AI Breaks a 90-Year Math Problem, Life’s Alphabet in Space, and Science Funding

Intro

0:00The recipe for life [music] may not be uniquely terrestrial. And we found all five letters of the genetic code floating around [music] in deep space with our first astrobiology story. Congress decided where the money went. It was it's law. It's not like an opinion. And the White House has begun to say, "Hey, [music] this is you're not getting any more money." He drops this tweet. Hello there. The Jacobian conjecture is false. Thanks. And >> [clears throat] >> the KNX. >> Hello internet. This is your captain speaking Lester Narre and I'm joined as always by my co-host and our resident PhD Christian Chowdery. This week we

0:44have a rundown episode with three very intriguing stories. We're starting off with astrobiology where new evidence suggests that asteroids possibly could have seated early Earth with life's molecular building blocks. 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 that ultimately makes all of the great stories that we cover on this pod happen. Who gets to

1:25decide who gets the money and how? um the process by which it's awarded and distributed is important and so we want to make sure we touched on this because this show would not exist without our current ecosystem. And we will lastly end with a new math discovery that has rocked social media uh 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 progress rapid progress in AI

2:06in the near future and along the way. I think we might learn a little lesson in 3D calculus. Uh I'm going to need that one. Um for the AI story, this is going to be a little bit more of a deep dive in this rundown because it's it's pretty important. But as always, we're going to talk about the science from the ground up today because this is from First Principles.

Life’s alphabet on asteroid Ryugu

2:41>> [music] >> So today we are going to start off with my favorite subject, aliens, as many of you longtime listeners know. Uh but what's interesting about this is the recipe for life may not be uniquely terrestrial. And we found all five letters of the genetic code floating around in deep space with our first astrobiology story. Yeah. Um, 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

3:21on Earth with all of life, there's four in DNA, A, T, G, and C. And then there's one that RNA uses. It uses the U uricil instead of thamine that's used in DNA. They found all five on an asteroid. 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. And um it's it's hugely

Abiogenesis and delivery from space

3:45important for origin of life debate, things like that. It's hugely important for this theory called abiogenesis which scientists have debated for a very long time. It's this idea that the process by which life arises naturally can be from outer space right and actually from non-living matter. That's that's the idea right abiogenesis means from non-living to living a bio. 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. Okay, it's like there

4:25was liquid water and then immediately we have fossils of tiny bacteria looking thingies. And the question is how come it happens so fast? And what's the necessary chemical ingredients that can assemble so quickly to create something like life? One important hypothesis is exogenous delivery. meaning asteroids and comets. We already know that comets brought most of the water on Earth. Um, 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. Let's not worry about how it

5:06originated somewhere else. But it originates somewhere else and then an asteroid gets kicked out of that planet or whatever thing has the life. It roams around space for millions maybe billions of years and then it finds its way on Earth. as an example that we've seen recently that we've covered on the show is, you know, the interstellar object of three Atlas, you know, that came into our neighborhood that was one of these travelers that was roaming around. Obviously, he did not make impact >> Yeah. >> with Earth. But we've also covered the crater in Arizona. >> Yes, we've covered the crater in Arizona. We've covered um how like actual Mars a Mars meteorite could

5:49impact Mars, get travel to Earth and then get on Earth and that entire process can be just enough like just um just the right conditions where life under the surface of the asteroid could survive. Maybe not on the surface because it's got to survive re-entry and all that kind of stuff, but like just under the surface, if there's life, it could survive the impact and it could actually create life on Earth. We we've we've had several stories in the past that talk about this. So, scientists have found individual organic compounds in meteorites that fell on Earth, right? But those samples always carried a major asterisk, which is mainly that you can have terrestrial contamination. The most

6:31famous of this is in the 1990s when Bill Clinton um he got in front of the press and I >> think it was in the Rose Garden maybe. >> Yeah, he was in the Rose Garden and he said, "I did not have [laughter] >> I always do this because we've covered this so many times. It's like a running joke now, but he he was actually announcing a science paper by um

The contamination problem

6:54scientists at JPL that showed that a Martian meteorite that they had found in Antarctica might have had some bacterial looking things. And then immediately the entire world community of scientists was like, well, no, that that could be because you've either contaminated and also just cuz it looks like a bacterial looking thing doesn't mean that it's actually bacteria from Mars, right? Um, so there's been a lot of debate about whether we can even make those kinds of arguments, right? Because once a space rock lands on Earth, it immediately gets contaminated by Earth biology. >> So here the hero of the story is a

7:34satellite/bombing system by the Japanese space agency called Hayabusa 2. And what it did was go to an asteroid called Ryu, R Y u guu.

How Hayabusa2 collected the sample

7:50>> Hopefully I'm pronouncing that right. What it does is it gets near this asteroid. It deploys a kind of remote bomb that >> pulverizes part of that asteroid and then it goes back to that site, >> goes down and then picks up the fragments of the pulverized rock. This is the alternative uh plot to Armageddon. Instead of going on to it and drilling, they did it from an at from the atmosphere. >> Yeah. Yeah. And and they did it remotely. Now we now we can do that, right? Um so >> that that's the the premise of the whole thing. The next video actually shows this is a video from the Jaxa probe and it's showing you you're lower down into

8:32the part of the asteroid that's been pulverized and you grab it. You grab that part and you come out. It's pretty cool that we can successfully like chase this spinning top asteroid, blast a crater, and then bring bring back pristine dirt from an asteroid back home. Who was the one who was like, "Guys, I have an idea. Let's take a bomb." >> Yeah. >> Let's explode it just above the surface so that we can then go down and >> Yeah. And then there's a bunch of dirt instead of rocks, >> right? Because the the idea is the surface is solid. It's a little bit You can't >> Yeah. And the rocks are kind of big. >> Big. And you don't want to have to have a drilling mechanism on the whatever probe you have. >> It's clever. I'm I'm very curious how

9:13that conversation went in the the the proposal meeting. >> Yeah. And the other thing I'm actually curious about is um if the if that little bomb changed the orbit of Ryu, right? Cuz we've we've talked about how planetary defense, one of the biggest things that we can do is just like >> put a minor explosion to change the momentum of that asteroid so that it changes the trajectory around the sun. 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. We've covered this in the past. So um Jaxa and the Hayabusa team they came out with a paper in nature in nature astro a complete set of canonical nucleiobases

9:56in the carbonatous asteroid which means they have found a gct and they found you which is the alphabet of life in some sense right >> it's all the ingredients we need to ratatouille ourself into the way of life >> 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. So there you can see the dirt from the asteroid. What they did was take that sample, 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. This is a it's a big it's

10:36a 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. >> Yeah, we've removed the contamination hypothesis and it's the same nucleotide basis that we have, >> which is kind of like like it's kind of weird that like aliens would also have the same [laughter]

Ryugu, Bennu, and cosmic chemistry

10:58nucleotides that we did. >> So, you're saying there's a chance? >> Yeah, [laughter] it's it's it's kind of crazy, right? And the big picture is, so this isn't actually the first time, okay? Um NASA >> had a similar experiment in 2023 where they went to the asteroid Ben New >> and they found samples that had the five nucleotide bases. What's cool about this is one that wasn't a fluke. The fact that it's happened again on a different asteroid means that this is something that is probably ubiquitous >> in asteroids and comets, >> which is incredible. >> Which is cool. Yeah, >> that's incredible. It's not a one-off thing. It's not like a special asteroid. >> We got lucky one time. >> Yeah. Yeah. >> That's not what we're talking about.

11:38>> Twice and and we and we find nucleotide bases. >> Not that this and there's another variable in a short time frame as well. It's not this is every hundred years, every 300 years, every thousand years, something happens to come around that has it. This was in a >> Yeah. And this is just local to our solar system, which is also great. >> Yeah. I I think I think I think it's really cool. And what you can do now and what the paper did is specifically compare the ryu composition of nucleotides with the benu composition of nucleotides and they actually have different ratios. >> Okay. >> So both asteroids have the same building blocks of adinine, guanine, cytosine, thymine and uricil. The same nucleotides

12:19that we have but the proportions are completely different. In Benu, it's heavily skewed towards the paramedines, which are the cytosine, the thymine, and the uricil. But in Ryu, they're more balanced. Now, what does that mean? That means that for one, there's complex chemistry happening, right? 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. Um, it could mean that it has something to do with the water rock interactions that are on the the asteroid, the internal

12:59temperatures. But the main thing is it's a lot more complicated up there, right? There's like actual chemistry happening. >> We have >> and it doesn't have to be on a planet. We have a cosmic oven, a cosmic easy bake oven that is our galact intergalactic postal service. Yeah. Like you sort of mentioned which and I think part of the point what 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 what's happening on earth is happening everywhere else. We like to see it. Yeah. Before we believe it. >> Exactly. >> Sure. Maybe it's happening other places,

13:42but let's actually get not only one use case, Bennu, but multiple use cases. And the idea that we're having now um a catalog. Now, we now say there's a catalog because it's more than one. [laughter] >> Yes. And as we do more and more, I'm sure we're going to find more and more asteroids with these building blocks, right? It's like the universe is like inherently preloaded >> with the chemical ingredients that are necessary for life. It's kind of insane. >> It's it's really fascinating. This is re and I mean there's so many again it feels like people well we've already known this like why does it matter? Um but as we know with the process of science like being able to have

14:23replicability uh matters a lot. These these missions are not cheap. >> No. >> Right. To to pull this off. >> Yeah. Um, >> and this is two this is two different space agencies doing it. Right. Right. So there's no conspiracy unless Japan is in on it. >> Yeah. Right. Right. Which look look we'll put the >> I'm just I know every Oh, of course. Yeah. Of course they're collaborating to conspire here. But it's not even something that's that, you know, quote unquote sexy to like even waste the time to do so. Um I this is a really big deal. And I do think the mix difference is also quite interesting because I imagine that there are several follow-up inquiries into

15:06seeing their journey. >> Mhm. >> And understanding based on that that journey, why is that mix different? >> Yeah. >> Um, fascinating.

