Latest / From First Principles / AI Breaks a 90-Year Math Problem, Life’s Alphabet in Space, and Science Funding (EP 50)

Transcript

AI Breaks a 90-Year Math Problem, Life’s Alphabet in Space, and Science Funding (EP 50) · From First Principles

• 0:00
  The recipe for life may not be uniquely terrestrial.
• 0:04
  And we found all five letters of the genetic code floating around in deep space with our first astrobiology story.
• 0:13
  Congress decided where the money went.
• 0:16
  It's law, it's not like an opinion.
• 0:19
  And the White House has begun to say, hey, you're not getting any more money.
• 0:24
  He drops this tweet.
• 0:26
  Hello there.
• 0:26
  The Jacobian conjecture is false.
• 0:28
  Thanks.
• 0:28
  And T-H-A-N-X.
• 0:31
  Hello, Internet.
• 0:32
  This is your captain speaking, Lester Nare, and I'm joined, as always, by my co-host and our resident PhD, Krishna Chowdhury.
• 0:42
  This week, we have a rundown episode with three very intriguing stories.
• 0:50
  We're starting off with astrobiology, where new evidence suggests that asteroids are...
• 0:56
  possibly could have seeded early earth with life's molecular building blocks.
• 1:02
  We'll then cover an update on the ongoing funding battle between the White House's Office of Management and Budget, Congress, and the scientific research community on a new funding policy change that would rewrite how federal research grants and the money
• 1:19
  that ultimately makes all of the great stories that we cover on this pod happen.
• 1:24
  Who gets to decide who gets the money and how?
• 1:29
  The process by which it's awarded and distributed is important.
• 1:32
  And so we want to make sure we touched on this because this show would not exist without our current ecosystem.
• 1:38
  And we will lastly end with a new math discovery that has rocked
• 1:44
  social media this week as AI cracked yet another 100-year-old problem, leaving mathematics to grapple with the very rapid and very troubling and unsettling change in the landscape as open problems continue to be knocked down by frontier models and what this means for rapid progress in
• 2:08
  And along the way, I think we might learn a little lesson in 3D calculus.
• 2:13
  I'm going to need that one.
• 2:15
  For the AI story, this is going to be a little bit more of a deep dive in this rundown because it's pretty important.
• 2:21
  But as always, we're going to talk about the science from the ground up today because this is from First Principles.
• 2:43
  So today we are going to start off with my favorite subject, aliens, as many of you longtime listeners know.
• 2:51
  But what's interesting about this is the recipe for life may not be uniquely terrestrial.
• 2:57
  And we found all five letters of the genetic code floating around in deep space with our first astrobiology story.
• 3:06
  Yeah, this was super interesting to me because the Japanese space agency, JAXA, put out this paper showing that all five of the nucleotides that we know and love here on Earth with all of life, there's four in DNA, A, T, G, and C, and then there's one that RNA uses.
• 3:28
  It uses the U, uracil, instead of thiamine that's used in DNA.
• 3:32
  They found all five on an asteroid.
• 3:35
  This isn't the first time that this has happened, but the fact that it's happened over and over is, I think, a very big deal.
• 3:42
  And it's hugely important for origin of life debate, things like that.
• 3:49
  It's hugely important for this theory called abiogenesis, which scientists have debated for a very long time.
• 3:55
  It's this idea that the process by which life arises naturally can be from outer space, right?
• 4:04
  And actually from non-living matter.
• 4:07
  That's the idea, right?
• 4:08
  Abiogenesis means from non-living to living, a bio.
• 4:12
  Now the question is, on early Earth, which is around 4 billion years ago, this is right after the Earth formed and it cooled down to the point where it could support liquid water, life emerged remarkably fast.
• 4:25
  It's like there was liquid water and then immediately we have fossils of tiny bacteria-looking thingies.
• 4:33
  And the question is, how come it happened so fast?
• 4:36
  And what's the necessary chemical ingredients that...
• 4:38
  that can assemble so quickly to create something like life.
• 4:42
  One important hypothesis is exogenous delivery, meaning asteroids and comets.
• 4:47
  We already know that comets brought most of the water on Earth.
• 4:53
  Asteroids and comets, maybe they have genetic material and material for life, and they act like a galactic postal service where life originates somewhere else.
• 5:04
  Let's not worry about how it originated somewhere else,
• 5:09
  It originates somewhere else, and then an asteroid gets kicked out of that planet or whatever thing has the life.
• 5:17
  It roams around space for millions, maybe billions of years, and then it finds its way on Earth.
• 5:22
  As an example that we've seen recently that we've covered on the show is the interstellar object of 3i Atlas that came into our neighborhood that was one of these travelers that was roaming around.
• 5:34
  Obviously, it did not make impact with Earth, but we've also covered the crater in Arizona.
• 5:40
  Yes, we've covered the crater in Arizona.
• 5:42
  We've covered how...
• 5:45
  like actual Mars, a Mars meteorite could...
• 5:49
  impact Mars, travel to Earth, and then get on Earth, and that entire process can be just enough, like just the right conditions where life under the surface of the asteroid could survive.
• 6:05
  Maybe not on the surface because it's got to survive re-entry and all that kind of stuff, but just under the surface, if there's life, it could survive the impact, and it could actually create life on Earth.
• 6:15
  We've had several stories in the past that talk about this.
• 6:18
  So scientists have found individual organic compounds in meteorites that fell on Earth, right?
• 6:24
  But those samples always carried a major asterisk, which is mainly that you can have terrestrial contamination.
• 6:31
  The most famous of this is in the 1990s when Bill Clinton...
• 6:35
  And he got in front of the press.
• 6:37
  I think it was in the Rose Garden, maybe?
• 6:39
  Yeah, he was in the Rose Garden, and he said, I did not have...
• 6:44
  I always do this because we've covered this so many times.
• 6:47
  It's like a running joke now.
• 6:48
  But he was actually announcing a science paper
• 6:52
  by scientists at JPL that showed that a Martian meteorite that they had found in Antarctica might have had some bacteria-looking things.
• 7:02
  And then immediately the entire world community of scientists was like, well, no, that could be because you've either contaminated, and also just because it looks like a bacteria-looking thing doesn't mean that it's actually bacteria from Mars, right?
• 7:16
  So there's been a lot of debate about whether we can even...
• 7:20
  make those kinds of arguments, right?
• 7:22
  Because once a space rock lands on Earth, it immediately gets contaminated by Earth biology.
• 7:29
  So here, the hero of this story is a satellite slash bombing system by the Japanese space agency called Hayabusa 2.
• 7:42
  And what it did was go to an asteroid called NASA,
• 7:48
  R-Y-U-G-U.
• 7:50
  Hopefully I'm pronouncing that right.
• 7:51
  What it does is it gets near this asteroid.
• 7:54
  It deploys a kind of remote bomb that pulverizes part of that asteroid.
• 8:01
  And then it goes back to that site.
• 8:03
  goes down, and then picks up the fragments of the pulverized rock.
• 8:09
  This is the alternative plot to Armageddon.
• 8:13
  Instead of going onto it and drilling, they did it from the atmosphere.
• 8:18
  Yeah, and they did it remotely.
• 8:19
  Now we can do that, right?
• 8:21
  So...
• 8:23
  That's the premise of the whole thing.
• 8:25
  The next video actually shows, this is a video from the JAXA probe, and it's showing you're lower down into the part of the asteroid that's been pulverized, and you grab it, you grab that part, and you come out.
• 8:38
  It's pretty cool that we can successfully chase this spinning top asteroid, blast a crater, and then bring back pristine dirt from an asteroid back home.
• 8:49
  Who was the one who was like, guys, I have an idea?
• 8:52
  Let's take a bomb.
• 8:54
  Yeah.
• 8:55
  Let's explode it just above the surface so that we can then go down.
• 8:59
  Yeah.
• 9:00
  And then there's a bunch of dirt instead of rocks.
• 9:02
  Right.
• 9:02
  Because the idea is the surface is solid.
• 9:04
  It's a little bit.
• 9:04
  You can't.
• 9:05
  Yeah.
• 9:05
  And the rocks are kind of big and you don't want to have to have a drilling mechanism on the whatever probe you have.
• 9:10
  Yeah.
• 9:11
  It's clever.
• 9:12
  I'm very curious how that conversation went in the proposal meeting.
• 9:16
  Yeah, and the other thing I'm actually curious about is if that little bomb changed the orbit of Raigü, right?
• 9:22
  Because we've talked about how planetary defense, one of the biggest things that we can do is just like...
• 9:29
  Put a minor explosion to change the momentum of that asteroid so that it changes the trajectory around the sun.
• 9:34
  I'm sure they're actually doing follow-up experiments to see if they can detect that tiny shift in momentum, which is extremely difficult.
• 9:42
  We've covered this in the past.
• 9:44
  So JAXA and the Hayabusa team, they came out with a paper in Nature Astro.
• 9:53
  a complete set of canonical nucleobases in the carbonaceous asteroid which means they have found A, G, C, T and they found U which is the alphabet of life in some sense right?
• 10:05
  It's all the ingredients we need to ratatouille ourself into the way of life.
• 10:09
  Yes exactly and if we go inside the lab you actually have a photo of the sample that they've got back from the asteroid.
• 10:17
  So there you can see the dirt from the asteroid.
• 10:19
  What they did was take that sample
• 10:22
  run it through mass spectrometry, chemical analysis, and they could actually identify individual nucleotides, the same ones that make up our DNA in that dirt, which is, I think, pretty crazy.
• 10:36
  It's a big deal for a number of reasons, not only the execution of it, but to your point earlier, we've removed the contamination hypothesis from the list of options.
• 10:45
  Yeah.
• 10:46
  We've removed the contamination hypothesis and it's the same nucleotide bases that we have, which is kind of like like it's kind of weird that like aliens would also have the same nucleotides that we did.
• 11:00
  So you're saying there's a chance.
• 11:01
  Yeah.
• 11:02
  It's kind of crazy.
• 11:03
  Right.
• 11:04
  And the big picture is so this isn't actually the first time.
• 11:07
  OK. NASA.
• 11:09
  had a similar experiment in 2023 where they went to the asteroid Bennu and they found samples that had the five nucleotide bases.
• 11:19
  What's cool about this is, one, that wasn't a fluke.
• 11:22
  The fact that it's happened again on a different asteroid means that this is something that is probably ubiquitous.
• 11:28
  in asteroids and comets.
• 11:30
  Which is incredible.
• 11:31
  Which is cool.
• 11:31
  Yeah.
• 11:32
  That's incredible.
• 11:32
  It's not a one-off thing.
• 11:33
  It's not like a special asteroid.
• 11:34
  We got lucky one time.
• 11:36
  Yeah.
• 11:36
  That's not what we're talking about.
• 11:38
  Twice and we find nucleotide bases.
• 11:41
  Not that this... And there's another variable.
• 11:42
  In a short time frame as well.
• 11:45
  It's not this is every 100 years, every 300 years, every 1,000 years.
• 11:49
  Something happens to come around that has it.
• 11:51
  This was in a... Yeah, this is just local to our solar system.
• 11:55
  Which is also crazy.
• 11:56
  Yeah.
• 11:56
  Yeah.
• 11:57
  I think it's really cool.
• 11:59
  And what you can do now, and what the paper did, is specifically compare the Raigü composition of nucleotides with the Bennu composition of nucleotides.
• 12:10
  And they actually have different ratios.
• 12:12
  Okay, that's interesting.
• 12:12
  So both asteroids have the same building blocks of adenine, guanine, cytosine, thiamine, and uracil, the same nucleotides that we have.
• 12:20
  But the proportions are completely different.
• 12:23
  In Benyu, it's heavily skewed towards the pyramidines, which are the cytosine, the thiamine, and the uracil.
• 12:29
  But in Rhaegu, they're more balanced.
• 12:30
  Now, what does that mean?
• 12:31
  That means that, for one, there is complex chemistry happening.
• 12:36
  Right.
• 12:37
  Either it's happening at the genesis of when the asteroid like formed or it's happening out in deep space when this stuff interacts with the ammonia that's on the asteroid, all the other stuff that's on the asteroid and with the solar wind and with radiation from all over the place like cosmic rays.
• 12:53
  It could mean that it has something to do with the water rock interactions that are on the asteroid, the internal temperatures.
• 13:00
  But the main thing is it's a lot more complicated up there, right?
• 13:05
  There's like actual chemistry happening.
• 13:07
  And it doesn't have to be on a planet.
• 13:09
  We have a cosmic oven, a cosmic easy bake oven that is our intergalactic postal service, like you sort of mentioned.
• 13:18
  And I think part of the point, what I'm getting from the point you're bringing up here is, you know, we like to keep our assumptions conservative when it comes to particularly things like...
• 13:34
  What's happening on earth is happening everywhere else.
• 13:36
  We like to see it before we believe it.
• 13:39
  Sure, maybe it's happening other places, but let's actually get not only one use case, Bennu, but multiple use cases.
• 13:48
  And the idea that we're having now...
• 13:53
  A catalog.
• 13:53
  Now we now say there's a catalog.
• 13:55
  Yeah.
• 13:55
  Because it's more than one.
• 13:56
  Yes.
• 13:57
  And as we do more and more, I'm sure we're going to find more and more asteroids with these building blocks, right?
• 14:03
  It's like the universe is like inherently preloaded with the chemical ingredients that are necessary for life.
• 14:08
  It's kind of insane.
• 14:09
  It's really fascinating.
• 14:11
  This is...
• 14:12
  And there's so many, again, it feels like, well, we've already known this.
• 14:16
  Like, why does it matter?
• 14:18
  But as we know with the process of science, like being able to have replicability matters a lot.
• 14:27
  These missions are not cheap.
• 14:28
  No.
• 14:29
  Right?
• 14:29
  No.
• 14:30
  To pull this off.
• 14:31
  Yeah.
• 14:32
  And this is two different space agencies doing it.
• 14:34
  Right.
• 14:35
  Right?
• 14:35
  So there's no conspiracy unless Japan is in on it.
• 14:38
  Right.
• 14:38
  Somehow.
• 14:39
  Right.
• 14:39
  Which...
• 14:40
  Look, look, we'll put the I'm just I know.
• 14:43
  Oh, I know.
• 14:44
  Yeah, of course.
• 14:45
  They're collaborating to conspire here.
• 14:47
  But it's not even something that's that, you know, quote unquote, sexy.
• 14:50
  Yeah.
• 14:50
  Like even waste the time to do so.
• 14:53
  This is a really big deal.
• 14:54
  And I do think the mix difference is also quite interesting because I imagine that there are several follow up inquiries into.
• 15:06
  seeing their journey and understanding based on that journey, why is that mix different?
• 15:12
  Yeah.
