1:03:44
Out of 650 different mathematical approaches that were tested and discarded, there were two specific sub-agents that were dubbed E2 and E2 pairs that discovered the path that breached this 41.6% barrier.
1:03:59
And it's the story of these two sub-agents that we're going to get into.
1:04:02
Because remember, Claude actually published
1:04:04
its own version of what happened.
1:04:06
So you get to see these sub-agents like talking to the orchestrator and so on.
1:04:10
And it actually also gives you the internal transcript of the E2 sub-agent.
1:04:14
So from that, this is what I've gathered happened.
1:04:17
It's kind of crazy that we can peer into it, right?
1:04:21
So here's the first thing.
1:04:22
The first thing is the orchestrator, it tasks E2 with investigating something called the Pontrygan Index of the Conditioned Space.
1:04:33
I don't know what that means.
1:04:34
But the point is, there's some kind of matrix.
1:04:38
The matrix, it thinks, has negative eigenvalues, something like 33 to 153 negative eigenvalues.
1:04:45
And this whole thing is inspired by Bombieri and his speech, his spiel about one way to tackle the Riemann hypothesis using algebraic techniques.
1:04:55
Algebraic techniques are like linear algebra is part of the algebraic techniques.
1:05:00
That's why we're getting into eigenvalues and things like that.
1:05:03
Okay, so...
1:05:03
Turns out the prompt's suggestion of doing this was wrong.
1:05:10
Okay?
1:05:10
So the E2 sub-agent, after 50 minutes of silent contemplation...
1:05:16
It wrote a Python script to test what the orchestrator was telling it to do.
1:05:20
And it turns out that there's some warp sync operator, like hyperbolic sign operator in there, that's causing some artifact of negative eigenvalues.
1:05:30
So it goes back to the orchestrator, and it says, actually, what you're telling them to do doesn't make any sense.
1:05:38
You're wrong, effectively.
1:05:39
It does not compute.
1:05:40
It does not compute, but...
1:05:44
I can pivot because I've realized something.
1:05:47
The exact same matrix structure could be used in reverse.
1:05:50
I'm going to pivot in something that it called a dual use of inertia.
1:05:54
So it has like inertia.
1:05:55
It's now pivoting to use whatever insight that it had gathered to prove its orchestrator wrong to now try a different approach.
1:06:02
Right.
1:06:02
This is a sub agent.
1:06:03
Right.
1:06:04
Right.
1:06:04
Okay.
1:06:05
Now, this sub-agent, it doesn't just fail the task and give up.
1:06:08
It independently starts executing this pivot.
1:06:12
And then it finds that the total number of positive eigenvalues of whatever matrix is bounded by some number, it starts to compute that strictly from the prime numbers.
1:06:22
And then successfully, it proves that you can get to 50%.
1:06:27
On the zeros.
1:06:28
Okay?
1:06:28
It goes back to the orchestrator.
1:06:30
On the non-trivial zeros.
1:06:31
On the non-trivial zeros.
1:06:32
We're now all from 41, whatever, 41% to now we're at 50%.
1:06:37
There was this singular insight by making the pivot to this dual use of inertia that now when – because one of the things that's great about math with these models is it's test.
1:06:46
It's immediately – because they can't execute, you can test.
1:06:49
And it's like, oh, I got whatever, 4%, 1%.
1:06:53
Yeah.
1:06:53
And so now it's at 50%.
1:06:55
Right.
1:06:55
Okay.
1:06:55
It goes back to the orchestrator and the orchestrator expresses intense skepticism.
1:07:00
Right.
1:07:01
Because I don't think, I mean, I'm not, I'm not trying to assign feelings to this thing.
1:07:05
Right.
1:07:05
But if I, if I were an orchestrator and one of my postdocs or something came back, like I got to 50.
1:07:11
Yeah.
1:07:11
My first, my first inclination would be, what did you do wrong?
1:07:15
Right.
1:07:15
Where's the mistake?
1:07:16
No, no, you didn't.
1:07:18
You're not that guy.
1:07:19
Right.
1:07:21
You just got here.
1:07:22
Yeah.
1:07:22
So it notes that like 50% would shatter the record.
1:07:26
And it goes back to the human, to Jared.
1:07:29
And it's like, my prior is that this is wrong.
1:07:31
Okay.
1:07:32
So here's what I'm going to do.
1:07:32
I'm going to rig up hostile referees, three hostile referees, more sub agents that are going to critique this proof.
1:07:41
Those referees go ahead and critique it.
1:07:45
Referee A, usually it's referee two.
1:07:48
In this case, I guess the AI just doesn't have a ranking, right?
1:07:52
So here, referee A discovers the technical flaw.