Support the show

15:15>> Very very cool. >> This is fasc. So, you know, as always, like, you know, I'm the 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. But before we do so, we're going to do a quick piece of housekeeping for those listening to us on Spotify and Apple. We greatly appreciate your ears in your commute to work as you're working through cleaning up the house, whatever it might be. One of the ways you can help support the continued coverage that we do on this show is a quick five-star review, sharing it with a friend, bring it to Journal Club. For those of you watching, we always, if you can, recommend

15:56watching because each episode we go through 25 to 65 different overlays where we actually show what's happening and it really helps to understand what's going on when you can see it. But we are trying to work on making sure for our audio listeners that we describe as much as possible. Again, a like, a share, a comment. Any of these engagements helps us win the battle against the billionaire algorithm. If you'd like to support the pod more directly, you can either donate directly at ffpod.com/donate or you can go to our new shop where you can buy some of our great merch. We are

16:38going to be dawning that merch more frequently on the pod so we can shill accordingly. But as you all know, it's the two of us here bringing the greatest and the best newest science discoveries to you every week. And any support that we get from our great FFP nation is much appreciated. Speaking of money as it relates to science, our second story is going to be about funding policy in the United States as it relates to science. Every time we cover a study on the show, um could be genetics, medicine, physics,

17:19space, we always talk about the final product, >> which is the published paper and usually the history and the buildup to that. Um, 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. The number one funer of basic research for the American ecosystem is the federal government and it's what makes it possible. And there's a recent controversy that's risen around this. But 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

17:59works. So in the US, Congress has what's known as the power of the purse or if you're from Boston, pocketbook, >> but for the rest of us, the purse. The president proposes a budget. Every year at the beginning of the year they he proposes a b he or she proposes a budget but Congress ultimately decides how federal agencies are legally permitted to spend money and for what purposes. So the executive branch has all these responsibilities but money and where it gets spent. Congress ultimately is decided there. OM which is the white house office of management and budget helps to execute the budget that is fundamentally decided by Congress. a

18:40portions funding to executive agencies and sets governmentwide rules about how that money can be spent. So, Congress says here's what here's the money and who can spend it and then OM kind of just distributes. >> Yeah. >> Uh how that goes and this is how federal grants especially in research but not only in research are administered. >> Okay. Um,

How federal science grants normally work

19:02normally in the history of this process, the office of man of management and budget doesn't select individual science awards or grants. It's not in the process of selecting who gets what money. um that happens at each particular agency, National Institutes of Health, National Science Foundation, NASA, Department of Energy where the experts are >> they are the ones who get to decide uh how that happens and the process is the agencies will announce a funding opportunity research researchers will submit proposals experts evaluate their merit. >> Yep. uh as you know very well in this process. Yeah. And the agency officials ultimately decide which projects get to

19:45receive awards. >> Yeah. And shout out to my dad. He's 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. >> And an expert being on that is something you absolutely >> want to be there. And then from there the university or research institution that receives the funding um employs the researchers. They operate the labs, they conduct the work and eventually produce what we talk about on the pod, which is the paper. So yeah, Congress funds it. >> OM establishes the rules. Agencies select and manage the grants. Researchers do the science. That's how the process has worked forever. Um,

The fight over research funding

20:27but now you know we're going to get into the current fight. But 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 uh closely, particularly uh Jeffrey Mvvis, Jocelyn Kaiser, Lauren Martin Aguello, Michael Gresko, and David Malikov at both science and 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 has has for people. And this is really again a great source for us as we try to begin

21:08to now incorporate science funding policy into our discussions because we would not be able to again have a show >> without this. So now we get to the current fight and it's always over who has the power to decide.

OMB’s proposed grant overhaul

21:24Um so I'm going to kind of read through a couple of the details here which are a little bit in the weeds. So on May 29th, OM or can only decide the rules, right? OM released this 412 page proposal to sort of revamp their role in distributing and managing grants and awards. >> And this is the White House effectively. >> This is the White House. And so this is a very that's a very important note. It's not some it's literally the White House's Office of Management and Budget. Um 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

22:05starts in January and February of every year. Um so this new proposal had until July 13th to provide comments >> and OM said that it wanted to sort of finalize these new rules by October 1st. The proposal applies to all federal grants, not just specifically science, but several of the provisions are kind of important. So, we're going to kind of just briefly walk through a few of these. So, first, senior political appointees could independently review discretionary grant decisions before awards were issued, right? Even before they were actually issued to people.

22:45Number one. Number two, peer review would quote remain a part of the process, but the proposal makes clear that scientific reviewers

Political appointees and peer review

22:53recommendations like your dad as you just mentioned would be advisory and not controlling. >> Okay. >> Right. Currently, they have the controlling what they say kind of goes. They're saying that's this more of just an opinion and we get to decide >> and and we being these political >> appointees, right? Who >> Great. Nice. >> Don't really have >> uh any scientific Yeah. >> background. 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. >> Okay. >> And then fourth, the proposal would impose major restrictions on

23:35international research collaborations with countries that are deemed to be national security risk. The fourth one, >> fourth one. I get >> that one has some some standing and you could argue the details there. >> Yeah. >> Okay. So, what is the argument from the White House, right? They're basically saying that they want to ensure that taxpayer money is lawfully, you know, executed on and that it's aligned with our priorities and there's no the favorite phrase of DC, waste, fraud, and abuse. >> Right? 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 abuse. But they argue that the peerreview process has become insular and the final

24:17responsibility should remain to politically accountable appointees. That's the argument being made for why this proposal is being place. And you know, in a in a vacuum, you would maybe say,

The scientific community responds

24:33okay, maybe there's an argument here. Um, but obviously the scientific community has reacted very strongly about this going well beyond 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 zeicist as it has since co happened in my view frankly at least in American culture. Yeah, I agree. >> Um where it's really become now a culture war issue in a way that it really never was before. >> So obviously the scientific community is arguing this is going to allow ideological nonsense

25:13>> to get involved in the process. So the the uh Association of American Universities submitted a 47page response which I might add is the longest public comment uh in the organization's 126 year history. So clearly uh it struck a nerve because everyone understands what's going on here. Um, and the the primary there's so many arguments to be made here, 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 agencies. >> That's right. >> Has not given OMB the authority to decide what categories of scientific research may receive federal funding.

25:54That power lies purely with inside the agencies and their statutory directives. So effectively the universities are saying the the universities are using the constitution to try and defend the way that we fund science >> which it is in the constitution. Yeah literally as well as the actual mechanism of the movement of and and how money is budgeted >> like you know Congress decides how money is spent. Yeah. >> Period. the executive branch is effectively trying now to insert themsself in the process of how money is spent. >> Yeah. >> Um so there's to your point there's a constitutional level issue here beyond

26:37even some of the practical matters. Um >> what's interesting is this proposal generated almost 500,000 497,000 public recorded submissions. So anytime there's proposals like these at different agencies or offices within um all the branches of government, they open up it to public comment. Oh >> yeah. >> So almost half a million comments were like obviously this is ridiculous. >> So again, I know this is dense, but I think this is really important because the implications here are could not be more uh important to cover. So that's

Existing grants and legal challenges

27:13what's currently happening in terms of OM trying to change the process. 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. >> Yeah. >> So, Congress decided where the money went. It was it's law. It's not like an opinion. 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, this is you're not getting any more money." >> Yeah. >> Something you may or may not have some

27:55context for. >> Oh, yeah. I mean, I have several friends in the university ecosystem that are very worried about their future and their present scientific endeavors, right? like they're they're they're in the middle of science projects that they're and they have a few experiments left before they publish the big paper in nature and science and cell and all this other stuff and yeah all of a sudden the grant funding is gone and you haven't quite finished the story what do you do right it's it's really tragic so as we have two sides of the story here there's the proposal that's happening but there's action that have already been being taken since January