• 15:14
  Fascinating.
• 15:15
  Very, very cool.
• 15:16
  This is fascinating.
• 15:17
  So, you know, as always, like, you know, I'm the alien guy, and this is great, but we're going to move quickly because this is a rundown episode, and we have a big pseudo deep dive at the end.
• 15:28
  But before we do so, we're going to do a quick piece of housekeeping.
• 15:31
  Yeah.
• 15:32
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• 15:42
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• 15:52
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• 15:54
  We always, if you can, recommend watching because each episode we go through 25 to 65 different overlays where we actually show what's happening.
• 16:06
  And it really helps to understand what's going on when you can see it.
• 16:09
  But we are trying to work on making sure for our audio listeners that we describe as much as possible.
• 16:15
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• 16:18
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• 16:23
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• 16:31
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• 16:37
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• 16:43
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• 16:52
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• 17:00
  as it relates to science.
• 17:03
  Our second story is going to be about funding policy in the United States as it relates to science.
• 17:12
  Every time we cover a study on the show, um, it could be genetics, medicine, physics, space.
• 17:20
  We always talk about the final product, which is the published paper and usually the history and the buildup to that.
• 17:26
  Um,
• 17:27
  But what we don't talk about or see is the federal funding system that actually enables a lot of this fundamental research to happen.
• 17:35
  The number one funder of basic research for the American ecosystem is the federal government, and it's what makes it possible.
• 17:45
  And there's a recent controversy that's arisen around this, but...
• 17:50
  Before we get there, I thought it might be helpful to briefly talk about the basic process because many folks might not understand how this works.
• 18:00
  So in the U.S., Congress has what's known as the power of the purse, or if you're from Boston, Congress.
• 18:07
  Pocket book.
• 18:08
  But for the rest of us, the purse.
• 18:09
  The president proposes a budget every year.
• 18:12
  At the beginning of the year, he or she proposes a budget.
• 18:15
  But Congress ultimately decides how federal agencies are legally permitted to spend money and for what purposes.
• 18:23
  So the executive branch has all these responsibilities.
• 18:25
  But money and where it gets spent, Congress ultimately is decided there.
• 18:30
  OMB, which is the White House Office of Management and Budget, helps to execute the budget that is fundamentally decided by Congress.
• 18:39
  It apportions funding to executive agencies and sets government-wide rules about how that money can be spent.
• 18:47
  So Congress says, here's the money and who can spend it, and then OMB kind of just distributes it.
• 18:54
  how that goes.
• 18:54
  And this is how federal grants, especially in research, but not only in research, are administered.
• 19:00
  Okay.
• 19:02
  Normally, in the history of this process, the Office of Management and Budget doesn't select individual science awards or grants.
• 19:13
  It's not in the process of selecting who gets what money.
• 19:17
  That happens at each particular agency, National Institutes of Health, National Science Foundation, NASA, Department of Energy, where the experts are.
• 19:26
  They are the ones who get to decide how to.
• 19:29
  That happens.
• 19:29
  And the process is the agencies will announce a funding opportunity.
• 19:33
  Researchers will submit proposals.
• 19:36
  Experts evaluate their merit.
• 19:37
  Yep.
• 19:38
  As you know very well in this process.
• 19:41
  Yeah.
• 19:41
  And the agency officials ultimately decide which projects get to receive.
• 19:46
  Yeah.
• 19:46
  And shout out to my dad.
• 19:48
  He's one of the astrophysicists that's on a lot of these panels that like decides which grants get awarded and things like that.
• 19:55
  And an expert being on that is something you absolutely want to be there.
• 20:00
  And then from there, the university or research institution that receives the funding employs the researchers.
• 20:07
  They operate the labs.
• 20:08
  They conduct the work.
• 20:09
  And eventually produce what we talk about on the pod, which is the paper.
• 20:13
  So Congress funds it.
• 20:16
  OMB establishes the rules.
• 20:18
  Agencies select and manage the grants.
• 20:20
  Researchers do the science.
• 20:22
  That's how the process has worked forever.
• 20:27
  But now, you know, we're going to get into the current fight.
• 20:30
  Right.
• 20:31
  Before we do, I do want to give a lot of credit here to many of the science policy journalists who have been doggedly covering this story closely, particularly Jeffrey Mervis, Jocelyn Kaiser, Lauren Martin Agudelo, Michael Greshko, and David Malikoff at both Science and AAAS.
• 20:49
  Yes.
• 20:50
  Um, this is sort of in the weeds and they've taken a very complex bureaucratic funding and policy process and have tried to make it connect to the actual science and impact that it has for people.
• 21:04
  And this is really, again, a great source for us as we try to begin to now incorporate science funding policy into our discussions because we would not be able to, again, have a show without this.
• 21:16
  Yeah.
• 21:17
  So now we get to the current fight, and it's always over who has the power to decide.
• 21:24
  So I'm going to kind of read through a couple of the details here, which are a little bit in the weeds.
• 21:28
  But so on May 29th, OMB can only decide the rules, right?
• 21:35
  Right.
• 21:36
  OMB released this 412-page proposal to sort of revamp their role in distributing and managing grants and awards.
• 21:45
  And this is the White House effectively.
• 21:47
  This is the White House.
• 21:48
  And so this is a very important note.
• 21:50
  It's not some – it's literally the White House's Office of Management and Budget.
• 21:56
  Granted, they have a role in this process because they are the ones who initially derive and drive the national priorities when the budget process starts in January and February of every year.
• 22:09
  So this new proposal had until July 13th to provide comments.
• 22:14
  And OMB said that it wanted to sort of finalize these proposals.
• 22:18
  new rules by October 1st.
• 22:21
  The proposal applies to all federal grants, not just specifically science, but several of the provisions are kind of important.
• 22:30
  So we're gonna kind of just briefly walk through a few of these.
• 22:33
  So first, senior political appointees
• 22:36
  could independently review discretionary grant decisions before awards were issued, right?
• 22:42
  Even before they were actually issued to people, number one.
• 22:45
  Number two, peer review would, quote, remain a part of the process, but the proposal makes clear that scientific reviewers' recommendations, like your dad, as you just mentioned, would be advisory recommendations,
• 22:58
  And not controlling.
• 22:59
  Okay.
• 23:00
  Right.
• 23:00
  Currently, they have the controlling.
• 23:02
  What they say kind of goes.
• 23:03
  They're saying that's just more of just an opinion.
• 23:05
  And we get to decide.
• 23:07
  And we being these political appointees.
• 23:09
  Right.
• 23:10
  Great.
• 23:10
  Nice.
• 23:11
  Don't really have any scientific background.
• 23:14
  Yeah.
• 23:15
  Third, the government would have broader authority to end existing grants if officials determine they no longer advance the current agenda of whatever party controls the White House at that time in terms of the, quote, national interest.
• 23:30
  Okay.
• 23:31
  And then fourth, the proposal would impose major restrictions on international research collaborations with countries that are deemed to be a national security risk.
• 23:40
  The fourth one –
• 23:42
  Fourth one I get.
• 23:44
  That one has some standing and you could argue the details there.
• 23:48
  Yeah.
• 23:48
  Okay.
• 23:49
  So what is the argument from the White House, right?
• 23:52
  They're basically saying that they want to ensure that taxpayer money is lawfully executed on and that it's aligned with our priorities and there's no – the favorite phrase of D.C. – waste, fraud, and abuse.
• 24:04
  Right.
• 24:05
  I don't think if you look at the grand scheme of the budget that scientific funding and grant funding is the biggest source of waste, fraud, and abuse.
• 24:10
  But they argue that the peer review process has become insular and the final responsibility should remain to politically accountable appointees.
• 24:21
  That's the argument being made for why this proposal is being put in place.
• 24:24
  Okay.
• 24:27
  In a vacuum, you would maybe say, okay, maybe there's an argument here.
• 24:36
  But obviously the scientific community has reacted very strongly about this going well beyond science.
• 24:42
  financial oversight this is clearly a battle about power and culture and culture yeah because science which has always kind of brushed at the edges of culture has never been quite as part of a zeitgeist the zeitgeist as it has since covet happened in my view frankly at least in american culture yeah um where it's really become now a culture war issue in a way that it really never was before yeah
• 25:07
  So obviously the scientific community is arguing this is going to allow ideological nonsense to get involved in the process.
• 25:14
  So the Association of American Universities submitted a 47-page response, which I might add is the longest public comment in the organization's 126-year history.
• 25:27
  So clearly it struck a nerve because everyone understands what's going on here.
• 25:33
  And the primary – there's so many arguments to be made here.
• 25:37
  But the primary argument that they're making is that Congress, power of the purse, they ultimately are the deciders of where the money gets ascribed in terms of agency.
• 25:46
  That's right.
• 25:47
  Has not given OMB the authority to decide what categories of scientific research exist.
• 25:52
  may receive federal funding.
• 25:54
  That power lies purely within the agencies and their statutory directives.
• 26:00
  So effectively, the universities are using the Constitution to try and defend the way that we fund science.
• 26:06
  Which is...
• 26:11
  It is in the Constitution.
• 26:12
  Yeah.
• 26:13
  Literally.
• 26:13
  As well as the actual mechanism of the movement of and how money is budgeted.
• 26:20
  Like Congress decides how money is spent.
• 26:23
  Yeah.
• 26:24
  Period.
• 26:24
  The executive branch is effectively trying now to insert themselves in the process of how money is spent.
• 26:31
  Yeah.
• 26:32
  So there's to your point, there's a constitutional level issue here beyond even some of the practical matters.
• 26:39
  Yeah.
• 26:40
  What's interesting is this proposal generated almost 500,000, 497,000 public recorded submissions.
• 26:48
  So anytime there's proposals like these at different agencies or offices within all the branches of government, they open up it to public comment.
• 26:57
  So almost half a million comments were like, obviously, this is ridiculous.
• 27:02
  So again, I know this is dense, but I think this is really important because the implications here could not be more important to cover.
• 27:11
  So that's what's currently happening in terms of OMB trying to change the process.
• 27:17
  But the administration that's currently in the White House has already terminated and suspended several existing grants that were appropriated by Congress last year.
• 27:31
  Yeah.
• 27:31
  So Congress decided where the money went.
• 27:34
  It's law.
• 27:35
  It's not like an opinion.
• 27:38
  And the White House has begun to say, even for either programs that have yet to receive allocations or some programs that have received their first batch, but not the second, to say, hey, you're not getting any more money.
• 27:52
  something you may or may not have some context for.
• 27:56
  Oh, yeah.
• 27:56
  I mean, I have several friends in the university ecosystem that are very worried about their future and their present scientific endeavors, right?
• 28:04
  Like they're in the middle of science projects that they have a few experiments left before they publish the big paper in Nature and Science and Cell and all this other stuff.
• 28:16
  And, yeah, yeah.
• 28:17
  All of a sudden, the grant funding is gone and you haven't quite finished the story.
• 28:23
  What do you do?
• 28:24
  Right.
• 28:24
  It's it's really tragic.
• 28:28
  So as we have two sides of the story here, there's the proposal that's happening, but there's actions that have already been being taken since January of last year.
• 28:38
  And on July 17th, the federal judge ruled that agencies cannot use newly adopted political or pragmatic priorities to retroactively cancel grants under different criteria.
• 28:52
  Yeah.
• 28:53
  So if in 2025, Congress appropriated money based on the national priorities at the time, and then in 2026, a new administration comes in and says, we don't want to do that anymore.
• 29:03
  You can't do that.
• 29:04
  So that's been established in federal court.
• 29:09
  Now, a caveat to that is the government may retain legitimate authority to terminate grants, which will still continue to be challenged.
• 29:18
  But the judge's central point was that it cannot change the rules after an award has already been made, which is where we get the proposal language, which says they can involve themselves prior to the awards being distributed.
• 29:31
  This has continued where the Department of Health and Human Services notified 70 researchers funded by the Agency of Healthcare and Research Quality that their projects would not receive their next annual funding installments.
• 29:43
  So this is not even just in things like physics and biology.
• 29:47
  This is also impacting technology.
• 29:49
  health research studies, those projects represented approximately $185 million in multi-year commitments.
• 29:56
  And so they've already received a distribution, and they're saying you're getting no more money.
• 29:59
  Yeah.
• 30:00
  Even though they've already received the money for it.
• 30:02
  Yeah.
• 30:04
  What they're trying to do now as the legal battle has continued is change the language about what it is that they're doing.
• 30:12
  So HHS said the grants were not technically terminated.
• 30:16
  They were simply not awarded continued funding because it's a multi.
• 30:21
  So rather than saying we're terminating funding for something that is a single allocation, we're saying, oh, this was a multi-year thing.
• 30:28
  And we're discontinuing the continued funding of what is a multi-year install.
• 30:34
  that doesn't make any sense are you serious so it's basically termination under a different name termination but they're trying to just voodoo magic they're saying they're not terminating they're just discontinuing right right which very nice i'm not the thesaurus but they may be synonyms in this context yeah um
• 30:55
  And this is so this is both a constitutional and budgetary dispute that that's important.
• 30:59
  So Congress does not vote on individual research projects.
• 31:02
  Right.
• 31:03
  Right.
• 31:03
  It just appropriates money to an agency.
• 31:05
  That agency chooses individual grants.
• 31:08
  Multi-year grants may also receive their funding in annual installments rather than all at once.
• 31:12
  So you have a program that gets a $50 million allocation in 2025, but it's spread over four years.
• 31:17
  Yeah, yeah.
• 31:18
  I mean, a lot of the, like, some of the grants that I was supported during my PhD were these multi-year grants where, like, my advisor or, like, the collaboration that I was a part of would get money every year, right?
• 31:32
  You don't get a lump sum.
• 31:33
  And so the debate is really around like the executive branch doesn't have unlimited discretion to decide their role in, again, managing the rules.
• 31:42
  Yeah.
• 31:43
  It's an administrative rule.
• 31:44
  Yeah, it's never been that way.
• 31:45
  It's not a discretionary rule.
• 31:47
  And this is a very, very contentious issue right now.
• 31:51
  So the central dispute is whether...
• 31:54
  the administration is simply managing the congressional funded programs or whether replacing Congress's priorities has, you know, their own right to do like, if they have their own right to basically displace what Congress has already decided to do.
• 32:11
  And the last note here before we wrap up on this is there's sort of three different fronts that this battle is being fought
• 32:23
  OMB, Office of Management and Budget, is trying to rewrite the government-wide rules that then gives them authority to say you can't get money.
• 32:31
  Congress is in the process of negotiating the next fiscal year's appropriation bills at this same time.
• 32:40
  And federal courts are weighing in on how much authorities the White House effectively has on canceling existing awards.
• 32:53
  Three branches of government, the executive, the legislative, and the judicial are concurrently engaging in a battle to really define what happens here.