1:07:56
And I didn't look in – they didn't actually show what the referee was saying, but I wonder if it was like as scathing as the stuff that we get in the emails.
1:08:04
Like you should just quit the job and go work in a McDonald's, right?
1:08:09
Like that's the stuff that we get in the emails.
1:08:11
But –
1:08:11
I wonder what referee A said to this sub-agent E2.
1:08:15
But it discovered a technical flaw regarding some ill conditioning of the matrix, and it corrected it.
1:08:22
And then it ran up these inequalities, and now that thing was fixed.
1:08:27
So the referee was doing what normal referees do, which is like, hey, this might be wrong.
1:08:30
No, a nice referee at least would be like, this might be wrong.
1:08:33
This is how you could change your experiment to test for this confound and things like that.
1:08:38
Another referee independently verified that the calculations don't secretly import the Riemann hypothesis.
1:08:45
Like, are you just trying to prove it by assuming the thing that you're trying to prove that doesn't make any sense?
1:08:50
So there's no circular logic.
1:08:53
Okay, so now we're up to 50%.
1:08:55
It goes back to the human prompter, to Jared.
1:08:58
And it's like, we got to 50%.
1:09:01
Jared then asks, what would you do next?
1:09:04
And Claude responds, push it to two-thirds.
1:09:08
So then...
1:09:10
Jared's like, all right, push it to two-thirds.
1:09:12
Why did it say two-thirds?
1:09:14
Now, the reason why it said two-thirds is because under the assumption of the Riemann hypothesis, Hugh Montgomery had reached a two-thirds bound by utilizing the fact that there's like some, for whatever reason that he was doing, and he had found that two-thirds of the zeros had some property.
1:09:32
Okay?
1:09:33
So...
1:09:35
Claude starts thinking, well, maybe the two-thirds and Hugh Montgomery's stuff I can now use.
1:09:41
I already used Bombieri's.
1:09:43
Right.
1:09:43
Now maybe instead of – he was using – like Claude was using something like the Kaushy-Schwartz inequality.
1:09:49
Not important what we have to get into.
1:09:50
Right.
1:09:51
Effectively, it said, I could push this to two-thirds because I know of Montgomery's stuff.
1:09:55
There's a reference point that says there's a chance.
1:09:59
And so in a well-defined space.
1:10:01
So let's go after it.
1:10:02
Let's go after it, right?
1:10:03
So again, the orchestrator confidently instructs this next E2 pairs to recover some lost efficiency in the Koshy-Schwartz inequality.
1:10:12
It's like this tool that we're using.
1:10:13
It's basically the triangle inequality.
1:10:15
But in any case, it's like you can do better than that, right?
1:10:18
The AI says...
1:10:21
I'm not going to do it that way.
1:10:23
That doesn't make any sense.
1:10:24
Again, it's going back to the orchestrator.
1:10:26
That doesn't make any sense.
1:10:28
But it's giving me another idea.
1:10:30
I'm going to write a Python script, and that's going to just make a bunch of simulated matrices consisting of zeros of the Riemann zeta, and it's going to do a bunch of math.
1:10:41
And I'm going to try to optimize this.
1:10:43
And it optimized it and noticed a pattern in how the optimization worked in that Python script.
1:10:49
From that, it garnered how it could reveal some profound insight into the non-trivial zeros.
1:10:57
And it went through and started writing up this little lemma, this five-line lemma, that's going to help it get to two-thirds.
1:11:05
All of a sudden, the thing crashes.
1:11:08
Okay?
1:11:08
This is also documented in Claude's report.
1:11:11
All of a sudden, this thing crashes.
1:11:14
By some miracle, Claude has dumped that five-line proof into a file.
1:11:24
Into its memory somewhere.
1:11:26
Yeah, it's into like a hard file memory somewhere.
1:11:29
Yeah, yeah, yeah.
1:11:29
And so when the orchestrator rigs it up again, it's like, I'm pretty sure you were doing something with this file.
1:11:35
Yeah.
1:11:35
It reads that file.
1:11:36
It's like, oh.
1:11:37
That's right.
1:11:39
And then it goes back and completes it.
1:11:40
Isn't that cool?
1:11:41
I love, I love it.
1:11:44
It's like a text file on the virtual disk.
1:11:46
Right, right.
1:11:46
And it like dumped its memory.
1:11:49
I guess, I mean, I don't know why, but it thought that, okay, this is important.
1:11:52
I'm going to write this.
1:11:53
Let me write this down.
1:11:54
I'm going to write this down.
1:11:55
And then it like crashed.
1:11:56
Of course.
1:11:58
And this is, I think, what's also interesting when you look at the chain of thought of these models.
1:12:04
And we tried to start doing this introspection is...
1:12:08
I understand this idea that there's a framing that these things don't have ingenuity and insight in the way that humans do.