28:37of last year. And on July 17th, a federal judge ruled that agencies cannot use newly adopted political or pragmatic priorities to retroactively cancel grants under different criteria. 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, it you can't do that. Right? So that's um you know been established in federal court. Now a caveat to that is the government may retain legitimate authority to terminate grants which will can still continue to be challenged. But the

29:18judge'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. 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. So this is not even just in things like physics and biology. This is also impacting health research studies. Those projects represented approximately $185 million in multi-year commitments. And so they've already received a distribution and they're saying you're

29:59getting no more money. >> Yeah. >> Even though they've already received the money for it. And 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. So HHS said the grants were not technically terminated. They were simply not awarded continued funding because it's a multi. So rather than saying we're terminating funding for something that is a single allocation, they're saying, "Oh, this was a multi-year uh thing and we're we're discontinuing the continued funding of what is a multi-year install." >> That doesn't make any sense. [laughter] >> Are you serious? So it's basically

30:40termination 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 >> I'm not the the the source, but they may be uh synonyms in this context. >> Yeah. >> Um >> and this is so this is both a constitutional and budgetary dispute that that's important. So, Congress does not vote on individual research projects. Right. Right. It just appropriates money to an agency. >> That agency chooses individual grants. Multi-year grants may also receive their funding in annual installments rather than all at once. So you could have a program that gets a $50 million allocation in 2025, but it's spread over four years. >> Yeah. Yeah. I mean, a lot of the like some of the grants that I was supported

31:22during my PhD were these multi-year grants where like my adviser or like the collaboration that I was a part of would get money every year, right? You don't get a lump sum. 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 and distribution. It's an administrative rule. It's never been that way. >> It's not a discretionary rule. And this is this is a very very contentious issue right now. So the central debate dispute is whether the administration is simply managing the the congressional funded programs or whether replacing Congress's priorities has you know their own right to do like

32:05if they have their own right to basically displace what Congress has already decided to do. 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 over. So OM office of office of management and

Three branches battle over funding

32:25budget is trying to rewrite the governmentwide rules that then gives them authority to say you can't get money. Congress is in the process of negotiating next the next fiscal year's appropriation bills right at the same time. and federal courts are weighing in on how much authorities the White House effectively has on cancelling existing awards. So all sort of three branches of government, the executive, the legislative, and the judicial are concurrently engaging in a battle to really define what happens here. >> The the comment period for the rule

33:06proposal changes is over. uh OM wants to set the statutory deadline for October 1st at which point this would then take plate this these new rules would go into place for fiscal year 2027. They're making the argument that this is about accountability. It is clearly not account about accountability because there's already systems in place to deal with that. Uh the real kind of issue is does the executive branch have the authority to make individual funding decisions within pro programs that Congress has already authorized? And again, this goes back to it's a

33:47fundamental constitutional question. >> Yeah. >> Because it literally goes counter to >> the structure of our co-equal branches of government. Yeah. So, will federal research continue to be evaluated under rules disclosed in advance, informed by expert review, and tied to programs funded by Congress, or will every grant now be contingent on the changing political priorities of whoever's in the White House? And it moves all the time. So, just because it stops for a 4year cycle or an 8year cycle does not mean it's then going to flip the other direction. And we cannot afford to have a seessaw problem in the what has made

34:31this country a huge part of what has made this country stable and successful uh and resilient is fact that how we won World War II is because we've done this funding uh in a way that's already been structured. >> Yeah. I [clears throat] mean you can go back to our America 250th anniversary episode, right? 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? And I'm sure a lot of those grants were awarded without political oversight. Okay? Because I doubt the the powers that be

35:14are going to understand why it's important to study bacterial immune systems. Okay? But then we get like restriction enzymes, genetic engineering crisper. Come on, guys. What are we doing? >> I don't want the people that did not understand that studies using transgenic mice >> Yeah. >> wasn't uh talking about the the gender of mice. >> Yeah. Yeah. Yeah. >> Like like you know what I mean? Like that's we can't >> Yeah. Transgen. And then they're like, >> "Guys, like, how they should not be in charge of of this kind of power. This is insane." >> We're going to keep you all updated on

35:54this process cuz it's happening in real time. Uh we will know more over the next couple of weeks. The one 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. >> Yes. Yeah. >> Period. And so it is now becoming a battle between the legislative and the executive. And it's a midterm election year. And so members of Congress are particularly

What the public can do

36:28sensitive to issues that are animating their constituents. 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. They're going to be having town halls. attend, call their offices, send them emails, they track this stuff. One of the greatest secrets ever is that people don't understand how much these members actually listen to people who reach out. The problem is no one reaches out. And so all these specialty issues get kind of overlooked because they just no one

37:10talks about it. So it's time to put the pressure on on this issue. Again, the system is working totally fine. We don't want the White House of any administration 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. And again, if you want changes around who those experts are in those agencies, there's a process for that. We don't need the White House involved. >> We really don't. >> Um, well, yeah, that's a and and you said this is a developing story, right? So the decision is moving to October 1st. So it's like right before the elections right? >> It's going to be a hot button issue and

37:52we need to make it >> a hot button issue >> 100%. >> Yeah. >> You know, um you guys like federal funding of science is the bedrock of American innovation and American innovation is the bedrock of why this country is so great. Okay. So we we need to get on it, guys. >> It matters a lot. Yeah, >> it matters a lot. And I I know some folks's eyes will glaze over, but again, we wouldn't be able to talk about Yeah. >> something like our next story without this ecosystem of funding because ultimately doing anything costs money. And I'm really excited about this story

38:32because it was all over my feed. >> Yes. >> All over my feed. >> Yeah. This was um this was kind of huge because it happened in the middle of the World Cup final. [laughter] If it was it was actually crazy. 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. >> Who who won that? I just I don't know. I can't remember. I I know it wasn't Argentina. It wasn't Argentina. That's right. >> That's right. It was not. >> It was not. >> It was not Argentina. >> That's right. Um,

39:13>> you love to see it. >> I just, you know, for everyone who suffered through our World Cup coverage, it's over, so we'll stop the sports ball. But I just I couldn't remember. >> Yeah. Yeah. It's good to It's good to remind oneself. >> Just just just to set the record straight. >> That's right. That's right. So, this next story is about AI and how it is encroaching on fundamental mathematics.

AI enters pure mathematics

39:35Um, it's very very cool because I mean it's it's completely obvious I think to everyone who lives under a 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. Okay, 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. Right? We can say that the act of understanding and the act of proving are one and the same thing. And a mathematical proof it's it's it's been here since the time

40:16of the ancient Greeks with Uklid. It's a certificate of the truth, right? 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. >> Very important point. >> Very important point. It's, you know, we've seen a lot of disciplines fall to computation. Chess >> crazy >> has fallen to computation. The game go has fallen to computation. protein folding has fall into computation and now we are finally seeing the the the the shining city on the hill

40:56which is pure mathematics it is starting to fall right there's crumbles in the infrastructure 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 yeah and breathe the same air that they did were the maths folks. >> Yeah, we're the maths folks. Yeah. >> Like 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 going to stick to [laughter] >> I'll stick to the physics department, which is right next door. I took a few

41:37math courses and it was it was extremely fun, but I could just never see myself like being a math major, right? And uh and I mean even in the physics department I was like man what am I doing [laughter] here? But the math the math department was its own little game. And so so this particular story has to do with an AI system. It was used by a researcher at Anthropic

The Jacobian conjecture

41:59um which is like the synonymous the the the company of Claude the the company that wants AI safety AI safety >> regulatory capture >> right 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 polomial maps for about 90 years. It's insane. It's not proved it in the conventional sense because what it's actually done is disproved the conjecture. It has provided a counter example which is um a mathematical object that violates whatever the rule is. It's a bit easier to do albeit than

42:42actually proving something. Right? 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." Because all you have to do is produce an example and boom, the debate is over. especially in mathematics. We're going to get into sort of 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 out of something that for 90 years the top mathematicians have been apparently wasting their time with. Yep. And trying

43:25to prove >> and this is the part of the ongoing debate about whether these AI systems are quote stochcastic parrots that >> don't have any original thought andor 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, this should still matter because for one, it's showing the power of AI to like think complex

44:07things >> in a very in a very real sense, right? AI is restructuring the labor markets. Um, and now it's going after really creative. Mathematics is a creative discipline, right? And a lot of people say that AI can't be creative. It's, as you said, a stoastic parrot. Well, in this case, nobody had a nobody had a proof of this, >> right? >> And it has a novel proof that the smartest people on the planet couldn't come up with >> for almost a century. >> Yeah. Um, it's it's huge, right? So, mathematics, right, as I said, it's been the gold standard of human reasoning. Um it's the language of physics, the foundations of electrical engineering,

44:47every other kind of engineering, economics, medicine. When we say [laughter] >> yeah, >> literally everything. >> Literally everything. And when we say that something is mathematically proven, we mean it is literally true, >> right? 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, which has happened time and time again. In mathematics, the decimals are inherent in your proof, right? It's like it's there and it's true. And if I've proven that it's true, the the the logic is eternal. >> 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