• 33:04
  The comment period for the rule proposal changes is over.
• 33:09
  uh, OMB wants to set the statutory deadline for October 1st, at which point this would then take play.
• 33:16
  This, these new rules would go into place for fiscal year, 2027.
• 33:21
  They're making the argument that this is about accountability.
• 33:25
  It is clearly not account about accountability because there's already systems in place to deal with that.
• 33:31
  Uh, the real kind of, uh,
• 33:34
  issue is does the executive branch have the authority to make individual funding decisions within programs that Congress has already authorized?
• 33:45
  And again, this goes back to it's a fundamental constitutional question.
• 33:49
  Because it literally goes counter to the structure of our co-equal branches of government.
• 33:57
  So will federal research continue to be evaluated under rules disclosed in advance, informed by expert review, and tied to programs funded by Congress?
• 34:08
  Or will every grant now be contingent on the changing political priorities of whoever's in the White House?
• 34:15
  Yeah.
• 34:17
  It moves all the time.
• 34:18
  So just because it stops for a four-year cycle or an eight-year cycle does not mean it's then going to flip the other direction.
• 34:24
  And we cannot afford to have a seesaw problem in what has made this country, a huge part of what has made this country stable and successful and resilient.
• 34:40
  is the fact that how we won World War II is because we've done this funding in a way that's already been structured.
• 34:47
  Yeah, I mean, you can go back to our America 250th anniversary episode, right?
• 34:52
  Like, I think something like 80% of the stories that we covered were because of federal funding or because of, like, funding that came from taxpayer money somehow, right?
• 35:04
  And I'm sure a lot of those grants were awarded...
• 35:09
  Without political oversight.
• 35:10
  Correct.
• 35:10
  Okay.
• 35:11
  Because I doubt the powers that be are going to understand why it's important to study bacterial immune systems.
• 35:19
  Okay.
• 35:19
  But then we get like restriction enzymes, genetic engineering, CRISPR.
• 35:25
  Come on, guys.
• 35:26
  What are we doing?
• 35:27
  I don't want the people that did not understand that studies using transgenic mice.
• 35:32
  Yeah.
• 35:33
  Wasn't talking about the.
• 35:37
  The gender of mice.
• 35:38
  Yeah, yeah, yeah.
• 35:39
  Like, you know what I mean?
• 35:40
  Like, that's... We can't... Yeah, transgen, and then they're like, Ugh!
• 35:45
  Guys.
• 35:46
  Like, how...
• 35:47
  They should not be in charge of this kind of power.
• 35:51
  This is insane.
• 35:52
  We're gonna keep you all updated on this process, because it's happening in real time.
• 35:57
  We will know more over the next couple of weeks.
• 35:59
  The one...
• 36:01
  note that I'll make is this is not we're not passengers to this process and Congress is not particularly happy about this because whether Congress cares about the underlying science or not, Congress is not in the business of allowing the executive branch to take power away from them.
• 36:15
  Yes.
• 36:15
  Yeah.
• 36:16
  Period.
• 36:17
  And so it is now becoming a battle between the legislative and the executive.
• 36:22
  And it's a midterm election year.
• 36:24
  And so members of Congress are particularly sensitive to issues that are animating their constituents.
• 36:32
  And this is something that if you believe that basic science research is something that's important and that you value and that you think should continue to exist in the construction that it does, you have the opportunity as we get into August recess, members of Congress are going to be going to their home districts.
• 36:52
  attend, call their offices, send them emails.
• 36:55
  They track this stuff.
• 36:57
  One of the greatest secrets ever is that people don't understand how much these members actually listen to people who reach out.
• 37:03
  The problem is no one reaches out.
• 37:04
  And so all these specialty issues get kind of overlooked because no one talks about it.
• 37:11
  So it's time to put the pressure on on this issue.
• 37:14
  Again, the system is working totally fine.
• 37:18
  We don't want the White House of any administration
• 37:21
  involving themselves in discretionary issues around funding as it relates to Congress's mandate and the mandate of the agencies who do have the experts to make these decisions.
• 37:31
  And again, if you want changes around who those experts are in those agencies, there's a process for that.
• 37:37
  We don't need the White House involved.
• 37:38
  We really don't.
• 37:41
  Well, yeah, that's a... And you said this is a developing story, right?
• 37:45
  So the decision is moving to October 1st.
• 37:48
  So it's like right before the elections.
• 37:50
  Right.
• 37:50
  It's going to be a hot-button issue, and we need to make it a hot-button issue.
• 37:54
  A hot-button issue.
• 37:55
  Yeah.
• 37:56
  You know...
• 37:58
  You guys, federal funding of science is the bedrock of American innovation.
• 38:04
  And American innovation is the bedrock of why this country is so great.
• 38:09
  Okay, so we need to get on it, guys.
• 38:12
  It matters a lot.
• 38:13
  Yeah.
• 38:14
  It matters a lot.
• 38:15
  And I know some folks' eyes will glaze over, but again, we wouldn't be able to talk about something like our next story without this ecosystem of funding because ultimately doing anything costs money.
• 38:29
  And I'm really excited about this story because it was all over my feed.
• 38:36
  All over my feed.
• 38:37
  Yeah, this was kind of huge because it happened in the middle of the World Cup final.
• 38:45
  It was actually crazy.
• 38:46
  The tweet that is at the center of this major, major mathematical breakthrough was tweeted in the middle of the World Cup final, which is when the entire world was fixated on something else.
• 39:01
  Who won that?
• 39:02
  I don't know.
• 39:03
  I can't remember.
• 39:04
  I know it wasn't Argentina.
• 39:06
  It wasn't Argentina.
• 39:07
  That's right.
• 39:08
  That's right.
• 39:08
  It was not.
• 39:09
  It was not.
• 39:10
  It was not Argentina.
• 39:12
  That's right.
• 39:12
  You love to see it.
• 39:14
  I just, you know, for everyone who suffered through our World Cup coverage, it's over.
• 39:18
  So we'll stop the sports ball.
• 39:20
  But I just, I couldn't remember.
• 39:21
  Yeah, yeah.
• 39:22
  It's good to remind oneself.
• 39:24
  Just to set the record straight.
• 39:26
  That's right.
• 39:27
  That's right.
• 39:28
  So this next story is about AI and how it is encroaching on fundamental mathematics.
• 39:35
  It's very, very cool because, I mean, it's completely obvious, I think, to everyone who does not live under a rock that we are living through one of the most disorienting times in the history of human intellectual life.
• 39:50
  OK, this is especially true when it comes to really, really hard intellectual disciplines like mathematics, because for thousands of years, mathematics has been the one domain where human beings can claim something that is very close to absolute certainty.
• 40:05
  Right.
• 40:05
  We can say that the act of understanding and the act of proving are one and the same thing.
• 40:12
  and a mathematical proof, it's been here since the time of the ancient Greeks with Euclid.
• 40:19
  It's a certificate of the truth, right?
• 40:23
  And it's a map as to why something has to be true, and very crucially, for the longest time, it's been something that only human minds can produce.
• 40:33
  Very important point.
• 40:34
  Very important point.
• 40:35
  It's, you know, we've seen a lot of disciplines
• 40:39
  fall to computation.
• 40:42
  Chess has fallen to computation.
• 40:44
  The game Go has fallen to computation.
• 40:47
  Protein folding has fallen to computation.
• 40:50
  And now we are finally seeing the...
• 40:55
  The shining city on the hill, which is pure mathematics, it is starting to fall, right?
• 41:00
  There's crumbles in the infrastructure.
• 41:04
  I just want to briefly note that, you know, in our time at Princeton, some of the smartest people we had the pleasure of being able to be around and breathe the same air that they did were smart.
• 41:17
  the maths folks yeah we're the maths folks yeah like yeah like they were a level of smart that is really hard to wrap my own head around yeah yeah yeah same for me i mean i i i took some courses with them right and i was just like i'm gonna stick to it
• 41:34
  I'll stick to the physics department, which is right next door.
• 41:37
  I took a few math courses and it was extremely fun, but I could just never see myself being a math major.
• 41:43
  I mean, even in the physics department, I was like, man, what am I doing here?
• 41:47
  But the math department was its own little game.
• 41:51
  And so this particular story has to do with an AI system.
• 41:55
  It was used by a researcher at Anthropic, which is like the synonymous, the company of Claude, the company that wants AI safety, AI safety.
• 42:08
  Regulatory capture.
• 42:09
  Right.
• 42:09
  And it has apparently disproved something called the Jacobian conjecture, which is a problem that has been at the center of algebraic geometry and the theory of polynomial maps for about 90 years.
• 42:24
  It's insane.
• 42:25
  It's not proved it in the conventional sense because what it's actually done is disproved.
• 42:30
  the conjecture.
• 42:31
  It is provided a counter example which is a mathematical object that violates whatever the rule is.
• 42:39
  It's a bit easier to do, albeit, than actually proving something, right?
• 42:43
  If you if you say a statement, proving the statement is a lot harder than just saying, hey, here's an example that shows it's not true.
• 42:50
  Because all you have to do is produce an example and boom, the debate is over, especially in mathematics.
• 42:55
  We're going to get into sort of
• 42:57
  how that works and why it's kind of easier than proving, but I think it's still an insane achievement that a machine, effectively, a neural network that is just doing multiplications of matrices at the end of the day, is creating this kind of truth.
• 43:17
  Out of something that for 90 years the top mathematicians have been apparently wasting their time with.
• 43:23
  Yep.
• 43:24
  Trying to prove.
• 43:25
  And this is part of the ongoing debate about whether these AI systems are, quote, stochastic parrots that...
• 43:34
  don't have any original thought and or anything that they do that's original is just by gathering all the information on the internet and or brute forcing and i understand that sentiment but i think it is a little bit more nuanced than the way in which it's presented in this black and white way exactly and i think like if you're not a mathematician i
• 43:59
  this should still matter because for one, it's showing the power of AI to like think complex things in a very, in a very real sense, right?
• 44:10
  AI is restructuring the labor markets.
• 44:13
  And now it's going after really creative mathematics is a creative discipline, right?
• 44:20
  And a lot of people say that AI can't be creative.
• 44:22
  It's as you said, a stochastic parrot.
• 44:24
  Well, in this case, nobody had a, nobody had a proof of this.
• 44:28
  And it has a novel proof that the smartest people on the planet couldn't come up with.
• 44:32
  For almost a century.
• 44:34
  Yeah.
• 44:36
  It's huge, right?
• 44:37
  So mathematics, right, as I said, it's been the gold standard of human reasoning.
• 44:43
  It's the language of physics, the foundations of electrical engineering, every other kind of engineering, economics, medicine.
• 44:50
  When we say...
• 44:52
  Literally everything.
• 44:53
  Literally everything.
• 44:54
  And when we say that something is mathematically proven, we mean it is literally true.
• 44:58
  Right?
• 44:59
  It's not something out of physics where like you can do an experiment and then 400 years later somebody can disprove the experiment because they went farther in the decimal places to measure something.
• 45:09
  Which has happened time and time again.
• 45:10
  In mathematics, the decimals are inherent in your proof.
• 45:14
  Right?
• 45:15
  It's like it's there and it's true.
• 45:16
  And if I've proven that it's true, the logic is eternal.
• 45:22
  Would it be fair to say that mathematics is one of, if not the only thing that we have where you can say that there is an objective truth when it comes to a mathematical proof?
• 45:31
  Yes, it is the pedestal of objective truth, right?
• 45:37
  You go from there are integers...
• 45:40
  And I can add and multiply them to then all of the stuff, right?
• 45:44
  Euclid's theorem that there are an infinite number of prime numbers, that still holds true today.
• 45:50
  Because no one can poke a hole into it.
• 45:52
  Why?
• 45:52
  Because it's a proof.
• 45:54
  They couldn't poke a hole back then because the logic was sound.
• 45:58
  And logic is something that is eternal.
• 46:00
  And so it's really cool.
• 46:03
  And when we start thinking about AI starting to do mathematics, not just assisting and not just suggesting, but actually proving and disproving things, then we start asking philosophical existential questions.
• 46:17
  And those become urgent, right?
• 46:18
  What is the role of the human mathematician?
• 46:21
  What is the role of the human mind?
• 46:23
  What does understanding mean if a machine can generate a proof that is true, that we can verify somehow through some kind of computation to be true, but we can't even understand it, right?
• 46:35
  These are now becoming real questions.
• 46:38
  It's kind of crazy.
• 46:39
  Yeah.
• 46:39
  So this is going to be a deep dive where I'm going to talk about the Jacobian conjecture, which is the thing that this AI model has disproven.
• 46:48
  We're really going to get into the weeds of it, and I think you're going to be able to understand exactly what it's done.
• 46:53
  And then we're going to talk about the modern phenomenon of AI and mathematics and why everyone, including mathematicians, are very worried.
• 47:06
  Okay?
• 47:07
  Okay.
• 47:08
  So let's start with the Jacobian conjecture.
• 47:10
  It's something that was proposed by Heinrich Keller in 1939, so nearly 100 years ago.
• 47:16
  And it addresses a fundamental local-to-global paradigm.
• 47:20
  It has to do with maps.
• 47:23
  Okay?
• 47:24
  By maps, I mean...
• 47:27
  going from one set of coordinates to another set of coordinates.
• 47:31
  So in order to understand the Jacobian conjecture, we have to first understand functional maps, which in physics language we like to call coordinate transformations.
• 47:41
  Here's the idea.
• 47:43
  A map...
• 47:45
  in this mathematical sense, is when you take a coordinate plane or a hyperplane or many-dimensional plane, and all of the points in your original space get mapped to a new space.
• 47:59
  In this case, what we're seeing is a 2D plane
• 48:03
  being mapped to another 2D plane.
• 48:05
  Okay?
• 48:06
  There's some kind of nonlinear transformation where the new x-coordinate and the new y-coordinate depend on the old x-coordinate and the old y-coordinate, right?
• 48:14
  In this case, a square is taken into a parallelogram.
• 48:19
  Notice that the new X can depend on both the old X and Y, not just the X.
• 48:27
  And the new Y can depend on both sets of coordinates.
• 48:30
  This can happen in higher dimensions as well.
• 48:33
  The Jacobian is the ratio between the new area and the old area.
• 48:40
  Which can also scale from 2D.
• 48:43
  Yes.
• 48:44
  You can have volumes, for example.
• 48:45
  You can have a volume in 3D that goes to a new volume in 3D, right?
• 48:50
  A cube can become like a weird kind of rhomboid, whatever, 3D prism is what it's called, I think.
• 48:57
  And then you're comparing the ratio between the volume or the area, depending on what dimensional space you're talking about.
• 49:02
  Between the two is where the Jacobian comes in.