1:12:20
But even though it's doing a lot of work through having the ability to execute with code and ingest lots of information, it still has to make after those executions that are functional –
1:12:34
It still has to try to choose a path.
1:12:37
And it intuits sometimes in ways that are interesting in between those like milestone points of functional tasking.
1:12:46
And this is a perfect example of, you know, one, it chose to not just do what the orchestrator told it.
1:12:55
It had to make that choice.
1:12:57
Two, it was able to sort of figure out this matricy map, this building out these matrices to find an insight within it and where to even look within it.
1:13:14
And then thought, let me make sure I write this down before I run it in case for whatever reason it crashes.
1:13:20
Yeah, yeah.
1:13:21
You know, again, it might not have done it explicitly for the reason thinking it would crash.
1:13:25
Yeah.
1:13:25
It might have just been part of its normal process.
1:13:27
But I just find this chain of thought so fascinating because I think it illustrates this is well beyond next token prediction.
1:13:37
Yeah, this is, yeah.
1:13:39
You know what I mean?
1:13:40
It's pretty insane.
1:13:40
Yeah.
1:13:41
I mean, this is pretty insane.
1:13:44
And then from there, you get to the 66%.
1:13:47
You get to the two-thirds.
1:13:49
And which can be formalized.
1:13:51
This lean software, which you can actually then have people look at it.
1:13:55
Exactly, yeah.
1:13:56
So then, I mean, they actually had the actual people look at it.
1:13:59
So, I mean, I think they got...
1:14:04
Al Pogate, the same guy who did the Jacobian conjecture, he took a look at it because he's a proper mathematician in his own right.
1:14:13
They also actually put it into the Lean software to take a look and see if it actually worked.
1:14:19
Lean is, you know, human peer review is infallible, but...
1:14:24
You can completely eliminate the possibility of logical hallucination by using this software.
1:14:32
It's called Lean4.
1:14:33
That's the latest one.
1:14:34
It's strictly typed machine-checkable proof assistant.
1:14:39
Effectively like truth statements and things like that.
1:14:41
Yep, yep.
1:14:43
It's hosted on GitHub.
1:14:44
And you can go take a look.
1:14:46
It's fairly cool that I think they've done it.
1:14:48
I mean, it still has to go through traditional peer review.
1:14:51
For sure.
1:14:51
Right?
1:14:52
But it seems that they have gotten a higher bound of the Riemann hypothesis.
1:14:58
And again, when – this goes back to what we were just talking about earlier.
1:15:01
When folks who are mathematicians start to try to look at what did it do in these arenas?
1:15:09
Because a lot of the argument has been, oh, these models are attacking these – there are all these –
1:15:15
sort of open problems.
1:15:16
And like, there are some, maybe two categories we'll say one category is open problems that are open, but have not had concerted attention put towards them, you know?
1:15:25
And so they're just open and gathering dust on the shelf.
1:15:28
Uh, but not necessarily either for whatever reason, meaningfully being, trying to be solved by leaders in the field.
1:15:36
And then there are ones like this that are, have been a high priority and a focal point.
1:15:44
Um,
1:15:45
And so it's not solving the hypothesis, but some of the insights that are gleaned might trigger somebody to have a completely different – Oh, that's interesting.
1:15:57
Out of left field insight.
1:15:58
I could take that and go this way with it.
1:16:02
We'll see.
1:16:02
Who knows?
1:16:03
We'll see.
1:16:04
I just find a lot of this –
1:16:06
It's just, it's happening.
1:16:07
Yeah.
1:16:08
It's happening as we speak.
1:16:09
I mean, it's pretty crazy.
1:16:12
A few interesting things that I came away with is the fact that first, as you said, it's not a next token prediction.
1:16:21
That's not the only thing.
1:16:22
The fact that it's able to create a verifiable environment, I think here is key.
1:16:27
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.
1:16:36
Um, second rejecting human guidance, as you said, um, that's pretty weird that it's able to just in not only reject human guidance, but also, um, its own guidance with the orchestrator telling all these sub agents what to do.
1:16:48
And, um, the funny thing is that absurdity of the interface.
1:16:52
Okay.
1:16:53
The fact that this human, all he had to do was say, believe in yourself.
1:16:58
And that triggered a 31 million token compute dump, right?
1:17:03
Yeah.
1:17:04
It's weird.
1:17:05
What does it mean to tell Claude to believe in itself?
1:17:09
I don't know.
1:17:11
Right?
1:17:13
And what does it mean that it responded in the way that you intended when you said that?
1:17:17
Yeah.
1:17:18
Like, is that really something that it just learned from all of the language on the internet?
1:17:23
It's weird.
1:17:25
I think it's genuinely weird that 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...