45:28objective truth. Yeah. >> When you come to when it comes to like a mathematical proof, >> yes, it is it is it is it is I think the pedestal of objective truth, right? You go from there are integers and I can add and multiply them to then all of the stuff right Uklid's theorem that there are are an infinite number of prime numbers that still holds true today because no one can poke a hole into it. Why? Because it's a proof. They couldn't poke a hole back then because the logic was sound and logic is something that is eternal, right? And so it's it's really cool. And when when when we start thinking about AI starting to do mathematics, not just assisting and not just suggesting but actually proving and

46:10disproving things, right? Then we start asking philosophical existential questions and those become urgent, right? What is the role of the human mathematician? What is the role of the human mind? 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 >> these are these are now becoming real questions it's kind of crazy 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 going to get into the weeds of it and I think you're going to

46:50be able to understand exactly what it's done >> and then we're going to talk about the modern phenomenon of AI and mathematics and why everyone including mathematicians are very worried. >> Okay, so let's start with a Jacobian conjecture. It's something that was proposed by Hinrich Keller in 1939. So nearly 100 years ago. >> Um and it addresses a fundamental local to global paradigm. It has to do with maps. Okay. By maps I mean going from one set of coordinates to another set of coordinates. So in order

Coordinate transformations and Jacobians

47:32to understand the Jacobian conjecture we have to first understand functional maps which in physics language we like to call coordinate transformations. Here's the idea. A map in this mathematical sense is when you take a coordinate plane or uh hyper plane or many dimensional plane and all of the points in your original space get mapped to a new space. In this case, what we're seeing is a 2D plane being mapped to another 2D plane. Okay, there's some kind of nonlinear transformation where the new x coordinate and the new y-coordinate depend on the old x

48:12coordinate and the old ycoordinate. Right? In this case, a square is taken into a parallelogram. Notice that the new x can depend on both the old x and y, not just the x. And the new y can depend on both sets of coordinates. This can happen in higher dimensions as well. The Jacobian is the ratio between the new area and the old area >> and which can also scale from 2D. >> Yes. You can have volumes. For example, you can have a volume in 3D that goes to a new volume in 3D, right? A cube can become like a weird >> kind of uh romboid what whatever 3D

48:55prism is what it's called, I And then you're comparing the ratio between the volume or the area depending on what dimensional space you're talking about between the two is where the Jacobian comes in. >> Yes. Exactly. And in this case the I think I think from from my eye the area is about the same. So the Jacobian in this transform would be one. Got it. >> Okay. Right. >> That's what the Jacobian means. Effectively it's saying okay I have a a map and that map is preserving area if the Jacobian is one. If the Jacobian is greater than one, then it's going to larger cases, right? And like when I think about maps, the reason why it's called a map is because it it really starts from this concept of like maps of the earth, right? You can think about

49:36the Mercada projection or these other projections that change the area and change the shape but keep the sorry, they change the shape but they keep the area the same. For example, with the Mercada projection, right? We all know that areas near the poles get larger. Mhm. >> So the Jacobian near the poles is greater than one. >> Right? That's why Africa looks small and Greenland looks the size of Africa >> because the Jacobian near Greenland is much larger than the Jacobian near Africa when it comes to the Mercada projection. 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. >> So and people would argue which is

50:18better when you talk about geographic maps. Is it better to keep the Jacobian closer to one? >> Yeah. >> Uh where you're going to get some shape distortion or is it better to have it have shape uh clarity but you have distortions at some extremes >> as it rel Okay. >> Yeah. So that that's what we're talking about when we talk about Jacobians. Okay. It's really it's a it's a ratio of how my transformation is changing little tiny areas, infinite decimal areas, right? Because in calculus we think about like there's continuous transformations. So the Jacobian here can be different from the Jacobian there and so on and so forth. >> It's not a family in the game of thrones is what you're saying. >> No, not at all. And so let let's let's do a very simple case. Okay. A linear

51:00transformation. >> Yes. >> Now in this case this is a very simple matrix 2112. Okay. When we multiply that by any 2D vector, we're going to get a new vector. For example, the red vector is getting slightly skewed like that. So the red vector is really a one zero. That's the x-axis. just uh one on the x- axis, zero on the on the y axis that gets transformed to two on the x axis and one on the y on the y- axis. Right? Similarly, the y unit vector which is 01 that gets transformed to one comma 2 which means one on the x axis, two on the y axis. That's why the green arrow is slanted more towards the y- ais, the red arrow is slanted more towards the x

51:41axis. And in this case, the determinant of that matrix, if you were to do it, it's 4 - 1. So that's three. You can see that the square becomes a rhombus that's the size of three and it it has to do with the the determinant of that matrix. So when we say a Jacobian determinant, that's what we're talking about. Okay. >> Yes. >> Okay. So this is a very simple case. Now let's give you um kind of a more complicated case but one of my favorite coordinate transformations as a physicist which is the Lorent transformation. This is the transformation that we know and love from relativity. >> Right. >> Right. In special relativity, we have the Lorent transformation. You might have heard that when we move

52:22fast, length contracts and time dilates. So the clock runs slower and length gets shorter. That's two different things,

Special relativity and the Lorentz transformation

52:32right? Our coordinate in this case is let's just think about 1D, right? There's a there's a length I'm like moving in this direction. So there's a length forward and backwards in the direction of my movement. And um there's also time. I've got a I've got a time on my watch. Now time gets dilated. What does that mean? That means that the clock is running slower, which means that a second is getting longer, but my length is getting shorter. So one coordinate is getting shorter and the other coordinate, the time coordinate, is getting longer. That is central to the physics of relativity. And let me

53:12show you by considering this plot. So let's consider this this plot. Um this is a plot that um almost every physics undergrad has seen. On the x-axis is space. On the y-axis is time. So on the y- axis we've got 1 second, 2 second, 3 second, 4 second. On the x-axis is meters, but actually the one is not 1 meter. The one is one light second. >> So it's really like 300,000 km, 600,000 km. So it's it's the speed of light multiplied by 1 second, 2 seconds, 3 seconds, and 4 seconds. Why do we want to do that? Well, if we if we represent the x-axis that way, then the speed of light is a diagonal. >> Yeah. >> Right. Because it's like one one. It's

53:54like, okay, in 1 second it travels one light second. In 2 seconds it travels two light seconds. So you've got a nice the dotted green line is what light would do in this coordinate um system. Okay. So that's that's usually how we represent things in relativity. >> Mhm. >> All right. >> 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. Yes. >> When they're on the wave planet, the guy is on the ship still and they have that time dilation issue. >> Uh because of exactly what we're talking about. >> Because of exactly what we're talking about. Exactly. And so we've considered that plot. Now let's consider for

54:35example we're in this plot right? It's the two of us. >> Yes. >> Okay. We're both stationary. We're both in the same reference frame. >> May I got a lot of motion. I don't know what you're talking about. >> But as of now I'm I'm going to I'm going to have you move. I'm going to have you move in the next plot. But in this plot >> Yeah. I don't have motion right now, but I will in the future. >> I want to I want to give you a little sense of how relativity is so cool. Okay. >> So let's consider we've got um we're sitting here. This is x equals z this table. Okay. And we set the time. We we have a stopwatch. I've got a stopwatch on my iPhone. I set it to time equals z. And let's say there are three explosions that happen. Okay? Or like three light

55:17switches. The the in the physics textbooks, it's always like light switches. Let's do let's do explosions. Explosions. >> But but we can survive the explosions. Okay. There's an explosion that happens here where we're sitting, but in 2 seconds time. Okay. There's an explosions that happens in front of us um two light seconds away. So 600,000 km away. >> Real close. >> And there's a explosion that happens behind us. Two light seconds behind us. Okay. So 600,000 kilometers behind us. But they all happen at the same time because we're sitting here and we see the we we see the the the explosions happen at the

55:58same time. Okay. So we're like, "Okay, cool. The coordinates of these explosions in spaceime." >> Now we're thinking about coordinates in space time, not just space. The coordinates in spaceime are the explosion that happens here is at space equals 0 cuz it's right here and the time equals 2 seconds. The the explosion in front of us is space equals 2. Yes. >> And the time equals 2 because it happened at the same time. And back there it's -2 time equals 2. >> When you say space time in this example, we're talking about two axes. One is space and one is time. And so you can represent space time >> on a on a two-dimensional xy graph in this example. And so it's not this m

56:40just it's >> no it's literal. Yeah. It's literal on a coordinate axis. This is the genius of Einstein right? >> Right. is he's like putting this stuff on a coordinate plane and he's asking right and this is actually Lorent came up with this stuff even before Einstein he was he was thinking about Lorent transformations and Einstein's the real guy to be like let's just take it seriously guys [laughter] >> right >> okay so now we ask the following we're both we're both sitting here and we agree if we were both sitting here these explosions would happen at the same time now suppose you were moving >> Mhm. that way. Yes. To the front. Yes. At at very close to the speed of light. >> Okay.