• 49:05
  Yes, exactly.
• 49:06
  And in this case, I think from my eye, the area is about the same.
• 49:11
  So the Jacobian in this transform would be one.
• 49:15
  Okay?
• 49:16
  That's what the Jacobian means.
• 49:17
  Effectively, it's saying, okay, I have a map, and that map is preserving area if the Jacobian is one.
• 49:23
  If the Jacobian is greater than one, then it's going to larger cases, right?
• 49:27
  Right.
• 49:28
  And like when I think about maps, the reason why it's called a map is because it really starts from this concept of like maps of the earth.
• 49:36
  Right.
• 49:36
  You can think about the Mercator projection or these other projections that change the area and change the shape.
• 49:43
  But keep sorry, they change the shape, but they keep the area the same.
• 49:46
  For example, with the Mercator projection.
• 49:48
  Right.
• 49:49
  We all know that areas near the poles get larger.
• 49:52
  So the Jacobian near the poles is greater than one.
• 49:56
  Right?
• 49:57
  That's why Africa looks small and Greenland looks the size of Africa.
• 50:01
  Because the Jacobian near Greenland is much larger than the Jacobian near Africa when it comes to the Mercator projection.
• 50:08
  But there's other projections where they try to preserve the Jacobian everywhere, but then that's going to distort the shape in some sense.
• 50:15
  Exactly.
• 50:15
  So people would argue which is better when you talk about geographic maps.
• 50:20
  Is it better to keep the Jacobian closer to one?
• 50:22
  Yeah.
• 50:23
  Where you're going to get some shape distortion or is it better to have it have shape clarity?
• 50:29
  Yeah.
• 50:30
  But you have distortions at some extremes as it relates.
• 50:34
  Very.
• 50:34
  OK. Yeah.
• 50:35
  So that's what we're talking about when we talk about Jacobians.
• 50:38
  OK, it's really it's a it's a ratio of how my transformation is changing little tiny areas, infinitesimal areas.
• 50:46
  Right.
• 50:46
  Because in calculus, we think about like there's continuous transformation.
• 50:49
  So the Jacobian here can be different from the Jacobian there and so on and so forth.
• 50:53
  It's not a family in the Game of Thrones.
• 50:55
  No, no, not at all.
• 50:56
  And so let's do a very simple case, okay?
• 50:59
  A linear transformation.
• 51:00
  Yes.
• 51:01
  Now, in this case, this is a very simple matrix, 2, 1, 1, 2, okay?
• 51:05
  When we multiply that by any 2D vector...
• 51:10
  we're going to get a new vector.
• 51:11
  For example, the red vector is getting slightly skewed like that.
• 51:14
  So the red vector is really a 1, 0.
• 51:16
  That's the x-axis, just 1 on the x-axis, 0 on the y-axis.
• 51:21
  That gets transformed to 2 on the x-axis and 1 on the y-axis, right?
• 51:27
  Similarly, the y...
• 51:29
  unit vector, which is 0, 1, that gets transformed to 1, 2, which means 1 on the x-axis, 2 on the y-axis.
• 51:35
  That's why the green arrow is slanted more towards the y-axis, the red arrow is slanted more towards the x-axis.
• 51:41
  And in this case, the determinant of that matrix, if you were to do it, it's 4 minus 1, so that's 3.
• 51:47
  You can see that the square becomes a rhombus that's the size of 3.
• 51:52
  And it has to do with the determinant of that matrix.
• 51:56
  So when we say a Jacobian determinant, that's what we're talking about.
• 51:59
  Okay?
• 51:59
  Yes.
• 52:00
  Okay, so this is a very simple case.
• 52:02
  Now let's give you kind of a more complicated case, but one of my favorite coordinate transformations as a physicist, which is the Lorentz transformation.
• 52:10
  This is the transformation that we know and love from relativity.
• 52:14
  Right.
• 52:15
  Right?
• 52:15
  In special relativity, we have the Lorentz transformation.
• 52:19
  You might have heard that...
• 52:21
  When we move fast, length contracts and time dilates.
• 52:26
  So the clock runs slower and length gets shorter.
• 52:30
  That's two different things, right?
• 52:33
  Our coordinate in this case is, let's just think about 1D, right?
• 52:36
  There's a length.
• 52:37
  I'm like moving in this direction.
• 52:38
  So there's a length forward and backwards in the direction of my movement.
• 52:43
  And there's also time.
• 52:45
  I've got a time on my watch.
• 52:48
  Now, time gets dilated.
• 52:51
  What does that mean?
• 52:51
  That means that the clock is running slower, which means that a second is getting longer.
• 52:58
  Yeah.
• 52:59
  but my length is getting shorter.
• 53:02
  So one coordinate is getting shorter, and the other coordinate, the time coordinate, is getting longer.
• 53:09
  That is central to the physics of relativity.
• 53:12
  And let me show you by considering this plot.
• 53:15
  So let's consider this plot.
• 53:17
  This is a plot that almost every physics undergrad has seen.
• 53:21
  On the x-axis is space, on the y-axis is time.
• 53:26
  So on the y-axis, we've got 1 second, 2 second, 3 second, 4 second.
• 53:29
  On the x-axis is meters, but actually the 1 is not 1 meter.
• 53:33
  The 1 is 1 light second.
• 53:36
  So it's really like 300,000 kilometers, 600,000 kilometers.
• 53:40
  So it's the speed of light multiplied by 1 second, 2 seconds, 3 seconds, and 4 seconds.
• 53:44
  Why do we want to do that?
• 53:46
  Well, if we represent the x-axis that way, then the speed of light is a diagonal.
• 53:52
  Yeah.
• 53:52
  Right.
• 53:53
  Because it's like one to one.
• 53:54
  It's like, OK, in one second, it travels one light second.
• 53:57
  In two seconds, it travels two light seconds.
• 53:59
  So you've got a nice the dotted green line is what light would do in this coordinate system.
• 54:05
  OK, so that's that's usually how we represent things in relativity.
• 54:11
  All right.
• 54:11
  For many folks who math may not be something you do very often, the closest thing you may have seen that touches this concept is interstellar.
• 54:20
  Yes.
• 54:20
  When they're on the wave planet, the guy is on the ship still, and they have that time dilation issue because of exactly what we're talking about.
• 54:28
  Because of exactly what we're talking about.
• 54:30
  Exactly.
• 54:30
  And so we've considered that plot.
• 54:33
  Now let's consider, for example...
• 54:36
  We're in this plot, right?
• 54:39
  It's the two of us.
• 54:40
  Yes.
• 54:40
  Okay?
• 54:41
  We're both stationary.
• 54:42
  We're both in the same reference frame.
• 54:44
  Mate, I got a lot of motion.
• 54:45
  I don't know what you're talking about.
• 54:47
  But as of now, I'm going to have you move.
• 54:49
  I'm going to have you move in the next plot.
• 54:51
  But in this plot.
• 54:52
  I don't have motion right now, but I will in the future.
• 54:54
  I want to give you a little sense of how relativity is so cool.
• 54:58
  Okay?
• 54:59
  So let's consider we've got we're sitting here.
• 55:02
  This is X equals zero, this table.
• 55:04
  OK. OK. And we set the time.
• 55:07
  We have a stopwatch.
• 55:08
  I've got a stopwatch on my iPhone.
• 55:10
  I set it to time equals zero.
• 55:12
  And let's say there are three explosions that happen.
• 55:15
  OK. Or like three light switches.
• 55:18
  In the physics textbooks, it's always like light switches.
• 55:20
  Right.
• 55:21
  Let's do explosions.
• 55:23
  I like explosions.
• 55:24
  But we can survive the explosions.
• 55:26
  Okay.
• 55:26
  There's an explosion that happens here where we're sitting, but in two seconds time.
• 55:33
  Okay.
• 55:33
  There's an explosion that happens in front of us.
• 55:37
  Two light seconds away, so 600,000 kilometers away.
• 55:41
  Real close.
• 55:42
  And there's an explosion that happens behind us.
• 55:45
  Two light seconds behind us.
• 55:47
  Okay.
• 55:47
  So 600,000 kilometers behind us.
• 55:50
  But they all happen at the same time because we're sitting here.
• 55:53
  And we see the explosions happen at the same time.
• 55:58
  Okay.
• 55:59
  Okay.
• 56:00
  So we're like, okay, cool.
• 56:01
  The coordinates of these explosions in space-time, now we're thinking about coordinates in space-time, not just space.
• 56:09
  The coordinates in space-time are, the explosion that happens here is at space equals zero, because it's right here, and the time equals two seconds.
• 56:16
  The coordinates...
• 56:17
  the explosion in front of us is space equals two yes and the time equals two because it happened at the same time and back there it's negative two time equals two when you say space time in this example we're talking about two axes yeah and one is space and one is time and so you can represent space time
• 56:34
  on a two-dimensional X, Y graph in this example.
• 56:39
  And it's not this... No, it's literal now.
• 56:42
  It's literal on a coordinate axis.
• 56:44
  This is the genius of Einstein.
• 56:45
  Right, okay.
• 56:46
  Right?
• 56:46
  He's like putting this stuff on a coordinate plane, and he's asking, right?
• 56:50
  And this is actually, Lorenz came up with this stuff even before Einstein.
• 56:54
  He was thinking about Lorenz transformations, and Einstein's the real guy to be like, let's just take it seriously, guys.
• 56:59
  Right.
• 57:00
  Okay?
• 57:01
  So now we ask the following.
• 57:04
  We're both sitting here and we agree.
• 57:06
  If we were both sitting here, these explosions would happen at the same time.
• 57:10
  Now, suppose you were moving that way to the front at very close to the speed of light.
• 57:19
  Okay?
• 57:21
  The three events that happen simultaneously for me, they are not going to happen simultaneously for you.
• 57:28
  Why?
• 57:29
  The explosion that's in front of us
• 57:32
  But before you, yeah, so this is me, right?
• 57:35
  This is my...
• 57:36
  Yes.
• 57:37
  So to me, everything's happening at the same time, which is why everyone's in the same time axis.
• 57:42
  Yes.
• 57:42
  Now, before we consider the next plot, let's just think about what would happen.
• 57:46
  You're moving towards the explosion in front of me at near the speed of light.
• 57:51
  Which is at 2-2.
• 57:52
  Yep, which is at 2-2 for me.
• 57:54
  For you.
• 57:54
  For me.
• 57:55
  But I'm moving towards it.
• 57:56
  But you're moving towards it, right?
• 57:57
  At the speed of light.
• 57:59
  Near the speed of light.
• 58:00
  Near the speed of light.
• 58:02
  Sue me.
• 58:02
  Yeah, okay.
• 58:03
  But if you're moving out there near the speed of light, now, from Einstein's relativity, we know that length is going to contract for you.
• 58:10
  So the distance to that explosion is going to go down, and the time is going to get a little bit slower.
• 58:17
  But at the same time, you're moving towards...
• 58:22
  That explosion.
• 58:22
  Yeah.
• 58:23
  So the light from that explosion is going to come meet you.
• 58:26
  Mm-hmm.
• 58:27
  Because we're basically, it's like two trains coming at each other.
• 58:29
  Yes.
• 58:30
  But because of the speed at which I'm moving towards it, from your perspective, it's going to take time for that light.
• 58:37
  It's going to take two seconds.
• 58:39
  But because I'm basically meeting the light before it gets to you at that speed, it's going to happen.
• 58:44
  Yeah.
• 58:45
  And so for you, that event is going to happen sooner than two seconds.
• 58:50
  Yeah.
• 58:51
  And it's going to happen closer than two light seconds because the length is contracted.
• 58:57
  Similarly, the event that happens here is going to take a little bit longer than two seconds because this thing happens for me at two seconds, but then the light has to go to wherever you are.
• 59:06
  So it's going to happen a little bit longer than two seconds.
• 59:09
  And the explosion that was way back there, that's going to take even longer.
• 59:12
  So what we should see when we look at your perspective...
• 59:17
  If we look at your perspective, the event that's in front of you should go down on the y-axis and down on the x-axis for your x and y.
• 59:27
  My x and y is stationary.
• 59:30
  And when we say down on the y-axis, because our y...
• 59:33
  So it's sooner in time.
• 59:35
  It's sooner in time.
• 59:35
  And it's sooner in space.
• 59:37
  Right.
• 59:37
  And getting closer to the origin.
• 59:39
  Yep.
• 59:40
  Yep.
• 59:40
  Yep.
• 59:40
  Okay.
• 59:41
  Got it.
• 59:42
  So now let's see how we did, right, with that logic.
• 59:45
  So we go to the next plot.
• 59:47
  In this plot, nothing has happened so far.
• 59:50
  I'm giving you some analysis of what we're talking about.
• 59:55
  As I said, the diagonals are light, right?
• 59:58
  So the diagonals are what light would travel, like the path that light would travel in my space time.
• 1:00:05
  Because, again, our x-axis is light seconds, and our y-axis is just time.
• 1:00:12
  Right.
• 1:00:13
  Right, and so that's why it's diagonal.
• 1:00:15
  And all of the dotted lines are diagonal because if a light beam started at T equals 1, then it would travel like at the...
• 1:00:22
  It would be the dotted line that's just above the T equals 0.
• 1:00:25
  This is why we want to decrement the x-axis in this way so that it creates this simplicity to understand the point.
• 1:00:31
  Exactly, exactly.
• 1:00:32
  Now, one of the... And the other big thing is there's two points and there's actually a line, a black line, right on the...
• 1:00:41
  right on the y-axis here, which says I'm not moving, right?
• 1:00:46
  Because you're the one who's moving.
• 1:00:47
  So to me, I'm just stationary at x equals zero for all of time.
• 1:00:52
  Yes.
• 1:00:52
  Right?
• 1:00:53
  So no matter what time it is, I'm always at x equals zero.
• 1:00:57
  OK?
• 1:00:57
  So now what would you see for me?
• 1:00:59
  Well, you would see me, if you're coming from back there, you'd see me with a positive x-axis, and then you'd see me go backwards, right?
• 1:01:06
  Because to you, I'm coming from up there and then going backwards.
• 1:01:10
  Okay?
• 1:01:10
  Now let's go ahead and do the Lorentz transformation to see what you would see.
• 1:01:15
  Okay.
• 1:01:16
  Okay?
• 1:01:16
  What would you see?
• 1:01:18
  What you would see is the following.
• 1:01:20
  You'd see...
• 1:01:21
  the coordinate axes change.
• 1:01:24
  And notice, first thing we want to notice is the three red dots are doing exactly what we had predicted.
• 1:01:31
  The event that's in front is getting closer to you.
• 1:01:34
  So the time is decreasing and the X is decreasing.
• 1:01:38
  The event that's happening right here
• 1:01:40
  is going forward in time because it's not two seconds.