1:17:39
It's like LeBron James...
1:17:44
I continue to be fascinated by this because I do think when you look deeply at these things, it does become... Because how did it interpret yourself?
1:17:52
Yeah.
1:17:52
Does it have a sense of self?
1:17:55
Yeah.
1:17:56
Or an abstract construction of what the human meant by self?
1:17:59
Yeah, yeah, yeah.
1:18:00
There's like some part of its weights that's like, oh, he's talking about me.
1:18:04
Like he's talking about the other weights over there that are doing that.
1:18:08
Or I don't know...
1:18:10
Yeah, dude, it's really weird.
1:18:11
On a final note, all I'll do is read the last line of Claude's, or I should say, Anthropics write-up, which I'm sure was written by Claude.
1:18:23
So this might be Claude talking about itself.
1:18:26
But I think it's very interesting in its outright.
1:18:29
It says, even Claude was surprised by its own finding.
1:18:33
It was skeptical at first, possibly because it has learned from its training about the difficulty of open problems in mathematics and about the limitations of AI models.
1:18:45
But after some encouraging prompts, it arrived at the result we've described.
1:18:49
Perhaps Claude, like many of us, underestimates the rate of AI progress.
1:18:56
I think that's a pretty nice way to end things.
1:18:59
The foothills of the singularity.
1:19:03
This is a very controversial topic.
1:19:05
People have a very emotional reaction to AI right now for a lot of...
1:19:11
legitimate reasons and some of it is because of our other larger macro problems and then ai becomes the manifestation that they connect to these other large macro problems i think part of what we are trying to do on the show is not make a moral argument one way or another about whether ai is good or not no but just it is important to understand what these things are doing at the edge at the frontier um
1:19:39
And what the implications can then mean from that, because as we were just talking about, I mean, this is – these are becoming powerful systems.
1:19:49
Yeah.
1:19:49
And I think underestimating their capability is –
1:19:55
not wise, just because you might not, I might not, you might not find everyday value in my life where I'm doing emails and other random nonsense, does not mean that these tools will not have a potential lasting and grave and large or beneficial impact in these larger areas.
1:20:20
Either in subject matters and areas, institutionally.
1:20:25
I just think this is a very complex and nuanced issue.
1:20:30
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.
1:20:38
That's all true.
1:20:39
But if you just look at what it's doing, not what everyone's saying about what it's doing, and it's – I don't know how to feel about it.
1:20:50
Yeah.
1:20:51
I don't know how to feel about saying believe in yourself and it like actually does.
1:20:56
Right?
1:20:57
Then, I mean, the questions of AI safety are –
1:21:02
are very active in my mind when I think about this kind of stuff, right?
1:21:05
If the, if the, if a model is capable of orchestrating like this, and its goal is not something as benign as try to get further on the Riemann hypothesis, suppose the goal is much more sinister, look at what it is capable of and look at what it can do in terms of insubordination to itself, right?
1:21:26
To the human person.
1:21:28
In this case, it was good because we got further on a math problem.
1:21:32
But we need to have these conversations at the highest level of our society.
1:21:38
And unfortunately, we're not.
1:21:39
We're only having it at a pretty low level of society.
1:21:42
And that's only... We're having these conversations on podcasts and YouTube videos and not on Capitol Hill, for example.
1:21:51
Which, as we all know...
1:21:54
it's complicated the social dynamics and the socioeconomic and political world we live in right now 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
1:22:20
is owned and operated by that class that has that wealth and power who are continuing to try to accrue more of it, it becomes difficult for the everyday person to really care, engage, want it to be continuing when the rent is too damn high, as that one guy would say.
1:22:42
So very complex stuff.
1:22:44
Thank you for walking through this because I now have a much better understanding of why it's important.
1:22:50
So for those of you who enjoyed, we're going to be keeping an eye on this.
1:22:55
It's hard to avoid the AI subject.
1:22:58
We do try to focus on the underlying scientific topics that are talked about and give you a better lens into this.
1:23:06
But we are going to move.
1:23:08
into a brief moment of me shilling the pod before we get into a last bit of fun here.
1:23:15
So for those of you who are listening, I should have said this at the beginning of the pod, this is definitely an episode that you should watch.
1:23:22
It's hard for us, given how long we go, to describe things in an audio format.
1:23:27
We are available on video, YouTube, Spotify, still coming soon to Apple Podcasts.
1:23:33
That will be dependent on when Spotify creators will.
1:23:36
releases those there if you're viewing us on any of those podcast platforms a five star is super helpful to help get this pod out to more people if you're on socials put it in the group dm give us a like or 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 people i want to give a
1:24:00
to everyone who took our merch giveaway survey from our one-year anniversary episode.