57:21The three events that happen simultaneously for me, they are not going to happen simultaneously for you. >> Why? The explosion that's in front of us before you before you Yeah. So, so this is me, right? This is this is my So, so to me, everything's happening at the same time, which is why everyone's in the same time axis. Yes. >> Now let's before we consider the next plot, let's just think about what would happen. You're moving towards >> the explosion in front of me at near >> which is at 22. >> Yep. Which is at 22 for me. >> For you. >> For me. >> But I'm moving towards >> But you're moving towards it. Right. >> At the speed of light. >> At the at near the speed of light. Near the speed of light.

58:02>> Sue me. >> Yeah. Okay. [laughter] 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. Mhm. >> So the distance to that explosion is going to go down >> and the time is going to get a little bit slower, >> but at the same time you're moving towards that explosion. So the light from that explosion is going to come meet you. >> Mhm. >> Because because we're basically it's like two trains coming at each other. >> Yes. >> But because of the speed at which I'm moving towards it >> from your perspective, uh it's going to take time for that light. >> It's going to take two seconds. But because I'm basically meeting the light before it gets to you at that speed,

58:43it's it's going to happen. >> Yeah. And so for you, that event is going to happen sooner than 2 seconds, right? And it's going to happen closer than two light seconds because the length is contracted. Similarly, the event that happens here is going to take a little bit longer than 2 seconds because this thing happens for me at 2 seconds, but then the light has to go to wherever you are. [clears throat] >> So, it's going to happen a little bit longer than 2 seconds. and the guy and the explosion that was way back there. That's going to take even longer. >> So, what we should see when we look at your perspective, right, and 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, right? For your x

59:26and y. >> Yes. >> My x and y is is is stationary. >> And and when we say down on the yaxis, cuz our y >> So, it's sooner in time. >> Sooner in time, >> and it's sooner in space. >> Space, right? and and getting closer to the origin. Yep. Yep. Yep. Okay. Got it. So now let let's let's see how let's see how we did right with that logic. >> So we go to the next plot. >> In this plot nothing has happened so far. I'm g giving you some analysis of of what we're talking about. The as I said the diagonals are light >> right? [clears throat] So the diagonals are what light would travel like the path that light would travel in my spaceime >> because again our our x-axis is light uh

1:00:08light seconds >> and our yaxis >> is just time >> time >> seconds right and so that's why it's diagonal and all of the dotted lines are diagonal because if a light beam started at t= 1 then it would travel like at the it would be the dotted line that's just above the t equals z. This is why we want to decrement the x-axis in this way so that it creates >> Yes. >> this simplicity to understand the >> Exactly. Exactly. Now, one of the one of the and the other big thing is um there's there's two points and there's actually a line a black line right on the right on the um y-axis here which says I'm not moving, >> right? Because you're the one who's moving. So to me, I'm just stationary at

1:00:49x equals z. >> Yes. >> For all of time, right? So no matter what time it is, I'm always at xals 0. Okay. So now what would what would you see for me? Well, you would see me if you're coming from back there, you'd see me with a positive x-axis and then and then you'd see me go backwards, right? Because to you, I'm coming from up there and then going backwards. Okay. Now, let's go ahead and do the Laurent transformation to see what you would see. >> Okay. >> Okay. What would you see? What you would see is the following. you'd see the coordinate axes change. >> Yeah. >> Okay. And notice, first thing we want to notice is the three red dots >> are doing [clears throat] exactly what

1:01:29we had predicted. >> The event that's in front is getting closer to you. So the time is decreasing and the x is decreasing. The event that's happening right here >> is going forward in time because it's not 2 seconds, it's a little bit more because the the light has to catch up to you for you to be like, "Ah, there was an explosion." And the event that happened way back there is happening way farther at 2 seconds. Right? >> But notice two things. And and and for one, the line that is me, >> which is usually time tals 0, xals uh time time equals whatever x equals 0. That thing is shifting to where at negative time. So before time t equals 0, I'm I'm ahead of you and then I go

1:02:11behind you. Right? That's the black line. Yep. >> Here's the key thing about this coordinate transformation, which is the genius of the Lorren transformation. All of the diagonals are still diagonal. >> Mhm. >> What does that mean? That means even in your reference frame, light is still moving at the same speed. >> It is still moving at the speed of light. >> Right? What's really happening is the length is contracting and the time is contra is is getting is getting faster, but they're happening at the same rate such that the diagonals are always diagonal. >> Yeah. They're they're just shifting on the same ratio of proportionality to each other. >> Yes. Yes. By the lance factor actually

1:02:53is is what is what is what it is. And and so it preserves relativity because one of the tenants of relativity is that the speed of light is the same in any reference frame. It's constant. >> So, so even when we go into your reference frame, the diagonals are still diagonal but >> the space has changed and the time has changed. >> The other cool thing about this is the volume of space-time is always the same. The the area of the square becomes a rectangle, but the area of each rectangle is exactly the same. >> Yes. Right. >> Again, because all everything is proport it's a shift that's happening. It goes back to this mapping point that you're bringing up earlier. We are making this

1:03:33transformation in our map but fundamentally our Jacobian remains as one the entire time. >> Yes, that's exactly right. And what that means is if like the volume of spaceime is always preserved. >> Yes. Yes. >> In the Lorren transformation. Okay. That's a central tenant of relativity. >> Okay. That ensures that the speed of light is always the same for the both of us. >> And really what it's kind of crazy when you think about it. 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 >> because neither of us are right. >> Correct. >> Right. Your >> perspective is the same as my perspective. So the speed of light better be the same. And the universe is

1:04:15doing all this nonsense of contracting length and making time go slower just so that the diagonals remain diagonal. I'm not saying it's a simulation, but if you wanted to make it a simulation, >> yeah, this would be a rule. This would be it would make sense that that's and I'm being facitious here, but I think the point you're bringing up is important, which is that the constant of the speed of light uh requires the universe to do all this other stuff. >> Yeah. >> So that regardless of your speed or your position in the universe, it's it remain that constant remains the same. Exactly. >> Which requires it to do all this other stuff. all the other things. >> So, so it's almost like a it's like the

1:04:55the speed of light is a golden rule. >> It always must be followed. >> Yes. No matter where you are, no matter how fast you're going, because I don't know if you're moving or I'm moving, right? >> We'll figure out the other stuff to make it happen. >> Yeah. Exactly. And so that's a c central tenant of Einstein special relativity. Okay. And that's the Lorren transformation that 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 to get it very quickly so it's not it's not >> special relativity is like actually not

Nonlinear maps and calculus

1:05:32that bad when you like really just think about it it's like yeah >> kind of makes sense it's when we get to general when [laughter] it's like I don't whoa whoa whoa whoa slow down I I just got here [laughter] like so so that's that's special relativity right and the jacobian there is one Mhm. >> Meaning that the volume in spaceime is preserved no matter what reference frame you're in. >> Okay. >> And when you say reference frame, you mean both position and speed. >> Mhm. Uh and time. >> And time. So both position and time. >> Yeah. Yeah. Whatever volume in your position and time versus my position and time are going to be exactly the same. It's just that the each individual thing is going to be changed, but the product

1:06:13of the two are going to remain the same. Yep. Right. Because the Lorren can the Lorren factor is going to cancel out. Makes sense. >> Okay. And that happens in we showed a 1D case where we were only considering like >> you know forward and backwards where the boost is but this this applies in all three. In that case you get like the manowsky tensor and all and all sorts of stuff but that that that's effectively what's happening. So this is a coordinate transformation right spacetime gets distorted but the jacobian is one. So the volume of spaceime remains the same. Now let's talk about nonlinear transformations. Okay, this is an example of a nonlinear transformation. Here the x-axis goes to

1:06:53x + sin of y / 2 and the y-axis goes to y + sin of x over2. This is what I mean by now you can start mixing >> stuff right in the other one you were also mixing but here you're doing it in a nonlinear way. You're not just like adding uh some lorren factor times the constant. This is like sign which is an oscilly thing. And so you're seeing like stuff shift, right? Things are getting wiggly. >> Okay. In this case, what is the Jacobian going to do? Well, the Jacobian now is actually different >> in different spots. You can see in some cases the areas are getting squished. In others cases, the areas are remaining about the same. >> Okay. And if we go

1:07:33>> and see and zoom in on a central area element >> of this transformation, what happens? Well, let's look at let's look at those let's look at those that that sector that's right in the middle of the one one box. Okay. Now the transformation is nonlinear. So it's curved right in every sense. But the beauty of calculus is the following. If I take a small enough element stuff is mostly going to be a line. That's that's all calculus is >> like the 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 >> just zoom yeah no exactly just zoom in and the square becomes a parallelogram it's not becoming a weird uh curved shape >> it goes back to the example we just talked about previously but it's all about what is your uh perspective of the >> or or or context by which you're making your calculation. >> Yeah, exactly. 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. Okay, in this case it looks like one because like it's getting squished and this like the rhombus looks about the same. So in that locality the Jacobian is one but somewhere else it could be two,