• 1:01:44
  It's a little bit more because the light has to catch up to you for you to be like, ah, there was an explosion.
• 1:01:48
  And the event that happened way back there is happening way farther at two seconds.
• 1:01:52
  Right.
• 1:01:54
  But notice two things.
• 1:01:55
  And, and, and for one, the line that is me,
• 1:01:59
  which is usually time t equals zero, x equals, time, time equals whatever, x equals zero.
• 1:02:04
  That thing is shifting to where at negative time, so before time t equals zero, I'm ahead of you, and then I go behind you, right?
• 1:02:12
  That's the black line.
• 1:02:13
  Yep.
• 1:02:14
  Here's the key thing about this coordinate transformation, which is the genius of the Lorentz transformation.
• 1:02:19
  All of the diagonals are still diagonal.
• 1:02:23
  What does that mean?
• 1:02:24
  That means even in your reference frame, light is still moving at the same speed.
• 1:02:30
  It is still moving at the speed of light.
• 1:02:33
  Mm-hmm.
• 1:02:33
  Right?
• 1:02:34
  What's really happening is the length is contracting and the time is getting faster.
• 1:02:40
  But they're happening at the same rate such that the diagonals are always diagonal.
• 1:02:45
  Yeah, they're just shifting on the same ratio of proportionality to each other.
• 1:02:50
  Yes, yes.
• 1:02:51
  By the Lorentz factor, actually, is what it is.
• 1:02:53
  And so...
• 1:02:57
  It preserves relativity because one of the tenets of relativity is that the speed of light is the same in any reference frame.
• 1:03:03
  It's constant.
• 1:03:04
  So even when we go into your reference frame, the diagonals are still diagonal, but...
• 1:03:10
  The space has changed and the time has changed.
• 1:03:13
  The other cool thing about this is the volume of space-time is always the same.
• 1:03:17
  The area of the square becomes a rectangle, but the area of each rectangle is exactly the same.
• 1:03:24
  Yes.
• 1:03:25
  Again, because everything is – it's a shift that's happening.
• 1:03:30
  It goes back to this mapping point that you were bringing up earlier.
• 1:03:32
  We are making this transformation in our map, but fundamentally our Jacobian remains as one the entire time.
• 1:03:38
  Yes.
• 1:03:39
  Yes.
• 1:03:39
  That's exactly right.
• 1:03:41
  And what that means is if, like, the volume of space-time is always preserved.
• 1:03:45
  Yes.
• 1:03:46
  In the Lorentz transformation.
• 1:03:48
  Okay?
• 1:03:48
  That's the central tenet of relativity.
• 1:03:50
  Okay.
• 1:03:51
  Okay?
• 1:03:51
  That ensures that the speed of light is always the same for the both of us.
• 1:03:55
  And really, it's kind of crazy when you think about it.
• 1:03:58
  The universe is doing all this just to make sure that the speed of light is the same for me as it is for you.
• 1:04:05
  Because neither of us are right.
• 1:04:07
  Correct.
• 1:04:07
  Right?
• 1:04:08
  Your perspective is the same as my perspective.
• 1:04:12
  So the speed of light better be the same.
• 1:04:14
  And the universe is doing all this nonsense of contracting length and making time go slower just so that the diagonals remain diagonal.
• 1:04:23
  I'm not saying it's a simulation, but if you wanted to make it a simulation.
• 1:04:27
  Yeah, this would be a rule that I would code up.
• 1:04:29
  It would make sense.
• 1:04:31
  And I'm being facetious here, but I think the point you're bringing up is important, which is that the constant of the speed of light requires the universe to do all this other stuff.
• 1:04:40
  Yeah.
• 1:04:41
  So that regardless of your speed or your position in the universe, that constant remains the same.
• 1:04:50
  Exactly.
• 1:04:51
  Which requires it to do all this other stuff.
• 1:04:52
  All this other things.
• 1:04:54
  So it's almost like the speed of light is a golden rule.
• 1:04:56
  Yeah.
• 1:04:57
  It always must be followed.
• 1:04:58
  Yes.
• 1:04:59
  No matter where you are, no matter how fast you're going.
• 1:05:01
  Because I don't know if you're moving or I'm moving, right?
• 1:05:03
  We'll figure out the other stuff to make it happen.
• 1:05:05
  Yeah, exactly.
• 1:05:06
  And so that's a central tenet of Einstein's special relativity.
• 1:05:11
  Okay?
• 1:05:11
  And that's the Lorentz transformation that...
• 1:05:14
  gives rise to all of the rich behavior that is a consequence of special relativity so that again is just a that's a linear transformation as i said right it's um it's not too complicated i was able i was able to get it yeah yeah exactly it's not special relativity is like actually not that bad when you like really just think about it it's like
• 1:05:35
  Yeah, kind of makes sense.
• 1:05:37
  It's when we get to general, when it's like, whoa, whoa, whoa, whoa, slow down.
• 1:05:43
  I just got here.
• 1:05:45
  So that's special relativity, right?
• 1:05:47
  And the Jacobian there is one, meaning that the volume in space time is preserved no matter what reference frame you're in.
• 1:05:55
  Okay.
• 1:05:55
  And when you say reference frame, you mean both position and speed?
• 1:06:00
  And time.
• 1:06:00
  And time.
• 1:06:01
  So both position and time.
• 1:06:02
  Yeah, yeah.
• 1:06:02
  Whatever volume in your position and time versus my position and time are going to be exactly the same.
• 1:06:09
  It's just that each individual thing is going to be changed, but the product of the two are going to remain the same.
• 1:06:14
  Yep.
• 1:06:15
  Right?
• 1:06:15
  Because the Lorentz factor is going to cancel out.
• 1:06:18
  Makes sense.
• 1:06:18
  Okay.
• 1:06:19
  And that happens in, we showed a 1D case where we were only considering like, you know, forward and backwards where the boost is.
• 1:06:26
  But this applies in all three.
• 1:06:28
  In that case, you get like the Minkowski tensor and all sorts of stuff.
• 1:06:31
  But that's effectively what's happening.
• 1:06:34
  So this is a coordinate transformation, right?
• 1:06:36
  Space-time gets distorted, but the Jacobian is 1, so the volume of space-time remains the same.
• 1:06:43
  Now let's talk about nonlinear transformations, okay?
• 1:06:47
  This is an example of a nonlinear transformation.
• 1:06:50
  Here, the x-axis goes to x plus sine of y over 2, and the y-axis goes to y plus sine of x over 2.
• 1:06:59
  This is what I mean by now you can start mixing stuff, right?
• 1:07:02
  In the other one, you were also mixing, but here you're doing it in a nonlinear way.
• 1:07:05
  You're not just like adding some Lorentz factor times the constant.
• 1:07:10
  This is like sine, which is an oscillatory thing.
• 1:07:12
  And so you're seeing like stuff shift, right?
• 1:07:15
  Things are getting wiggly.
• 1:07:16
  Mm-hmm.
• 1:07:18
  In this case, what is the Jacobian going to do?
• 1:07:21
  Well, the Jacobian now is actually different in different spots.
• 1:07:25
  You can see in some cases the areas are getting squished.
• 1:07:28
  In other cases, the areas are remaining about the same.
• 1:07:31
  And if we go and see and zoom in on a central area element of this transformation, what happens?
• 1:07:39
  Well, let's look at those...
• 1:07:42
  Let's look at that sector that's right in the middle of the 1-1 box.
• 1:07:47
  Okay?
• 1:07:51
  The transformation is nonlinear, so it's curved in every sense.
• 1:07:55
  But the beauty of calculus is the following.
• 1:07:57
  If I take a small enough element, stuff is mostly going to be a line.
• 1:08:03
  That's all calculus is.
• 1:08:05
  The essence of calculus is weird stuff is happening, but if I take a small enough element, it's mostly just linear.
• 1:08:14
  Calculus, just zoom in.
• 1:08:15
  Just zoom.
• 1:08:16
  Yeah, no, exactly.
• 1:08:17
  Just zoom in and the square becomes a parallelogram.
• 1:08:20
  It's not becoming a weird curved shape.
• 1:08:23
  It goes back to the example we just talked about previously, but it's all about what is your perspective of the or or context by which you're making your calculations.
• 1:08:34
  Yeah, exactly.
• 1:08:35
  And in this case, we can understand what the Jacobian is because the Jacobian at that point is going to be what is the ratio of the square area to the new parallelogram area.
• 1:08:45
  Okay, in this case, it looks like 1 because it's getting squished and the rhombus looks about the same.
• 1:08:49
  So in that locality, the Jacobian is 1, but somewhere else it could be 2, somewhere else it could be less than 1, and so on and so forth.
• 1:08:57
  But that's the beauty of calculus is that you can zoom in and you can figure out a well-defined Jacobian at every given point.
• 1:09:03
  We have a larger complex system, but at sufficient zoom in, that complex system still comes back to the fundamental that we just talked about.
• 1:09:11
  Yeah, okay.
• 1:09:12
  Yeah, exactly.
• 1:09:12
  And so now we've got this core concept, which is the Jacobian matrix.
• 1:09:16
  The Jacobian matrix tells you how each of the directions sort of swish and become one another.
• 1:09:22
  And the determinant of that matrix tells you the ratio of the areas before and after.
• 1:09:28
  Mm-hmm.
• 1:09:29
  or I should say it's the ratio of areas after divided by before.
• 1:09:34
  So 2 means that the ratio went up.
• 1:09:36
  Less than 1 means the ratio went down.
• 1:09:38
  There's also negative Jacobians, which means that the axes got flipped.
• 1:09:43
  So it's like, I mean, it's just flipped.
• 1:09:46
  It's like it got flipped, right?
• 1:09:48
  Okay, so if the determinant at a given point is non-zero, as we just saw, then it signifies a linear approximation, right?
• 1:09:56
  And it doesn't collapse.
• 1:09:58
  Now here we've got an example that is at the center of the Jacobian conjecture.
• 1:10:05
  Okay?
• 1:10:05
  We've got to consider situations where the determinant is a non-zero constant.
• 1:10:11
  It's a constant everywhere.
• 1:10:13
  Meaning, no matter where I am in my map, the area is actually the same.
• 1:10:18
  Even though the upper part is getting more squished, you would think,
• 1:10:22
  The actual area, the ratio of the areas is actually exactly the same.
• 1:10:26
  This is a particular map that I came up with in Desmos.
• 1:10:29
  There's a lot of really cool Desmos pre-made applications where you can plug in your own custom linear map and you can visualize how the map is going to change the axis.
• 1:10:44
  So in this case, the new x-axis is x plus one-fourth y squared.
• 1:10:49
  And the y-axis is just the y-axis.
• 1:10:52
  So every y gets plotted to its own y, but the x gets shifted by a parabola.
• 1:10:58
  That's why these things are becoming a parabola.
• 1:11:01
  But the horizontal lines are remaining horizontal lines.
• 1:11:05
  It's going from lanes on a highway to a track, the elliptical of a track.
• 1:11:08
  Yeah, and the lanes on the highway is the X, right?
• 1:11:11
  So that's why that's getting shifted into a parabola.
• 1:11:13
  But the horizontal lines of latitude, so to speak, are remaining the same because the Y just gets plotted to the new Y.
• 1:11:21
  Nothing happens.
• 1:11:22
  Okay, in this case, if you were to plot that area, that little green area, that little green area becomes a parallelogram, but the area is preserved.
• 1:11:33
  The Jacobian here is one.
• 1:11:34
  If I were to take that area and put it up top,
• 1:11:37
  The parallelogram might get more squished, right?
• 1:11:40
  It might have a different shape, but the area is still going to be 1.
• 1:11:44
  This particular mapping is one where the Jacobian determinant is constant, meaning that the transformation of area everywhere to all of infinity in X and Y, the area is preserved, okay?
• 1:12:00
  Now, given this, one can ask, suppose I was given the end product, right?
• 1:12:08
  Okay?
• 1:12:08
  If I was given the end product of the last, like, the points in that after the transformation.
• 1:12:18
  Yes.
• 1:12:18
  Suppose I were given the points after the transformation.
• 1:12:21
  Can I go backwards?
• 1:12:23
  Right.
• 1:12:24
  Meaning, is the map invertible?
• 1:12:25
  Yes.
• 1:12:26
  Okay?
• 1:12:27
  In this case, it is.
• 1:12:27
  You can actually solve for it.
• 1:12:29
  If you give me the new...
• 1:12:31
  x and y coordinates in this case i'm going to call it u and v because their u is going to be my new x-axis v is going to be my new y-axis if i want to recover the old x and y-axis that like gave me that point i can just like do the mathematics and plug in it's actually u minus one fourth v squared so the the plus one fourth becomes a minus okay and then i can recover the old map
• 1:12:55
  Is it as a concept to help me grok this?
• 1:12:58
  Is it similar to when you talk about encryption and decryption?
• 1:13:00
  When you have the key in the middle.
• 1:13:02
  Very good.
• 1:13:03
  You're able to go back, even though it's jumbled.
• 1:13:05
  Yeah.
• 1:13:06
  Because you have the key that allows you to.
• 1:13:08
  Okay.
• 1:13:08
  Exactly.
• 1:13:09
  Exactly.
• 1:13:09
  And.
• 1:13:10
  In that particular transformation, because the Jacobian was not zero, I could go backwards.
• 1:13:16
  If the Jacobian is zero, what does that mean?
• 1:13:18
  That means an area collapses into a point or like a line, something with zero area.
• 1:13:23
  Well, what that means is now we've got a bunch of points where it's degenerate.
• 1:13:29
  Like let's say the extreme case where like a giant square collapsed into a single point.
• 1:13:33
  Well, now if you give me that single point, I don't know where in the square it came from.
• 1:13:37
  Right.
• 1:13:38
  Right.
• 1:13:39
  If you give me a line, for example, the entire square gets collapsed into a line.
• 1:13:42
  Well, that line is only 1D information.
• 1:13:44
  I don't know where in the 2D.
• 1:13:47
  Right.
• 1:13:47
  The thing came from.
• 1:13:48
  You need both.
• 1:13:49
  You need you need both.
• 1:13:50
  Yeah.
• 1:13:51
  You can't have only one and then zero on the other.
• 1:13:53
  Exactly.
• 1:13:54
  Right.
• 1:13:54
  And that happens when the Jacobian is zero.
• 1:13:55
  So when the Jacobian is zero, I can't go backwards.
• 1:13:58
  Trivial.
• 1:13:59
  It's not invertebrate.
• 1:13:59
  That makes sense, right?
• 1:14:01
  And that's like a well-known theorem in calculus, right?
• 1:14:04
  And where locally, if I have the Jacobian that's not zero, then I can go backwards.