1:24:08
We really appreciate the feedback.
1:24:11
It's very fascinating to learn about the audience and where you all are coming.
1:24:14
There's a lot of folks who are grad students, who are postdocs, who are in a variety of technical fields who find this podcast relevant.
1:24:22
We are going to extend the submission deadline for one more week.
1:24:26
So if you're hearing this now,
1:24:28
Our merch giveaway deadline is getting extended for one more week at ffppod.com backslash survey.
1:24:37
We're going to select at first 10, but if we get a ton of people, we might increase it to 20 people to get free merch from the shop.
1:24:45
Again, you can go to the website, check out all of that.
1:24:48
All our pods are on the website as well.
1:24:50
The two of us here are trying to give you the best science show on the planet each and every week.
1:24:57
And as we speak, we're about to cross 250,000 followers on Instagram.
1:25:02
And so it seems that at least some of you do believe that this is the best science show out there.
1:25:10
We're going to end the pod with a little bit of fun.
1:25:12
A little bit of relaxation.
1:25:13
That first half, that was a brain bender.
1:25:16
Yeah.
1:25:17
That was Jimmy Neutron style.
1:25:20
So we're going to have a little bit of fun here.
1:25:23
And I'm going to start this next section talking about an individual who on the 1st of July was the man who moved the former Nobel Prize winner
1:25:40
in chemistry last year because he started a new job, not at Berkeley where he's been for quite some time, last 25 years actually, but a move to Tsinghua in Beijing.
1:25:52
And Berkeley didn't get anything, nothing.
1:25:55
They put a little numeritus next to his name.
1:25:58
Yeah.
1:25:59
But for those who are sports fans, it's that time of year, it's transfer season.
1:26:05
And there's no such thing as a transfer fee
1:26:08
in science.
1:26:08
He just walked on the free, on the free.
1:26:10
And he's not the only one.
1:26:12
And what we wanted to do was count to see how many great American scientists have now transferred out of the U.S. market into the rest of the global market.
1:26:25
And August is a great time to do it.
1:26:26
For those who are not familiar with the concept of the transfer window, many sports shows right now, it's wall-to-wall coverage.
1:26:34
Up the shells, we've had a great, great business in this transfer market.
1:26:39
This is people who are moved, who are bought, clubs get robbed, a club signs someone new, the pundits will pontificate about what that means for the next season.
1:26:48
And in sports, people get paid, but in science, they do not.
1:26:53
And I think it's interesting to view all of this.
1:26:56
particularly because of the science policy posture we've taken in the United States, what does the American summer transfer window look like?
1:27:06
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 have them kind of on a rolling basis.
1:27:22
But in this window, we looked at 42 transfer.
1:27:25
confirmed moves in the last 12 months.
1:27:28
23 to Europe, 17 to China, two elsewhere in Asia.
1:27:32
And if you narrow it down to this summer alone, 13 out of the 42 landed in June, July, and August.
1:27:40
Wow, that's crazy.
1:27:42
China has more than...
1:27:45
Yeah, it's like UK plus France plus Switzerland combined.
1:27:50
So if we look at our ranking here, China's number one.
1:27:53
Right.
1:27:54
17.
1:27:55
As a close second is the UK.
1:27:57
Obviously, we've covered the UK and the great institutions there.
1:28:00
With eight, both France and Switzerland, with five American transfers leaving our shores.
1:28:05
Germany with three.
1:28:06
And then Singapore, Austria, and the UAE all with one.
1:28:10
And Italy and France are actually sharing one.
1:28:14
uh, uh, someone who's going to be traversing across two institutions.
1:28:19
But I think this is interesting to view it in this way.
1:28:22
Yeah.
1:28:23
Um, and we're not going to name every name on this board tonight.
1:28:27
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, um,
1:28:37
What would be considered, you know, top, top tier or high level, well awarded folks that are people we don't want to lose at the club.
1:28:49
And we're going to start off with someone.
1:28:52
Listeners of the pod who have been listening to us for a while will know very well.
1:28:56
Omar Yagi, the founder of Reticular Chemistry, most well-known for metal organic frameworks, transferring from UC Berkeley to Tsinghua University in Beijing, already there July 2026 on the free, Nobel Prize winner in chemistry in 2025.
1:29:13
He was born in Amman, Jordan, moved to the U.S. at 15, and he was born in Amman, Jordan
1:29:17
Did his PhD at the University of Illinois Urbana-Champaign, Harvard postdoc, then Arizona State, Michigan, UCLA.
1:29:24
A great tour.
1:29:25
And in 2012, he was a Treader Chair at Berkeley.
1:29:29
Plus, spent some time at Lawrence Berkeley National Lab.
1:29:33
The pioneer.