1:08:54somewhere else it could be less than one and so on and so forth. That's the beauty of calculus is that you can zoom in and you can figure out a well- definfined Jacobian at every given point. >> We have a larger complex system but at sufficient zoom in the that complex system still comes back to the fundamental that we just talked about previously. >> Okay. Yeah. Exactly. And so now we've got this core concept which is the Jacobian matrix. The Jacobian matrix tells you how each of the directions sort of swish and become one another. And the determinant of that matrix tells you the ratio of the areas before and after or I should say it's the ratio of it's the ratio of areas after divided by before. Okay. So two means that the

1:09:35ratio went up. Less than one means the ratio went down. There's also negative Jacobians which means that the the axis got flipped. Okay. So it's like I mean >> it's just yeah it's it's like it got it got flipped, right? >> Okay. So if the determinant at a given point is non zero as we just saw then it signifies a linear approximation right and it doesn't collapse. Now here we've got an example that is at the center of the Jacobian conjecture. Okay we got to consider situations where the determinant is a nonzero constant. It's a constant everywhere. >> Meaning no matter where I am in my map

1:10:16>> the area is actually the same. Even though the upper part is getting more squished, you would think the actual area, the ratio of the areas is actually exactly the same. This is a particular map that um I came up with in Desmos. There's a lot of really cool Desmos um >> 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. So in this case the new xaxis is x + 1/4 y^2. >> Mhm. >> And the y- ais is just a y y y y y y y y y y y y y y y y y y y y y ais. So every y gets plotted to it it's it its own y but the x gets shifted by a parabola.

1:10:58That's why the these things are becoming a parabola but like the the horizontal lines are remaining horizontal lines. >> It's going from lanes on a highway to a track. The elliptical of a track >> and so and the lanes on the highway is is the X, right? So that's why that's getting shifted into a parabola. But the the horizontal like lines of latitude so to speak are remaining the same because the y just gets plotted to the new y. Nothing happens. Okay. In this case, if you were to plot the if you were to plot that area, that little green area, >> that little green area becomes a parallelogram, but the area is preserved. >> The jacobian here is one. If I were to take that area and put it up top, >> the parallelogram might get more

1:11:39squished, right? It might have a different shape but the area is still going to be one. 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. Now given this one can ask suppose I was

Invertibility and the conjecture

1:12:05suppose I was given the end product. Okay. If I was given the end product of the of the last like the the the par the the the points in that um after the transformation. >> Yes. >> Suppose I were given the points after the transformation. Can I go backwards? >> Right. >> Meaning is the map invertible? >> Yes. >> Okay. >> In this case it is. You you can actually solve for it. If you give me the new x and y coordinates in this case I'm going to call it u and v because the u is going to be my new x ais. 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

1:12:45mathematics and plug in it's actually u - 1/4 v^2. So the the plus 1/4 becomes a minus. Okay. And then I can recover the old map. >> Is it is as a concept to help me gro this is is it similar to when you talk about encryption and decryption when you have the key in the middle. >> Very good. Um you're you know you're able to go back even though it's jumbled. Yeah. Because you have the key that is allows you to Okay. >> Exactly. Exactly. And >> in that in that particular transformation right because the Jacobian was not zero >> I could go backwards. If the Jacobian is zero what does that mean? That means an area collapses into a point or like a line something with zero area. Well what that means is now we've got a bunch of

1:13:25points where it's degenerate. Like let's say let's say the the extreme case where like a giant square collapsed into a single point. Well now if you give me that single point I don't know where in the square it came from. >> Right. If you give me a line for example the entire square gets collapsed into a line. Well that line is only 1D information. I don't know where in the 2D the thing came from. >> You you you need both. You need you need both. >> Yeah. >> You can't have only one and then zero on the other side. >> Exactly. Right. And that happens when the Jacobian is zero. So when the Jacobian is zero I can't go backwards. trivial invert. That makes sense, right? And that's and that's like a well-known theorem in calculus, right? And where locally if I have the Jacobian that's

1:14:06not zero, right? >> Then I can go backwards. >> That makes sense. >> Okay, now we can finally understand the conjecture. >> Okay, >> the Jacobian conjecture is the following. It says that if I have a map where the Jacobian is a constant everywhere, can I always go backwards? >> Right. >> Everywhere. >> Okay. [clears throat] If the constant is if if the Jacobian is constant everywhere, can I always create an invertible function? You give me a function where the Jacobian is constant everywhere. Can I always go backwards? Okay, the formal the formal statement is the following. Okay, let K have a characteristic of zero. K is a ring. A ring really just

1:14:46means like numbers. Okay, mathematicians have other types of rings, but for us, let's just say they're numbers. Okay, >> um they have a characteristic of zero. A characteristic of zero, a ring really means something that has addition and multiplication. Okay. Um, a characteristic of zero means that you know a ring has a one. It's some kind of element that's a one. For example, in the matrices, the one element is the identity. For numbers, it's just the number one. And a characteristic of zero means no matter how many times I add one to itself, I'm never going to get back zero. >> Okay, that just means to me normal numbers. Okay, mathematicians will come up with rings that have characteristics that are non zero where like I add the

1:15:28number one five times and I get zero for some reason. I don't know why you would want to study such objects but they find a way. Um in any case we're talking about normal things here. Okay. >> Um let K have a characteristic of zero. So this is just real numbers, complex numbers, things like that. 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. >> Mhm. >> That plots my n-dimensional space. Yep. >> To back to the n dimensional space. >> Yeah. Yeah. Yeah. Okay. >> Yeah. Yeah. >> And this thing is regular meaning that it's components are polomials.

1:16:08>> Okay. >> Okay. So if I have a polomial function that's going in polinomial meaning things like x cub plus x^2 y like powers of stuff y >> then the inverse map is also a polomial >> okay >> that is the jacobian conjecture. >> Okay so this is this is actually interesting because now now I can understand the formulation here of what we like what this open problem yeah >> is trying to suggest. Yes, it's trying to suggest that any time >> 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

1:16:49area it scales area but it scales it in the same way everywhere >> everywhere >> okay >> if this function scales it in the same way everywhere >> then I should always be able to go backwards >> I should always be able to decrypt my encryption >> yes okay as long as it preserves serves area in the same scale everywhere. Okay, that's the conjecture. It was formulated way back about 90 years ago and people have been trying to prove it ever since. >> Okay, that is the question. The problem was number 16 on Steven Smallley's 1998 list of important mathematical challenges and it has very famously

1:17:29defeated top top mathematicians. We're which we're going to get into later. >> Top 20, baby. >> Yeah, >> that's top 20. >> Top 20. top 20 open problems. >> Yeah, that's quite nice. >> Now, July 19th, 2026 in the middle of the World Cup final.

“The Jacobian conjecture is false”

1:17:46>> Okay. Lavant alpug, he is um part of the Harvard Society of Fellows and also one of the researchers at Anthropic. He drops this tweet. >> I saw it within 30 minutes of it going up. >> Yeah, because it it was insane. >> It it literally 30 minutes. >> Yeah. First of all, it's all lowercase. It's he's it looks like he's trolling. >> Yeah. Yeah. Yeah. Yeah. >> You know, hello there. The Jacobian conjecture is false. Thanks. And Tnx, >> this is very >> Thanks to my close friend Akil 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. And then he just and he just

1:18:28shows the counter example. He just writes it out. There's so many things about I'm not going to bring it up now, but >> and it could fit in a tweet. You know, there's like the 100 character whatever limit thingy. He just fit the entire proof in a tweet. That's all you need. >> Arguably, if NFTTS were still a thing, this NFT would be very valuable, >> bro. That is I would buy this NFT. >> This would be so valuable. >> Yeah, I would we we would get this NFT for the NFT. >> Exactly. because it there you know for those who may not understand Twitter culture and the it you know it is like it's a moment in time you had to be there and to see because it's real time and so many people especially in the AI research community are there and

1:19:10apparently also were not watching the World Cup final at the time >> the the immediate >> um conversation that arose literally within an hour >> of this being out there was as a bystand 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 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, you're talking

1:19:50about coordinate transformations, you're talking Okay, like it kind of makes sense to me." Okay. And then to to see this come out in a troll language, I think it's just so so funny. Okay. So, let let's talk about that exact solution that it has. Okay. And in the in the next in in the next overlay, we we have it type formatted because

How the counterexample works

1:20:09within hours it was on Wikipedia, >> which is >> right. You go into the Jacobian conjecture Wikipedia article, it says counter example and it shows that it's not true. >> You know that meme where of the we got him with with uh what's his name and the balloons and confetti? >> Yeah. >> They did it immediately. >> Yeah. Yeah. Immediately. Right. And this is a counterex example meaning he has shown a map where the Jacobian determinant is a constant. >> So this functional mapping this map between this is in 3D. So we're taking 3D space and plotting it to another 3D space. So volume is getting morphed, but the volume getting morphed is always at a constant of -2, meaning one of the