• 1:14:08
  That makes sense.
• 1:14:09
  Okay?
• 1:14:10
  Now we can finally understand the conjecture.
• 1:14:12
  Okay.
• 1:14:13
  The Jacobian conjecture is the following.
• 1:14:15
  It says that if I have a map where the Jacobian is a constant everywhere, can I always go backwards?
• 1:14:24
  Okay.
• 1:14:25
  Right.
• 1:14:25
  Everywhere.
• 1:14:26
  Okay?
• 1:14:27
  If the constant is...
• 1:14:29
  If the Jacobian is constant everywhere, can I always create an invertible function?
• 1:14:35
  You give me a function where the Jacobian is constant everywhere, can I always go backwards?
• 1:14:39
  Okay?
• 1:14:40
  The formal statement is the following.
• 1:14:42
  Okay?
• 1:14:42
  Let K have a characteristic of zero.
• 1:14:44
  K is a ring.
• 1:14:45
  A ring really just means like numbers.
• 1:14:48
  Okay?
• 1:14:48
  Mathematicians have other types of rings, but for us, let's just say they're numbers.
• 1:14:52
  Okay?
• 1:14:53
  Okay?
• 1:14:53
  Okay?
• 1:14:54
  they have a characteristic of zero.
• 1:14:56
  A characteristic of zero, a ring really means something that has addition and multiplication.
• 1:15:00
  Okay?
• 1:15:02
  A characteristic of zero means that, you know, a ring has a one.
• 1:15:05
  It's some kind of element that's a one.
• 1:15:07
  For example, in the matrices, the one element is the identity.
• 1:15:11
  For numbers, it's just the number one.
• 1:15:13
  And a characteristic of zero means no matter how many times I add one to itself, I'm never going to get back zero.
• 1:15:19
  Mm-hmm.
• 1:15:20
  That just means to me normal numbers.
• 1:15:22
  Mathematicians will come up with rings that have characteristics that are non-zero, where I add the number one five times and I get zero for some reason.
• 1:15:31
  I don't know why you would want to study such objects, but they find a way.
• 1:15:36
  In any case, we're talking about normal things here, okay?
• 1:15:40
  Let k have a characteristic of zero, so this is just real numbers, complex numbers, things like that.
• 1:15:45
  And if jf, which is the Jacobian determinant, is a non-zero constant of some functional map, then f, which is the functional map, has an inverse function g that plots my n-dimensional function,
• 1:15:59
  space to back to the n-dimensional space yeah yeah okay yeah and this thing is regular meaning that its components are polynomials okay okay so if i have a polynomial function that's going in polynomial meaning things like x cube plus x squared y like powers of stuff then the inverse map is also a polynomial okay
• 1:16:21
  Okay.
• 1:16:22
  That is the Jacobian conjecture.
• 1:16:24
  Okay.
• 1:16:24
  So this is actually interesting because now I can understand the formulation here of what this open problem is trying to...
• 1:16:36
  suggest yes it's trying to suggest that anytime right i have a function where the jacobian determinant is a constant everywhere meaning the the function preserves area preserves area i shouldn't say preserves area it scales area but it scales it in the same way everywhere yeah okay yeah if this function scales it in the same way everywhere
• 1:16:59
  Then I should always be able to go backwards.
• 1:17:01
  I should always be able to decrypt my encryption.
• 1:17:04
  Yes.
• 1:17:04
  Okay.
• 1:17:05
  As long as it preserves area in the same scale everywhere.
• 1:17:10
  Okay.
• 1:17:10
  That's the conjecture.
• 1:17:12
  It was formulated way back about 90 years ago, and people have been trying to prove it ever since.
• 1:17:19
  Okay.
• 1:17:20
  That is the question.
• 1:17:22
  The problem was number 16 on Stephen Smalley's 1998 list of important mathematical challenges.
• 1:17:28
  And it has very famously defeated top, top mathematicians, which we're going to get into later.
• 1:17:33
  Top 20, baby.
• 1:17:34
  Yeah.
• 1:17:34
  That's top 20.
• 1:17:35
  Top 20.
• 1:17:36
  Top 20 open problems.
• 1:17:38
  Yeah, that's quite nice.
• 1:17:39
  Now, July 19th, 2026, in the middle of the World Cup final.
• 1:17:46
  Okay.
• 1:17:47
  Levant Alpuge, he is part of the Harvard Society of Fellows and also one of the researchers at Anthropic.
• 1:17:55
  He drops this tweet.
• 1:17:58
  I saw it within 30 minutes of it going up.
• 1:17:59
  Yeah, because it was insane.
• 1:18:02
  Literally 30 minutes.
• 1:18:03
  Yeah.
• 1:18:03
  First of all, it's all lowercase.
• 1:18:05
  It looks like he's trolling.
• 1:18:07
  Yeah, yeah, yeah.
• 1:18:08
  You know?
• 1:18:09
  Hello there.
• 1:18:10
  The Jacobian conjecture is false.
• 1:18:11
  Thanks.
• 1:18:12
  And T-H-A-N-X.
• 1:18:15
  Thanks to my close friend Akil, we're going to get into who that is, for asking about it, and my other close friend Fable, which is the AI anthropic model, for working during the World Cup final.
• 1:18:27
  And then he just shows the counterexample.
• 1:18:29
  He just writes it out.
• 1:18:31
  There's so many things, but I'm not going to bring it up now.
• 1:18:33
  And it could fit in a tweet.
• 1:18:35
  You know, there's like the 100 character whatever limit thing.
• 1:18:38
  He just fit the entire proof in a tweet.
• 1:18:41
  That's all you need.
• 1:18:42
  Arguably, if NFTs were still a thing, this NFT would be very valuable.
• 1:18:46
  Bro, I would buy this NFT.
• 1:18:48
  This would be so valuable.
• 1:18:50
  Yeah, we would get this NFT for the FFP vault.
• 1:18:53
  Right, exactly.
• 1:18:54
  Because for those who may not understand Twitter culture,
• 1:19:02
  You had to be there.
• 1:19:04
  And to see, because it's real time and so many people, especially in the AI research community, are there and apparently also were not watching the World Cup final at the time.
• 1:19:13
  The immediate conversation that arose...
• 1:19:21
  literally within an hour yeah of this being out there was as a bystander as like an as an innocent bystander was one of the most fascinating things yeah this was insane when i saw it because i heard about the jacobian conjecture when when i was at princeton doing mathematics i took a course in like um algebra and they like dropped hints of it about how it was unsolved but it's like fairly easy to understand yeah yeah right yeah it's like
• 1:19:45
  If you don't get into the weird rings and all that kind of stuff, like as a physicist, I was like, oh, you're talking about numbers.
• 1:19:50
  You're talking about coordinate transformations.
• 1:19:52
  You're talking.
• 1:19:52
  OK, like it kind of makes sense to me.
• 1:19:54
  OK, and then to see this come out in a troll language, I think it's just so, so funny.
• 1:20:00
  OK, so let's talk about that exact solution that it has.
• 1:20:04
  OK, and in the next in the next overlay.
• 1:20:08
  We have it type formatted because within hours it was on Wikipedia.
• 1:20:12
  You go into the Jacobian Conjecture Wikipedia article, it says counterexample.
• 1:20:16
  And it shows that it's not true.
• 1:20:18
  You know that meme where we got him?
• 1:20:20
  We got him.
• 1:20:21
  With what's his name and the balloons and confetti?
• 1:20:24
  Yeah.
• 1:20:24
  They did it immediately.
• 1:20:25
  Yeah, yeah, immediately, right?
• 1:20:27
  And this is a counterexample, meaning he has shown a map where...
• 1:20:32
  The Jacobian determinant is a constant.
• 1:20:35
  So this functional mapping, this map between, this is in 3D.
• 1:20:40
  So we're taking 3D space and plotting it to another 3D space.
• 1:20:44
  So volume is getting morphed.
• 1:20:47
  But the volume getting morphed is always at a constant of negative 2, meaning one of the axes is flipped.
• 1:20:52
  So it's like getting flipped.
• 1:20:54
  And if you've got a cube of the size 1, it becomes a cube of size 2 in the new one.
• 1:21:00
  But one of the axes is a little bit...
• 1:21:02
  It is flipped.
• 1:21:03
  This is a functional map that shows that we've got a constant Jacobian determinant, but two different points are being plotted to the same resulting point.
• 1:21:16
  Aha.
• 1:21:17
  Meaning, if you gave me that resulting point...
• 1:21:20
  I couldn't tell you which of the two it came from.
• 1:21:23
  Yes.
• 1:21:25
  The point being, you don't have the ability to go from the result back to the original map because two points point to the same place.
• 1:21:36
  It could be one of two things.
• 1:21:37
  So it's inherently not invertible.
• 1:21:40
  It turns out there's a third point as well.
• 1:21:42
  And this is actually what's crazy about this is you can do this yourself.
• 1:21:47
  And I actually did it.
• 1:21:49
  Like, before we, like, as I was doing research for this episode, I went ahead and did it by hand.
• 1:21:57
  And I can show you my work.
• 1:22:00
  It's in the next one.
• 1:22:01
  So this is, I did it myself, right?
• 1:22:03
  I took the three...
• 1:22:06
  components of the function so the new x coordinate is some seven degree polynomial that has that mixes x y and z the new y coordinate is another i think seven degree polynomial and then the the z coordinate is not a seven degree polynomial it's a little bit simpler but you i've got this map where x x y and z goes to a new x new y new z and
• 1:22:27
  I calculated the derivatives, which are on the left.
• 1:22:30
  So that becomes a matrix, which you see on the top right.
• 1:22:34
  Right.
• 1:22:34
  And then I did a little bit of cheating because I decided I didn't want to multiply all those polynomials.
• 1:22:40
  Okay.
• 1:22:41
  I wanted to see if I could still do it, if I still got it.
• 1:22:44
  And I got halfway and I'm like...
• 1:22:46
  I still got it.
• 1:22:47
  So then I went on Wolfram Alpha, and I just multiplied all of those polynomials to calculate the determinant.
• 1:22:53
  And the determinant is negative 2.
• 1:22:55
  And then I went ahead and plugged in those two numbers.
• 1:22:59
  So fascinating.
• 1:23:00
  And they went to the same number, negative 1 fourth, 0, 0.
• 1:23:04
  There you go.
• 1:23:05
  I did it on a single piece of paper plus, okay, a little bit of Wolfram Alpha.
• 1:23:08
  Right?
• 1:23:08
  It would have involved another piece of paper for me to do it.
• 1:23:11
  But I was already, you can see the crossing out.
• 1:23:13
  I was like starting to make mistakes.
• 1:23:14
  I was like, let's.
• 1:23:16
  This is so interesting because, you know, and I think this is why it had the, you know, it comes back to your point earlier about why it's easier to disprove than to prove.
• 1:23:28
  Because especially given the structure of this open problem.
• 1:23:32
  And the nature of the disproof, if that's the right naming convention.
• 1:23:38
  The counter example.
• 1:23:39
  Sorry, the counter example.
• 1:23:44
  Anyone can do it and see that it is true, going back to our point about mathematics.
• 1:23:49
  Yeah.
• 1:23:49
  And the objective nature of it.
• 1:23:51
  Yeah, exactly.
• 1:23:52
  I did it.
• 1:23:53
  Anyone can do it.
• 1:23:54
  Right.
• 1:23:55
  This is something that algorithmically there's no AI here.
• 1:23:57
  Right.
• 1:23:58
  Right.
• 1:23:58
  It's just it's an algorithmic process of taking derivatives, taking the determinant of the Jacobian.
• 1:24:02
  There you go.
• 1:24:03
  Boom.
• 1:24:04
  There's a constant determinant and two points point to the same thing.
• 1:24:08
  So I can't go backwards.
• 1:24:09
  I can't go backwards.
• 1:24:10
  And it's not invertible, which means the Jacobian conjecture fundamentally not a real thing.
• 1:24:15
  It's not a real thing.
• 1:24:16
  Right.
• 1:24:17
  It's not a real thing.
• 1:24:18
  And it's a counter example in degree, well, it's in degree 7, but it's in three dimensions, right?
• 1:24:25
  Which means for all higher dimensions, it's also not true.
• 1:24:28
  Because I can just take, like, consider the trivial map where I take a function of four dimensions.
• 1:24:35
  Let's say X, Y, Z, and T. T is our fourth dimension in a la relativity.
• 1:24:42
  X, Y, and Z do what that function was doing, and T just points to T, right?
• 1:24:46
  So the time doesn't change.
• 1:24:48
  That is going to have the same Jacobian determinant, and the things are going to point to the same point in 3D space, and the time doesn't change.
• 1:24:58
  Right?
• 1:24:58
  I can do that for n equals 5, n equals 6, any dimension greater than 3 now.
• 1:25:03
  So this proved all dimensions greater than 3.
• 1:25:05
  Right.
• 1:25:06
  So it's only true in 2.
• 1:25:08
  Yeah.
• 1:25:08
  Now it's like, now it's an open question.
• 1:25:10
  Oh, right, because we don't know.
• 1:25:11
  We don't know.
• 1:25:11
  Yeah.
• 1:25:13
  Right?
• 1:25:14
  So now the Jacobian conjecture is, yes, it's only in two dimensions.
• 1:25:18
  Now it's still an open problem.
• 1:25:19
  Still an open problem.
• 1:25:21
  Not yet solved.
• 1:25:21
  Okay.
• 1:25:22
  But that's interesting.
• 1:25:24
  But like for all the infinity of dimensions above.
• 1:25:27
  Which is like really a much more meaningful.
• 1:25:30
  Yeah, yeah.
• 1:25:31
  It's crazy.
• 1:25:32
  No, this.
• 1:25:32
  And I think what's so interesting is that, again, and we might get to this, but one of the things that people always want to view a solution that has an AI system involved as being there has to be some, you know, it's a magic show.
• 1:25:49
  So there has to be some trick somewhere.
• 1:25:51
  Yeah.
• 1:25:51
  Oh, you just ask Fable, the latest frontier model from Anthropik, to just brute force it.
• 1:25:57
  And if you look at the amount of time required to brute force, it's just, it's like to the power of...
• 1:26:07
  To the power of crazy.
• 1:26:08
  And that's actually my next point.
• 1:26:10
  Okay.
• 1:26:11
  We can actually ask, did Fable just brute force?
• 1:26:13
  Right, right.
• 1:26:14
  Did it just do like supercomputer nonsense?
• 1:26:17
  Because they have a lot of computers.
• 1:26:18
  Yeah, and there's history behind supercomputer nonsense, okay?
• 1:26:21
  So let's talk about, like, this is a counterexample, okay?
• 1:26:24
  And historically, computation has been very good at producing counterexamples, okay?