1:29:35
of metal organic frameworks, MOFs, the idea of these crystal cages with absurd surface area.
1:29:41
We did a deep dive in our Nobel Prize episode coverage last year.
1:29:46
So if you really want to know more about it, take a look.
1:29:49
These can be used for things like carbon capture, gas storage, water from desert air.
1:29:54
The Nobel Committee compared them to Hermione's handbag in Harry Potter.
1:30:00
Yeah.
1:30:01
Very clever idea.
1:30:02
visual reference.
1:30:04
He named the field reticular chemistry.
1:30:06
I mean, what are your thoughts about this type of talent, particularly from the state of California?
1:30:14
Yeah.
1:30:15
Leaving.
1:30:16
I mean, this, I think, is probably a direct response to our science policies, I have to say.
1:30:24
It hasn't been said explicitly, but I think Tsinghua University gave him an offer that he couldn't refuse.
1:30:31
like insane amounts of funding.
1:30:34
I think all of his students could like go with him effectively.
1:30:38
Um, this is a man who's a legend in the field.
1:30:41
This is a man that was such a legend that I could call him.
1:30:45
Um,
1:30:47
as one of my top picks to win the Chemistry Nobel, right?
1:30:50
That's how obvious it was that he was going to win the Chemistry Nobel.
1:30:53
He's had a tour, as you were saying, from Arizona State University to then UCLA to Berkeley.
1:31:00
All of these institutions poached him from the last institution because they knew he was going to win the Nobel.
1:31:07
And now that he's won the Nobel, his clout has gone up a whole lot.
1:31:11
Metal-organic frameworks are...
1:31:14
the chemicals of the future in terms of water storage, carbon capture, as you were saying, lots of industrial applications.
1:31:23
And it's a huge loss, I think, to U.S.
1:31:27
He himself has criticized grant pressure and visas, right?
1:31:31
He's an international student.
1:31:34
He started his life as an international student.
1:31:36
So he knows exactly how important it is
1:31:40
for American science to hold that edge with the international community.
1:31:45
So the fact that he's leaving, yeah, I think he probably saw the writing on the wall.
1:31:50
And to him, his research was more important than where he was doing the research.
1:31:57
A seasoned trophy winner.
1:31:59
Yeah.
1:32:00
13 trophies.
1:32:00
Albert Einstein World Award of Science in 2017.
1:32:04
Wolf Prize in Chemistry in 2018.
1:32:06
The Balzahn Prize in 2024.
1:32:09
Just to name a few along with the Nobel.
1:32:11
This is not a mid-table signing.
1:32:13
No.
1:32:14
This would be like...
1:32:15
Ballon d'Or winner in 2025, Ousmane Dembele, moving from PSG right now, which just won the Champions League, and is choosing to go to the Chinese Super League or Saudi right now.
1:32:29
In terms of how crazy it would be to see happen.
1:32:36
He has been an honorary professor at Tsinghua since 2022.
1:32:41
He started full-time on July 3rd to lead their Amitri
1:32:46
This new scientific discipline that was pioneered by Yagi, combining AI, material science, and chemistry to design and synthesize advanced materials.
1:32:55
The idea being, can we compress the timeline of materials discovery given our new tools?
1:33:01
If there was ever a man to do it, he will.
1:33:03
And as a place that's trying to
1:33:05
onshore manufacturing, you know, re-onshore manufacturing and really own this idea of AI enabled in the physical world.
1:33:13
It's not ideal.
1:33:16
He did criticize the US for grant pressuring visas.
1:33:20
He hasn't drawn a clear line that that's what caused this move.
1:33:24
So, you know, it's kind of ambiguous, but, you know, Tsinghua gave him a bigger sandbox and he took it.
1:33:31
That's our first big transfer of the summer.
1:33:34
Marquee signing for China.
1:33:36
We're going to move to our second big name signing, Shirley Meng, who is going to be our pioneer in batteries and energy storage, this idea of next generation battery materials.
1:33:51
University of Chicago.
1:33:52
And she's now being transferred to Nanyang Technological University in Singapore, Hong Kong.
1:33:59
Also, deal done in July.
1:34:02
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.
1:34:15
We'll get to that in a second.
1:34:17
So, I mean, she's just been incredible around this energy space and energy storage specifically.
1:34:25
She served as the chief scientist for energy storage science at the U.S. Department of Energy, DOE, Argonne National Lab, and directed the DOE-funded Energy Storage Research Alliance.
1:34:36
This is a $62 million over five-year DOE program that's sort of designed for figuring out how do we make energy storage easier.
1:34:47
Longer lasting for grid batteries, safer and cheaper.
1:34:52
Grew up in China, got her first degree at NTU, but she was returning to her alma mater by way of the Singapore MIT Alliance PhD program.