1:20:51axes is flipped. So it's like getting flipped, but and and um if you've got a cube of the size one, it becomes a cube of size two, right? >> Okay, in the new one, but like one of the axes is a little bit >> is flipped. 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. >> Uhhuh. >> Meaning if you gave me that resulting point, I couldn't tell you which of the two it came from. >> Yes. Yes. The point being it's not you cannot you don't have the ability to go from the result back to the original map

1:21:33because two points uh point to the same >> it could be one of the it could be one of two things >> so it's [clears throat] inherently not invertible right it turns out there's a third point as well okay and this is actually what's crazy about this is you can do this yourself >> right >> and I actually did it like before we before we like as I was doing research for this for this episode I went ahead and did it by hand and you I could show you my work. It's in the next one. >> Yes. >> So this is I did it myself, right? I took the I took the the the three components of the function. So the new x

Verifying the result by hand

1:22:09coordinate is some seven degree polomial that has that mixes x y and z. The new y-coordinate is another I think seven degree polomial and then the the the z coordinate is not a seven degree polinomial. It's a little bit simpler but you you I've got this map where x x y and z goes to a new x new y new z. >> I calculated the derivatives which are on the left. So that becomes a matrix which you see on the top right. And then I did a little bit of cheating because I decided I didn't want to multiply all those polomials. Okay. I I I wanted to see if I could still do it if I still got it. And I went got halfway and I'm like [laughter] I still got it. So then so then I went on wolf frame alpha and I

1:22:49just multiplied all of those polomials to calculate the determinant and the determinant is -2 >> and then I went ahead and plugged in those two numbers. >> So fascinating >> and they went to the same number 1/4 0 0 >> there you go I did it on a single piece of paper plus okay a little bit of wolf frame alpha right it would have involved another piece of paper for me to do it but I I was already you can see the crossing out. I was like starting to make mistakes. I was like, let's >> 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 >> uh than to prove because especially given the structure of this open problem

1:23:32and the nature of the disproof if that's the right naming. >> The counter example. >> The count sorry, the counter example. Um you it does not it anyone can do it and see that it is true. Going back to our point about mathematics. >> Yeah. >> And the objective nature of it. Yeah. >> Structure. >> Yeah. Exactly. I I did it. Anyone can do it. Right. This is something that algorithmically there's no AI here. Right. >> Right. It's just it's an algorithmic process of taking derivatives taking the determinant of the Jacobian. There you go. Boom. >> The there's a constant determinant and two points point to the same thing. So I can't go backwards. >> I can't go backwards. And it's not invertible. Which means the chacobian

1:24:12conjecture fundamentally not a real thing. >> It's not a real thing. Right? >> It's not a real thing. And it's a it's a counter example in degree um well it's in degree seven but it's in three dimensions right which means for all higher dimensions it's also not true because I can just take like uh consider the trivial map where I take a function of four dimensions >> um let's say xyz and t is our fourth dimension in allah relativity um x y and z do what that function was doing and t just points to t right so the time doesn't change that is going to have the the same Jacobian determinant and the

1:24:54things are going to point to the same point in 3D space and the time doesn't change right I can do that for n equals 5 n equals 6 any dimension greater than three now so this is disproved all dimensions greater than three >> right so the it's only true in two >> yeah now it's like now it's an open question >> oh right because we don't know

What remains open

1:25:13>> right so now the jacobian conjecture is Yes. Yes. Only in two dimensions. >> Only in two dimensions. Now it's still an open problem. >> Still an open problem. Not not yet solved. Okay. >> But the but that's interesting. >> But like for for all the infinity of dimensions above >> above which is like really a much more meaningful >> Yeah. Yeah. It's crazy. Um, no this and it I think what's so interesting is that again and I was we might get to this but one of the things that people's 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 so there has to be some trick somewhere >> right so [clears throat] oh was you just ask fable the latest frontier model from

1:25:56anthropic to just brute force it and if you look at the uh amount of time required to brute force. It's just it's like to the power of >> to the power of a crazy. And that's

Did the AI brute-force it?

1:26:09actually my next point. Okay. >> We can actually we can actually ask did Fable just brute force. >> Right. Right. Which I >> didn't just do like supercomputer nonsense >> cuz they have a lot of computers. >> Yeah. And and there's history behind supercomputer nonsense. Okay. So let's talk about like this is a counter example. Okay. And historically computation has been very good at producing counter examples. Okay. way back in the 1960s actually. So let's talk about um Oilers's conjecture. Oilers's conjecture is the following that um you remember um we were talking about like from theorem and the Pythagorean theorem like the Pythagorean theorem is that like the the sum of two squares can equal another square. So like two >> two thingies to the power of two equals another thing to the power of two.

1:26:51Trivial example 32 + 42 = 52 because 9 + 16 = 25. Um Plato way back in the day had found another kind of Pythagorean looking thing where he showed that 3 cub + 4 cub + 5 cubed is actually 6 cubed >> which is kind of cool. >> It's called uh it's called Plato's number. >> Mhm. >> Um so all right the Oiler 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 >> Yes. >> to get to another cube. You're right. >> Okay. So, Oiler is like, what if I need n >> to the nth power to get to another nth

1:27:32power? I need at least n things to the nth power to get to another nth power. So, if I want to if I want something like a 4th plus b 4th plus c 4th, I would need another d 4th in order to get to an e to the 4th. >> He was trying to generalize number. >> Exactly. He's trying to general cuz pyagorean theorem already happened, right? 32 + 42= 52. Plato's number is saying okay I need three cubes to get to another cube so maybe I need four to the fourth power to get to a fourth I need five to the fifth power right and that's that's Oiler's conjecture remains untrue for a very long time lo and behold um Lander and Parkin in a very very famous mathematical paper one of the shortest

1:28:14math papers of all [laughter] time right they're like uh direct search with the CDC 6000 66000 which is one of the first successful supercomputers they just brute forced it. >> Y >> and they found a counter example. >> Um 27 to the 5ifth power plus 84 to the 5th power plus 110 to the 5ifth power plus 133 to the 5ifth power equals 144 to the fifth power. So I only needed four fifth power thingies to get to a fifth power. >> We don't like that. >> Yeah. And they were just like it's the smallest instance >> in which five fifth powers. There you go. >> Sorry. >> Sorry Oiler. You're wrong. crazy to say the oiler is [laughter] wrong,

1:28:56>> but but this this is a perfect point. This is why people immediately go to again, you know, the >> oh, they just brute forced it. >> Yeah. Right. Right. And this in this case, the brute force it could work because the sample space that they were they they were just like, okay, let's just go for fifth powers. Let's go all the way up to like 200 for all the numbers and let's just try it out. >> And it happened to be there. And the search space is small enough where with a big enough supercomput I can do it. >> And this was some some time ago. >> Yeah, this was in the '60s. And now we found more and more exceptions of Oilers's conjecture. It's rare. It's increasingly rare. And there's a lot of mathematics about how rare it is. And that's its own, you know, mathematical field. But let's consider could Fable

1:29:39have just like brute forced it, >> right? >> I don't think so. I think this is fundamentally different, right? The search space is actually I think too large because here what we'd have to do is test every single function >> in of polomials right there's x y and z. So let's just do like okay it found like the seven degree polinomial right there was an x to the seven there um >> if you just think about seven degree polomials and seven degree maps so you can have seven 7° in x 7° in y 7° in z right and each of those can have different numbers >> right so that's first of all that's 343

1:30:19different polomial terms each of those polomial terms can then make up 1300 cubic equations And then the coefficients of that equation, right? Is it 1 x^2 y? Is it 2x^2 y? Is it 1 x cubed y? Each of those coefficients to create individual functional maps. Even if you were to even if you were to test within a tiny integer range of let's say -10 to 10 >> and try all of the different coefficients, that would be something like 10 the 475 candidates. >> Right. >> Right. >> Right. even like a million cubit quantum computer if we were to create a quantum algorithm that could somehow do this which that's I'm just I'm just saying we don't even have one >> for another day.