• 1:26:28
  Way back in the 1960s, actually.
• 1:26:30
  So let's talk about Euler's conjecture.
• 1:26:33
  Euler's conjecture is the following.
• 1:26:34
  That you remember we were talking about like Fermat's last theorem and the Pythagorean theorem.
• 1:26:39
  Like the Pythagorean theorem is that like the sum of two squares can equal another square.
• 1:26:43
  Yes.
• 1:26:44
  So like two squares.
• 1:26:46
  Two thingies to the power of two equals another thing to the power of two.
• 1:26:51
  Trivial example, three squared plus four squared equals five squared because nine plus 16 equals 25.
• 1:26:57
  Plato, way back in the day, had found another kind of Pythagorean-looking thing where he showed that three cubed plus four cubed plus five cubed is actually six cubed, which is kind of cool.
• 1:27:10
  It's called Plato's Number.
• 1:27:11
  Mm-hmm.
• 1:27:12
  Um, so, all right.
• 1:27:14
  The Euler looked at this, looked at Plato's number, and he said, well, um, look, if, if, if I need, in that case, I needed three cubes.
• 1:27:24
  Yes.
• 1:27:25
  To get to another cube.
• 1:27:26
  Yeah, right.
• 1:27:26
  Okay.
• 1:27:27
  Right.
• 1:27:27
  So Euler's like, what if I need N...
• 1:27:29
  To the nth power to get to another nth power.
• 1:27:33
  I need at least n things to the nth power to get to another nth power.
• 1:27:38
  So if I want something like a to the fourth plus b to the fourth plus c to the fourth, I would need another d to the fourth in order to get to an e to the fourth.
• 1:27:46
  He was trying to generalize Plato's number.
• 1:27:49
  Exactly.
• 1:27:50
  He's trying to generalize, because Pythagorean theorem already happened, right?
• 1:27:53
  Three squared plus four squared equals five squared.
• 1:27:54
  Plato's number is saying, okay, I need three cubes to get to another cube.
• 1:27:58
  So maybe I need four to the fourth power to get to a fourth.
• 1:28:01
  I need five to the fifth power, right?
• 1:28:02
  And that's Euler's conjecture.
• 1:28:04
  Remains untrue for a very long time.
• 1:28:06
  Lo and behold, Lander and Parkin, in a very, very famous mathematical paper, one of the shortest math papers of all time,
• 1:28:15
  Right.
• 1:28:16
  They're like direct search with the CDC 6000 66000, which is one of the first successful supercomputers.
• 1:28:24
  They just brute forced it.
• 1:28:25
  Yeah.
• 1:28:26
  And they found a counter example.
• 1:28:30
  27 to the fifth power plus 84 to the fifth power plus 110 to the fifth power plus 133 to the fifth power equals 144 to the fifth power.
• 1:28:39
  So I only needed four fifth power thingies to get to a fifth power.
• 1:28:42
  We don't like that.
• 1:28:43
  Yeah.
• 1:28:44
  And they were just like, it's the smallest instance in which five fifth powers.
• 1:28:49
  There you go.
• 1:28:49
  Sorry.
• 1:28:50
  Sorry, Euler.
• 1:28:51
  You're wrong.
• 1:28:51
  Which is crazy to say.
• 1:28:52
  Right.
• 1:28:53
  Euler is wrong.
• 1:28:56
  But this is a perfect point.
• 1:28:58
  This is why people immediately go to, again, you know, the... Oh, they just brute force.
• 1:29:02
  Yeah, right.
• 1:29:03
  Right.
• 1:29:03
  And this, in this case, the brute force, it could work.
• 1:29:07
  Because the sample space that they were, they were just like, okay, let's just go for fifth powers.
• 1:29:12
  Let's go all the way up to like 200.
• 1:29:14
  200.
• 1:29:15
  For all the numbers, and let's just try it out.
• 1:29:17
  And it happened to be that.
• 1:29:18
  And the search space is small enough where with a big enough supercomputer, I can do it.
• 1:29:22
  And this was some time ago.
• 1:29:24
  Yeah, this was in the 60s.
• 1:29:26
  And now we've found more and more exceptions of Euler's conjecture.
• 1:29:29
  It's rare.
• 1:29:30
  It's increasingly rare.
• 1:29:31
  And there's a lot of mathematics about how rare it is.
• 1:29:34
  And that's its own mathematical field.
• 1:29:36
  Yes.
• 1:29:37
  But let's consider, could Fable have just brute forced it?
• 1:29:41
  Right.
• 1:29:41
  I don't think so.
• 1:29:42
  I think this is fundamentally different, right?
• 1:29:43
  The search space is actually, I think, too large.
• 1:29:46
  Because here, what we'd have to do is test every single function...
• 1:29:53
  of polynomials, right?
• 1:29:55
  There's X, Y, and Z.
• 1:29:56
  So let's just do like, okay, it found like the seven degree polynomial, right?
• 1:30:00
  There was an X to the seven there.
• 1:30:03
  If you just think about seven degree polynomials and seven degree maps, so you can have seven degrees in X, seven degrees in Y, seven degrees in Z.
• 1:30:13
  And each of those can have different numbers.
• 1:30:17
  So first of all, that's 343 different polynomial terms.
• 1:30:21
  Each of those polynomial terms can then make up 1,300 cubic equations.
• 1:30:26
  And then the coefficients of that equation, right?
• 1:30:28
  Is it 1x squared y?
• 1:30:30
  Is it 2x squared y?
• 1:30:32
  Is it 1x cubed y?
• 1:30:33
  Each of those coefficients to create individual functional maps, even if you were to test within a tiny integer range of, let's say, negative 10 to 10,
• 1:30:43
  and try all of the different coefficients, that would be something like 10 to the 475 candidates.
• 1:30:48
  Right.
• 1:30:48
  Right?
• 1:30:49
  Right.
• 1:30:50
  Even like a million-cubit quantum computer, if we were to create a quantum algorithm that could somehow do this, which I'm just saying, we don't even have one.
• 1:30:59
  For another day.
• 1:31:00
  Yeah, that's for another day.
• 1:31:01
  We don't even have one.
• 1:31:02
  But, like, you can't.
• 1:31:04
  You can't do that.
• 1:31:05
  This is not a brute force thing, right?
• 1:31:07
  Like Fable had to understand something about the structure of polynomial maps, understand something about the properties of polynomial maps where the Jacobian is constant, and then look for places where things would overlap.
• 1:31:24
  Because if the Jacobian is constant, that means that any given infinitesimal area is constant.
• 1:31:30
  is being put into the same thing.
• 1:31:31
  So it's not just trivially that the thing is going to the same spot.
• 1:31:35
  There's overlap happening where different parts of the space are getting mapped to the same thing in some weird way.
• 1:31:42
  But somehow the Jacobian determinant is still a constant.
• 1:31:46
  I think what's been interesting to hear about the discussion of open problems and AI systems, and this is, I think, is a perfect example.
• 1:31:53
  There was a discussion.
• 1:31:55
  It's going to escape me who it was with.
• 1:31:57
  But there were sort of three different there are three people in the field who were coming from three different perspectives about AI's potential in particularly this idea of novelty or or scientific discovery.
• 1:32:13
  Yeah.
• 1:32:13
  General, not well-defined term.
• 1:32:15
  And one of the arguments was that even if a system like Fable has scooped up every math paper ever.
• 1:32:29
  Yeah.
• 1:32:29
  Right?
• 1:32:29
  Right.
• 1:32:31
  It also has the benefit of having every physics paper, every biology paper.
• 1:32:37
  And because of the way structurally these fields are for human beings, although it's maybe changing in recent times, people who are very deep in a particular lane or expertise in math or in physics or in biology are...
• 1:32:54
  They may dip their toe in orthogonal fields, but they don't have the same level of expertise in their own field in that other field.
• 1:33:06
  And simply by just having that same level of expertise in a cross-functional, in a cross-field way...
• 1:33:16
  Because a lot of science is making these connections that are non-obvious.
• 1:33:19
  The ability for these systems to make connections that are non-obvious is just exponential.
• 1:33:24
  Yeah.
• 1:33:24
  And we may discover that there are a lot of solutions just simply in taking known information and finding things you can carry over from physics.
• 1:33:38
  that have an application in math that we've just not explored.
• 1:33:43
  Yeah, that's true.
• 1:33:44
  I mean, like in string theory, for example, string theory bleeds a lot into fundamental mathematics, right?
• 1:33:48
  Now, one of the things that is really unclear about this discovery is like what the prompt that was used, right?
• 1:33:56
  If the chain of thought has some insight into how it got to this particular functional map, right?
• 1:34:02
  And whether it used all of those...
• 1:34:05
  All of that knowledge.
• 1:34:06
  Like that was my that was my suspicion when there was that one discovery that we talked about with the the glue on tree and like how there's like a seven glue on Feynman diagram that it could calculate that's non zero.
• 1:34:19
  And that's something that we have to worry about.
• 1:34:21
  I was like, OK, maybe there's some mathematics that's happening elsewhere that is like.
• 1:34:25
  bleeding into this physics, right?
• 1:34:27
  It's, however, very rare to go up the chain of logical inquiry.
• 1:34:33
  And that's why I'm, like, freaked out, kind of.
• 1:34:35
  I think this is an important point because I wanted to lay that out there to this exact note, which is, like, even if that's true, what I just walked through,
• 1:34:46
  What we're talking about with this solution to the Jacobian conjecture seems to be outside of that, potentially outside of that sandbox.
• 1:34:53
  Yeah.
• 1:34:54
  Yeah.
• 1:34:54
  It seems.
• 1:34:55
  It seems.
• 1:34:55
  I don't know.
• 1:34:56
  I don't know.
• 1:34:57
  And I'd love to talk to Levant.
• 1:34:59
  Even the potential.
• 1:35:00
  Yeah.
• 1:35:00
  Yeah.
• 1:35:01
  That it's not obviously what we just talked about, which has been the case for other things.
• 1:35:04
  Yeah.
• 1:35:04
  It's like, oh, this is they made the connection here and you can draw the through line.
• 1:35:09
  Even just the potential that that's true.
• 1:35:11
  Yeah.
• 1:35:12
  Is, you know, and people need the part of it.
• 1:35:15
  You know, the Levant who discovered it is obviously one of the world's greatest mathematicians.
• 1:35:19
  It's been so for a long time.
• 1:35:20
  But he's not.
• 1:35:21
  Oh, he's not.
• 1:35:21
  No.
• 1:35:22
  I mean, look, no offense to Levant.
• 1:35:25
  Look, I don't want that heat.
• 1:35:26
  Look, look, no offense to Levant.
• 1:35:28
  He's got a PhD from Princeton, so he's clearly in the top 1% of math petitions, I'd say, right?
• 1:35:33
  So on and so forth.
• 1:35:34
  But he's not like an active researcher, I would say.
• 1:35:37
  Okay, that's fair.
• 1:35:38
  That's fair.
• 1:35:38
  And...
• 1:35:40
  But he knows clearly, I mean, he's a researcher in anthropic, and he's very good at understanding how to use and leverage anthropic, right?
• 1:35:48
  And clearly he knows a lot about mathematics, right?
• 1:35:51
  But this isn't like someone like Peter Sarnak, that's like a number theorist, using this stuff, right?
• 1:35:57
  Right.
• 1:35:58
  That's why I really want to know what the prompt was.
• 1:36:03
  Which almost makes it again even more.
• 1:36:05
  Yeah.
• 1:36:06
  Like he's clearly gifted, but he's not at that top level that's getting Fields medals.
• 1:36:11
  Fascinating.
• 1:36:13
  There's so many angles to this story that I think are, you know, again, a lot of the CEOs and people around who are at the frontier of these systems who have unlimited compute with them, by the way.
• 1:36:24
  Yeah.
• 1:36:24
  Which is part of the value out there.
• 1:36:27
  have been sort of the canary in the coal mine around that look they've been saying that this is where it's headed this is and everyone's kind of been like yeah yeah show me yeah well yeah and there's several examples right now before we get into the other examples one thing i wanted to talk about was like how humans have been approaching this problem right
• 1:36:47
  right?
• 1:36:48
  There's been this degree paradox where human researchers have spent decades brute-forcing variables.
• 1:36:54
  Like, we've been doing a lot of the brute-forcing, up to 16 variables, degree 100 plus.
• 1:36:59
  So for 2D, for example, we've been focusing as humans a lot on the 2D case, because we thought 2D would be where we could find, with the least amount of computation, some kind of
• 1:37:11
  counter example, right?
• 1:37:13
  But it turns out there's proofs that show that the smallest 2D case would have to have a polynomial of degree like 100, like X to the 100 plus X to the 99, da-da-da-da-da, right?
• 1:37:27
  And so we've been focusing on this like...
• 1:37:29
  Highly complicated thing.
• 1:37:30
  But here, Claude Fable V proves that a low degree, degree 7, it's just in a higher dimension.
• 1:37:36
  Yeah.
• 1:37:37
  Dimension 3.
• 1:37:38
  Right.
• 1:37:38
  That counterexample exists.
• 1:37:40
  It's kind of like a human blind spot that we didn't really consider.
• 1:37:44
  There's not a lot of papers about the Jacobian conjecture in degree 3.
• 1:37:49
  Sorry, in dimension 3.
• 1:37:50
  Yes.
• 1:37:50
  Yes.
• 1:37:51
  Yeah, which goes back to then what is the source material for the insight.
• 1:37:55
  And historically, there's kind of an irony.
• 1:37:58
  If you look at Yiteng Zhang, he's a very famous number theorist.
• 1:38:03
  He's one of the first guys to bring down the prime gaps to something like, I think it was like 79 million.
• 1:38:09
  Prime gaps meaning, you know, the twin prime conjecture where like there's an infinite number of primes that are two apart.
• 1:38:15
  three and five, 11 and 13, one-on-one, one-on-three.
• 1:38:18
  So we still don't know if there's an infinite number of those, but he's the first guy to bring down.
• 1:38:22
  And he said, there are an infinite number of primes that are like less than 70 million.
• 1:38:27
  And everyone was like, ah, right.
• 1:38:29
  But his story is insane because like during his PhD at Purdue university, he was under the advice of, um, so on seeing Mo and,
• 1:38:40
  And his work was on this very conjecture.
• 1:38:43
  It led to years of hardship.
• 1:38:44
  He couldn't get anywhere.
• 1:38:47
  It's a really kind of a sad story because his dissertation relied on this auxiliary corollary about the conjecture that was proved by his advisor, later discovered to be false under peer review.
• 1:38:58
  And so his entire PhD...
• 1:39:00
  Turns out to not be true because that's how math works.
• 1:39:03
  Right.
• 1:39:03
  And then kind of crazy.