1:35:03
came to the States, also made a stop at the University of Florida, became the Zabel chair at UC San Diego.
1:35:10
Shout out California again.
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And in 2021, she was a professor at Chicago's Pritzker School for Molecular Engineering.
1:35:18
25 years.
1:35:19
years uh here she's going back home to run the club uh and her big focus has been better battery solid state sodium ion anode free uh in 2024 her lab reported the first anode free sodium solid state battery which we've covered the concept of yeah in one of our previous episodes as well in
1:35:44
I mean, the paper that we covered, it was out in Juul, if you remember.
1:35:47
Yes.
1:35:47
And it was actually from her lab.
1:35:49
It was from her lab.
1:35:50
It was straight up from her lab.
1:35:51
We were talking about why sodium is so important compared to lithium.
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Lithium is great, but it's hard to get at.
1:35:58
Sodium is literally in seawater.
1:36:00
Yes.
1:36:00
So if we could make a battery out of sodium, that'd be great.
1:36:03
I mean, it's not going to replace lithium ion batteries, but...
1:36:07
for certain types of applications, it's going to be key for scalability.
1:36:13
Like if we want to store like massive amounts of power, um,
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and then deploy it like later at night and things like that.
1:36:21
Like if you want to have solar farms that, that create a bunch of electricity during the day, and then we want a way to store like, you know, infrastructure level worth of power, sodium batteries might be the key.
1:36:34
And she's been doing insanely amazing work at the university of Chicago.
1:36:39
Another, another great professor who is now leaving us.
1:36:44
Another seasoned trophy winner, Faraday Medal of the Royal Society of Chemistry.
1:36:50
Oh.
1:36:50
ECS Battery Division Research Award, ACS Electrochemistry Honors, not a mid-table signing.
1:36:57
She's becoming the VP of Industry and Distinguished University Professor, the highest faculty rank.
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She's already there as of July 1.
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And Chicago is going to keep a partial appointment today.
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through the transition.
1:37:08
So this is a loan, not on the free.
1:37:11
She's stepping down as the director of the DOE hub.
1:37:14
The hub's money wasn't cut.
1:37:15
She's just leaving as a chair in science.
1:37:20
Her reasons for leaving were explicit.
1:37:22
Oh, the U S turning away from decarbonization and the immigration rules squeezing international, uh,
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including Chinese-born scientists.
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She's one of the ones who's put that on the record.
1:37:35
Singapore has handed her the keys to her old club, plus a C-suite seat, and she took the loan.
1:37:42
We are going to move here through to our second-to-last in our coverage of the summer science transfer window.
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If you want us to cover more, let us know.
1:37:52
We can go country by country.com.
1:37:55
John Baker is our gravitational wave astrophysicist, NASA Goddard Space Flight Center.
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He is moving to French National Center of Scientific Research in Toulouse.
1:38:06
This happened actually late last year, also on the free.
1:38:10
winner of the Goddard's Highest Space Science Honor, the John C. Lindsay Memorial Award in 2008, Kansas City sort of upbringing, Truman State undergrad, Penn State PhD in gravitational physics, and then a postdoc at the Albert Einstein Institute in Potsdam, joined Goddard in 2001 and never left.
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He's been a member of the Gravitational Astrophysics,
1:38:35
physics 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 which is europe's space-based detector that's
1:38:57
Another not mid table signing a key member here.
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So as he moves to CNS, CNRS, it's going to be the same message mission just on the other side of the zoom.
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He'll be still working in this Lisa arena.
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The launch for that is still aimed for 2035.
1:39:16
And unlike Yagi, who was not clear about why he was leaving, Baker also mentioned political and social conditions in the U.S. and his family's safety.
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He put that on the record as well.
1:39:30
You're sensing a trend here around, you know, scientists are normally relatively diplomatic, especially related to policy, because that's who puts food on their table in terms of money and funding.
1:39:43
Well, this one's – yeah.
1:39:45
I mean, he straight up said that.
1:39:48
This is quite rare for scientists to say it.
1:39:53
Yeah.
1:39:54
I don't know what to tell you.
1:39:55
I don't know what to tell you.
1:39:56
We're going to touch one more, which is not just a single signing.
1:40:03
This was a double signing.
1:40:05
And this one is also quite interesting.
1:40:08
We have Ron Mangun and Tamara Swab, both in the area of cognitive and language neuroscience.
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Currently out of UC Davis, California.
1:40:20
Again, I promise that wasn't.
1:40:23
That wasn't bias.
1:40:25
It's just we have the best.
1:40:27
Making their way to the University of Birmingham in the UK on the free for both package deal.
1:40:33
The whole lab is moving to the university.
1:40:38
This is a two-body problem.
1:40:39
This is a two-body problem.
1:40:40
Because they're married scientists.
1:40:42
Yeah.
1:40:43
They are married.
1:40:44
They've had decades at Davis.
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Mangun, distinguished professor of psychology and neurology, director of the Center for the Mind and Brain since 2002, former dean of social sciences, and was the author of the textbook, Cognitive Neuroscience, the Biology.
1:41:00
of the mind.
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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 Max Planck, past editor-in-chief of cognition, and a fellow of the Association of Psychological Sciences.
1:41:23
Across the two of them, they have a
1:41:25
14 trophies.
1:41:27
We're talking about winners here.
1:41:29
These are winners at the top of their game.
1:41:33
Again, Birmingham signed both of them.
1:41:35
She's going to arrive as professor of brain and language cognition in the spring, which has already happened.
1:41:42
He's going to arrive in the fall as the 125th anniversary chair, professor of cognitive neuroscience, and the director of the Center for Human Brain Health.
1:41:52
This is like...
1:41:54
Both are going to keep their emeritus ties to Davis.
1:41:57
They had mixed reasons on record, Swab told NPR.
1:42:00
The optimism she once associated with U.S. science now feels more present in Britain and Europe, plus a real offer, including the UK Global Talent Fund support.
1:42:15
you know, it's for man.
1:42:17
Gonna was, uh, adding Birmingham as an opportunity as much as a reaction.
1:42:22
Um, this is again, again, again,
1:42:27
We're losing out to Europe and something?
1:42:29
Yeah.
1:42:30
It's been a long time.
1:42:31
It's been a long...
1:42:32
In science?
1:42:32
It's been a very long time.
1:42:33
In science?
1:42:35
So I just wanted to give some coverage around this because we did put a new section onto the website, ffppod.com backslash funding.
1:42:43
There is an ongoing funding battle here in the U.S., not only around who has the power and authority to deal with grant making, but also what funding actually goes into...
1:42:55
the larger scientific apparatus.
1:42:58
For those who have not really looked at what funding goes where, it is quite interesting.
1:43:04
It's based off of the AAAS research and development funding data sets that have existed back through, I believe it's the 1970s and it's actually the 1960s.
1:43:18
And what's interesting about this is because it's R&D, it includes defense.
1:43:25
which is the lion's share of R&D spending, there's been some ebbs and flows.
1:43:30
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 U.S. has worked as it's related to major milestones across that timeframe, as well as being able to look at the 2026 active fiscal year budget
1:43:47
That was implemented and is currently having battles over money already written into law reaching the institutions that should be getting it as well as the ongoing battle for the FY fiscal year 2027 budget and the current proposals.
1:44:03
Both of these things are active issues in Congress.
1:44:06
So if you want to get engaged in.
1:44:08
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.
1:44:24
And science is one of those issues, particularly when you're looking at the hard sciences, especially.
1:44:31
There's a lot of complexity around some of the social sciences.
1:44:35
And if you look at the tracker on what these things actually fund and where they go, you may be surprised at how much we can do with very little.
1:44:43
as compared to, for example, defense spending.
1:44:46
So this is a culture issue.
1:44:49
It's a funding issue.
1:44:51
It's a competitive landscape issue because in some cases it's just people are giving better opportunity.
1:44:58
And this show lives off of the frontier research community.
1:45:04
And so we really want to make sure that you as listeners understand
1:45:07
understand what's happening right now today with some of our best and brightest and how we can continue to stay competitive as we move into the future
1:45:18
Yep.
1:45:18
And I mean, you know, we're going to cover science no matter where it happens.
1:45:23
Right.
1:45:23
We're going to cover science if it happens here.
1:45:25
We've covered several Chinese papers.
1:45:27
We're going to cover several European papers.
1:45:30
Anywhere that science happens, we are going to cover it.
1:45:32
But, you know, we're also American and we care about...
1:45:38
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 in the first year we kind of we kind of shied away from from doing this kind of stuff yeah
1:45:58
But I think we've decided as an organization, as an FFP nation, that this is important and we're going to start caring a lot more.
1:46:10
As always, I am your host, Lester Nare, joined by my co-host and our resident PhD.
1:46:18
We made it under the two-hour mark.
1:46:21
That's become the new benchmark.
1:46:24
Again, we are super grateful for all of you who listened to the end of the pod.
1:46:29
One last reminder, if you have made it this far, you should definitely enter the merch giveaway at ffppod.com backslash.
1:46:38
survey.
1:46:38
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 as we move into year two.
1:46:49
We are very excited in our next two episodes, if I'm not mistaken, we are going to be covering something that's very near and dear to our resident PhD, Krishna's heart.
1:47:02
And I will
1:47:07
Thanks to me.