• 1:39:05
  He like he couldn't get a his advisor was kind of pissed off about that.
• 1:39:10
  Yeah.
• 1:39:11
  He couldn't get a recommendation letter.
• 1:39:13
  And he was he was a subway worker, minimum wage, and he lived out of his car.
• 1:39:18
  Jesus Christ.
• 1:39:19
  for a while before he proved the twin prime corollary and then became like a very famous mathematician.
• 1:39:28
  And now we've got something like AI that maybe can save mathematicians time, but at the same time it's like how much of this is AI going to do to the point where are mathematicians going to be left with a job?
• 1:39:40
  Or are they going to just be explaining stuff that AI is doing?
• 1:39:44
  That brings me to my next point.
• 1:39:46
  Which is about the...
• 1:39:47
  In 2020...
• 1:39:48
  This is part of a trend.
• 1:39:49
  Yes.
• 1:39:50
  In 2026, the OpenAI model disproved the Erdos planar unit distance conjecture that was conjectured by Erdos, Paul Erdos in 1946.
• 1:40:01
  This is the proof for it.
• 1:40:03
  It has to do with like the...
• 1:40:06
  it asks like the maximum number of pairs of points that can be exactly one unit apart in 2d.
• 1:40:12
  So like trivially, you could imagine a grid of points, right?
• 1:40:17
  A grid of points.
• 1:40:17
  All of these points are one distance apart.
• 1:40:20
  And you would think that that's like the best way to pack all the points, right?
• 1:40:23
  Airdosh showed that there's like an upper bound on how many points I can pack with this amount of distance.
• 1:40:29
  And,
• 1:40:31
  The OpenAI model disproved that bound and showed that there's a better bound.
• 1:40:36
  And then later on, mathematicians took that proof and made a short, digested, human-verified version of that counterexample.
• 1:40:45
  Again, it's a counterexample.
• 1:40:46
  But here, the authors are a who's who of mathematicians.
• 1:40:50
  We've got Timothy Gowers from Trinity College at the University of Cambridge.
• 1:40:55
  in the UK.
• 1:40:57
  He's a Fields Medalist.
• 1:40:58
  We've also got Jacob Zimmerman, which it's been leaked, is the Fields Medalist this year.
• 1:41:03
  If you go on Polymarket, it's all collapsed to 99% for people.
• 1:41:08
  That's like, okay.
• 1:41:09
  Somehow it got leaked, and now no one's betting anymore.
• 1:41:12
  Jacob Zimmerman, actually, I think he was my...
• 1:41:16
  He was my TA in freshman year.
• 1:41:18
  Oh, we love that.
• 1:41:19
  Yeah, because he was a PhD student of Peter Sarnak, who was teaching my... Come on the show.
• 1:41:24
  We'd love to talk to you about it.
• 1:41:25
  Yeah, we'd love to talk to you about how you gave me a... Never mind.
• 1:41:29
  Let's not talk about how you graded my midterm.
• 1:41:31
  But, you know, so this is a who's who, and what they're doing is...
• 1:41:35
  taking the work of open ai and trying to make it human digestible so is that what mathematicians are going to just end up doing right right right being translators of the machine genius yeah and and the funny thing about this right so um open ai solved that eros conjecture
• 1:41:52
  um levant our hero from this story at anthropic he tweeted again in all lower caps he's like um over the weekend i checked the obvious thing which is whether mythos is able to solve the erdos unit distance problem um aka erdos problem um the answer is yeah i love i think he's just trolling it's just all all lowercase
• 1:42:18
  Gen Z capitalization language.
• 1:42:20
  And then his colleague is like, you know, huge, huge credit to the OpenAI team.
• 1:42:25
  But he's like retweeting.
• 1:42:27
  But there are colleagues and he's like trying to make it a little bit more.
• 1:42:31
  Yeah, yeah, yeah.
• 1:42:32
  Corpo speak.
• 1:42:32
  Yeah, yeah, yeah.
• 1:42:33
  But it's just so funny.
• 1:42:35
  I think the guy's just trolling.
• 1:42:36
  Yeah.
• 1:42:37
  This is so...
• 1:42:38
  There's so much here.
• 1:42:39
  Sorry, continue though.
• 1:42:40
  Yeah.
• 1:42:41
  And the final thing I want to end with is this is clearly now becoming an existential crisis for mathematics, right?
• 1:42:47
  If AI is going to prove everything, what are pure mathematicians going to do?
• 1:42:52
  And it makes me kind of sad, if I'm not going to lie.
• 1:42:55
  It makes me happy that we're seeing such obvious counterexamples for things and stuff that I can understand.
• 1:43:00
  And who knows?
• 1:43:01
  We'll maybe...
• 1:43:03
  Find a proof for the Riemann Zeta, or maybe we'll find a counterexample for the Riemann Zeta hypothesis where it's going to just come up with a zero for the Riemann function that's like, I don't know, two billion on the one half real line.
• 1:43:19
  Who knows, right?
• 1:43:20
  Who knows?
• 1:43:21
  Probably not, because every mathematician that I know says that the Riemann hypothesis is true.
• 1:43:26
  But who knows, right?
• 1:43:27
  At this point, it would be kind of cool to find it, because that I could also check.
• 1:43:32
  Yeah.
• 1:43:32
  Like, not by hand, maybe, because I'd have to do it a bunch of times, but I could probably check on Wolfram, right?
• 1:43:38
  So it's concerning.
• 1:43:42
  It's concerning, to say the least, for pure mathematics.
• 1:43:45
  And in June 2026, there was a statement from the Lorentz Center regarding AI-generated proof.
• 1:43:50
  It was called the Leiden Declaration on Artificial Intelligence and Mathematics.
• 1:43:57
  A bunch of mathematicians came together, and they laid out their key concerns about this new shift
• 1:44:05
  In how we are starting to do mathematics as a discipline.
• 1:44:09
  There's a threat to open science, right?
• 1:44:11
  Because we don't know what the model weights are.
• 1:44:15
  Yes.
• 1:44:16
  Because they're closed source.
• 1:44:17
  Yeah, they're closed source.
• 1:44:18
  So I don't know what's happening inside.
• 1:44:21
  It's not really open.
• 1:44:22
  I'd have to pay to play.
• 1:44:24
  Mm-hmm.
• 1:44:25
  it's unsustainable journal review backlogs, because you can just now start making all sorts of proofs, submit it to the analysis of mathematics, and then what?
• 1:44:35
  The humans are going to... And then now what?
• 1:44:37
  We get AI reviewers?
• 1:44:39
  Yeah.
• 1:44:39
  Seems...
• 1:44:40
  I don't know.
• 1:44:41
  We don't like that.
• 1:44:41
  I don't know, right?
• 1:44:43
  And then there's divergent incentives, because you've got a commercial focus on benchmarks from all of these AI companies, versus the mathematicians who want to do deep theory.
• 1:44:53
  Mm-hmm.
• 1:44:54
  What are we going to do?
• 1:44:55
  And they made some recommendations.
• 1:44:56
  One was mandatory AI disclosure.
• 1:44:59
  We love disclosure.
• 1:45:00
  We love disclosure.
• 1:45:01
  Here on this podcast.
• 1:45:03
  Enhance scrutiny.
• 1:45:04
  And there's proper attribution for human research.
• 1:45:07
  Where did these AI models get the inspiration, right?
• 1:45:10
  We'd like to know in our workings.
• 1:45:12
  Bingo.
• 1:45:13
  It's going to be tough because a lot of these LLMs are black boxes at the end of the day.
• 1:45:16
  Claude, Anthropic is doing a great job trying to elucidate what is happening in the inner layers of their models.
• 1:45:24
  But at the end of the day, it is kind of...
• 1:45:28
  A black box, right?
• 1:45:30
  They're also trying to go for independent funding from these industries to support mathematicians because you just covered a story about how fundamental research funding is down.
• 1:45:40
  Well, mathematics research is also down.
• 1:45:43
  And I don't think it should be one of those things where, oh, AI can do it.
• 1:45:47
  We shouldn't have humans doing it.
• 1:45:48
  I mean, I think mathematics is one of the most beautiful human disciplines that we can ever take part in, right?
• 1:45:54
  It's something where we can say...
• 1:45:55
  a statement is true and it will live on forever as long as human civilization is alive.
• 1:46:00
  If we can engrave it in stone, right, and it outlasts human civilizations, and aliens came down a la Halo and saw the forerunners and saw Fermat's last theorem embedded in rock, they would be like, I kind of understand that, right?
• 1:46:18
  If they saw squares in Pythagorean theorem, like...
• 1:46:21
  That would be so cool because they have their own versions of that.
• 1:46:26
  And so I don't want this to be a death sentence for mathematics.
• 1:46:33
  It's really, I really hope that's not what's happening.
• 1:46:37
  I do think another kind of aspect to this that you bring up is mathematics as an understanding.
• 1:46:43
  If we want to retain human oversight over AI systems is fundamentally necessary.
• 1:46:50
  Yeah.
• 1:46:51
  Because the way these systems are built and work requires this as a discipline that is understood very well.
• 1:46:56
  Yes, logic, correctness, truth.
• 1:46:59
  And you cannot, we cannot say, oh, we'll have AGI and artificial general and artificial superintelligence.
• 1:47:07
  And again, the argument is, well, if it's superintelligent, humans can't keep up.
• 1:47:12
  We need to maintain this cohort of experts for as long as possible to manage whatever this transition is going to be.
• 1:47:19
  Because I think we're now, I think Sir Demas Asabas has said, we are on the shores.
• 1:47:26
  He has a great phraseology for this.
• 1:47:30
  Oh, we are on the shores of the singularity, or we are on basically the sort of the dawn of the singularity here.
• 1:47:37
  Yeah.
• 1:47:37
  And I know Ray Kurzweil is known as kind of the modern proponent for this concept.
• 1:47:44
  No one has ever been able to describe in words what it would feel like to be right there when the flip was about to switch.
• 1:47:54
  And to me, even as a layman.
• 1:48:00
  you can start to feel the ground moving yeah underneath our feet it's not to say the killer robot outcome and sky now that's not what we're talking about but if the raw power of machines at the edges of what human capability is is is here now what does that mean and there are so many ways in which
• 1:48:25
  mathematically, scientifically, engineering-wise, socially, politically, economically, religiously, all of the E's.
• 1:48:37
  You know, someone made the joke, it's like, when people say that they're, why do I care about politics, it doesn't affect me.
• 1:48:43
  Oh, that must mean you're not, you know, a public employee, a veteran, a young person, an old person, a disabled person.
• 1:48:49
  You come from wealth.
• 1:48:52
  It is.
• 1:48:53
  I know.
• 1:48:53
  I know that everyone is tired of hearing about AI.
• 1:48:58
  I know that.
• 1:48:58
  I get it.
• 1:48:59
  I totally understand it.
• 1:49:01
  However, if we could have stopped the bankers from doing 08 before it happened, would people say no to that?
• 1:49:09
  Yeah.
• 1:49:10
  And it feels like a similar situation where there's a runaway train here and we need to do something.
• 1:49:17
  Yeah.
• 1:49:18
  Yeah.
• 1:49:18
  I don't know what the answer is.
• 1:49:19
  Yeah.
• 1:49:19
  And maybe it's not to stop the progress because progress is always good.
• 1:49:23
  Right.
• 1:49:24
  But we got to talk about it.
• 1:49:26
  And we have to engage in it.
• 1:49:27
  Yeah.
• 1:49:28
  And we have to have these discussions.
• 1:49:29
  And we got to have some kind of policy, right?
• 1:49:32
  It can't just be, oh, it's like too late.
• 1:49:34
  Right.
• 1:49:34
  Right.
• 1:49:34
  Because everyone's unemployed.
• 1:49:36
  Because it's not.
• 1:49:36
  And now let's try to solve it.
• 1:49:38
  Because right now it's not too late.
• 1:49:39
  But very quickly it might be too late.
• 1:49:41
  For mathematicians.
• 1:49:42
  Yeah.
• 1:49:43
  Or for some of these industries.
• 1:49:44
  A lot of.
• 1:49:45
  Oh, it's the white collar jobs.
• 1:49:46
  Who cares?
• 1:49:47
  I'm blue collar.
• 1:49:47
  It's not going to impact me.
• 1:49:49
  Well.
• 1:49:49
  No.
• 1:49:50
  The humanoid robotics are coming.
• 1:49:51
  So it is, this is, you know, I know I'm making this larger philosophical and sociological point.
• 1:49:57
  But the detail of what we walked through today is the canary in the coal mine.
• 1:50:02
  The second time I'm saying this, this episode for, I think, a larger shift that is part of the reason we do this show, which is to make sure that the frontier of research in all of these industries is,
• 1:50:15
  does not get so far ahead of the everyday person that people don't understand what people are doing on their behalf, especially with taxpayer-funded money.
• 1:50:27
  This was so good, dude.
• 1:50:30
  This was really, really good.
• 1:50:31
  I can't believe I can go around and be like, oh, yeah, the Jacobian, it equals one.
• 1:50:36
  It's a ratio of areas.
• 1:50:38
  You should say volumes.
• 1:50:42
  Because to mathematicians, an area is really a 2D volume.
• 1:50:46
  Oh, those mathematicians.
• 1:50:48
  They crack me up.
• 1:50:49
  We love them.
• 1:50:50
  This was, again, we're having a problem staying under two hours.
• 1:50:53
  We're approaching two hours again this episode.
• 1:50:57
  One, it's because we love y'all so much, and we just can't stop because it's all so good.
• 1:51:02
  We touched on astrobiology.
• 1:51:03
  We talked on science funding, and we spent a long time...
• 1:51:07
  on what may be as we continue to see what transpires what was the prompt can we get a little bit more detail about how we got here what may be another watershed moment in the line of deep blue alpha fold alpha go uh the original i can't remember the name the points in the grid one that opening i did not that long ago no that was this year that was yeah yeah
• 1:51:33
  It's all happening this year.
• 1:51:34
  It's happening right now.
• 1:51:35
  We are so blessed to have you all listening.
• 1:51:37
  If you are still listening at this point in the pod, we need a quote, a comment for our deep, deep audience.
• 1:51:45
  Make fun of mathematicians.
• 1:51:47
  I love doing it as a physicist.
• 1:51:49
  Just say something that's funny about mathematicians.
• 1:51:53
  I am your host, Lester Nare, joined as always by my co-host, our Princeton undergraduate physics, UCLA, PhD physics, occasional mathematician, Krishna Chowdhury.
• 1:52:06
  Thank you all for joining us again.
• 1:52:08
  We are coming up on our one-year anniversary.
• 1:52:11
  It's incredible that it's only been a year.
• 1:52:13
  We have so many fun things planned for the future.
• 1:52:15
  We will see you all next week.