1:31:00>> Yeah, that's for another day. We don't even have one. But like you you can't you can't do that. This is not a brute force thing, right? Like Fable had to understand something about the structure of polomial maps. understand something about um the properties of polomial maps

Why brute force seems impossible

1:31:18where the Jacobian is constant >> and then look for places where things would overlap because if the Jacobian is constant that means that any given like infinite decimal area is is being put into the same thing. So it's not just trivially that like >> the thing is going to the same spot. It's like there's like overlap happening where like different parts of the of the space are getting mapped to the same thing in some weird way but somehow the Jacobian determinant is still a constant. >> I 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. Um there was a discussion it's going to escape me who it was with but there there were sort of

1:31:58three diff there were three people in the field who were coming from three different perspectives about AI's potential in particularly this idea of novelty >> um or or scientific discovery as a general not well- definfined term >> and one of the arguments was that even if a system like Fable has scooped up every math paper ever. >> Yeah. >> Right. It also has the benefit of having every physics paper, every biology paper and because of the way structurally

1:32:40um 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, they may 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. >> Yeah. >> And simply by just having that same level of expertise >> in a cross functional in a cross field way >> cuz a lot of science is making these connections that are non-obvious. The ability for these systems to make

1:33:21connections that are nonobvious is just exponential. >> Yeah. And we may discover that there are a lot of solutions just simply in taking known information and finding uh things you can carry over from physics that have an application in math that we've just not explored. Yeah, that's true. I mean like in string theory for example, string theory bleeds a lot into fundamental mathematics. Right now, one of the things that is really unclear about this discovery is like what the prompt that was used, right? Um if the chain of thought has

AI, novelty, and human blind spots

1:33:58some insight into how it got to this particular functional map, >> right? >> Right. and whether it used all of those all of that knowledge. Like that was my that was my suspicion when there was that one discovery that we talked about with the the gluon tree and like how there's like a seven gluon fineman diagram that it could calculate that's non zero and that's something that we have to worry about. I was like, okay, maybe there's some mathematics that's happening elsewhere that is like bleeding into this physics, right? It's however very rare to go up the chain of logical inquiry. >> And so, and to >> and that's why I'm like freaked out kind of. >> So, and this I think this is an important point because I wanted to lay that out there.

1:34:39>> Yeah. to this exact note which is like even if that's true what I just walked through >> what we're talking about with this solution to the Jacobian conjecture seems to be outside of that potentially outside of that sandbox. >> Yeah. Yeah. which seems it seems >> I don't know I don't know enough but see >> and I'd love to talk to Levant >> even the potential >> that it's not obviously what we just talked about which has been the case for other things where it's like oh this is they made the connection here and you can draw the through line >> even just the potential that that's true >> is um you know and people part of it's you know the leavant who discovered it is obviously one of the world's greatest mathematicians has been so for a long

1:35:20time >> but he's not >> oh he's not >> no >> I mean look no offense Leavant. >> Look, I I don't want that heat. I >> look Look, I love Professor Leavant. He's got a PhD from Princeton, so he's clearly like in the top 1% of mathematicians, I'd say, right? So on and so forth. But he he's a he's not like an active researcher, I would say. Right. Um and >> but he he knows clearly. I mean, he's he's a researcher and anthropic and he's very good at understanding how to use and leverage anthropic, right? And clearly he knows a lot about mathematics, right? But this isn't like someone like Peter Sarnac that's like a number theorist using this stuff, right? And that's what I really want to know what the what the um what the prompt was,

1:36:02>> which which almost makes it again even more. >> Yeah. >> Um >> like he's clearly gifted, but he's not at at that top level that's getting Fields medals. [sighs] >> Fascinating. I I just 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. Yeah. >> Which is part of part of the value ad there 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 show me. >> Yeah. Well, yeah. And there's several examples. Right. Now, before we get into the other examples, one thing I wanted

1:36:43to talk about was like how humans have been approaching this problem, right? Right. Um, there's been this degree paradox where human researchers have spent decades brute forcing variables. Like we've been doing a lot of the brute forcing up to 16 variables, degree 100 plus. 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 um counter example, right? But it turns out there's there's proofs that show that the that the smallest 2D case would have to have a polomial of degree like a 100 like x to the 100 plus x to the 99 da da da da

1:37:26da right and so we've been focusing on this like highly complicated thing but here claude fable 5 proves that a low degree degree 7 it's just in a higher dimension >> dimension three >> right >> that counter example exists it's kind of like a human blind spot that We didn't really consider there's not a lot of papers about the Jacobian conjecture in degree 3. Sorry, in dimension three. >> Yes. Yeah. Which which goes back to then what is the source material for the inside >> and and historically there's there's kind of an irony. If you look at um Yeetang, he's a very famous number theorist. Um he's one of the first guys to bring down the prime gaps to

1:38:07something like I think it was like 79 million. prime gaps meaning you know the twin prime conjecture where like there's an infinite number of primes that are two apart three and five 11 and 13 101 103 so we still don't know if there's an infinite number of those but he's the first guy to bring down and he said there are an infinite number of primes that are like less than 70 million and everyone's like ah right but his story is insane because like during his PhD um at Purdue University he was under the advice of um Tuang Singh Mo and his work

The human cost of open problems

1:38:41was on this very conjecture. It led to years of hardship. He couldn't get anywhere. Um it's a really kind of a sad story because his dissertation relied on this auxiliary correlary about the conjecture that was proved by his adviser later discovered to be false under peer review. And so his entire PhD turns out to not be true because that's how math works, right? And then um kind of crazy he like he couldn't get a ad he his adviser was kind of pissed off about that. >> He couldn't get a recommendation letter and he was he was a subway worker minimum wage and he lived out of his car. >> Jesus Christ >> for a while before he proved the twin

1:39:22the twin prime corollary and then became like a very famous mathematician. But like to you know and now we've got something like AI that maybe can save mathematicians time. Mhm. >> But at the same time, it's like how how much of this is AI going to do to the point where are mathematicians going to be left with the job or are they going to just be explaining stuff that AI is doing? That brings me to my next point >> which is about the the in 2020. This is part of a trend in 2026 the open AI model disproved the erdos planer unit distance conjecture that was that was conjectured by Erdos Paul Erdos in 1946.

OpenAI and the Erdős conjecture

1:40:01This is the proof for it. It has to do with like the it asks like the maximum number of pairs of points that can be exactly one unit apart >> in 2D. So like trivially you could imagine a grid of points, right? A grid of points. All of these points are one distance apart. And you would think that that's like the best way to pack all the points, right? Erdog showed that there's like an upper bound on how many points I can pack with this amount of distance. um and the open AI model disproved that bound and showed that there's a better bound. And then later on, mathematicians took that proof and made a short digested humanverified version of that

1:40:44counter example. Again, it's a counter example, but here the the authors are a who's who of mathematicians. We've got Timothy Gowowers from Trinity College at the University of Cambridge >> in the UK. Um he's a fields medalist. We've also got Jacob Zimmerman, which it's been leaked, is the Fields Medalist this year. If you go on Poly Market, it's all collapsed to 99% [laughter] for people. It's like, okay. So, somehow it got leaked and now no one's like betting anymore. Jacob Zimmerman, actually, I think he was my um my he was my TA in freshman year. >> Oh, we love that. >> Yeah, cuz he was a PhD student of Peter Sarn teaching. >> Come on the show. We'd love to talk to

1:41:24you about it. >> Yeah. Yeah. We'd love to talk to you about how you how you gave me a never mind. Let's not talk [laughter] about how you graded my midterm, but um you know, so this is a who's who and what they're doing is 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 funny thing about this, right? So um Open AI solved that Aeros conjecture. Um, Leavant, 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 Aeros unit distance

1:42:05problem. Um, aka Ardos problem. Um, the answer is yeah, [laughter] >> I love >> I think he's just trolling. It's just all all lowercase [laughter] >> gen Z capitalization. Then his colleague is like, you know, huge huge credit to the Open AI team and but he's like retweeting, but he they're are colleagues and he's like trying to make it a little bit more >> Yeah. Yeah. Good. [laughter] Corpo speaking. >> Yeah. Yeah. Yeah. But but the it's just so funny. I think the guy's just trolling. This is so there's so much here. Sorry. Continue though. >> Yeah. Um and the final thing I want to end with is this is clearly now becoming an existential crisis for mathematics,

1:42:47right? If AI is going to prove everything, what are pure mathematicians going to do? And it makes me kind of sad if I'm not going to lie. It makes me happy that we're we're seeing such obvious counter examples for things and like stuff that I can understand. And you know, who knows? We'll maybe find a proof for the reman zeta or maybe we'll find a counter example for the reman zeta hypothesis where um it's going to just come up with a a zero for the remon function that's like I don't know two billion on the 1/ half real line. Who knows, right? Probably not because every mathematician that I know says that the remon hypothesis is true. >> But who knows, right? At this point, it

1:43:28would be kind of cool to find it because that I could also check >> like not by hand maybe cuz I'd have to do it a bunch of times, but I could probably check on Wolf Ram, right? So,

The Leiden Declaration

1:43:39>> it's it's um it's concerning. >> It's concerning to say the least for pure mathematics. And in June 2026, there was a statement from the Lorent Center regarding AI generated proof. It was called the lightened declaration on artificial intelligence and mathematics. A bunch of mathematics came to mathematicians a bunch of mathematicians came together and they laid out their key concerns about this new shift in how we are starting to do mathematics as a discipline. There's a threat to open science, right? Because the we don't we don't know what the model weights are. >> Yes. >> Because they're closed source. >> Yeah, they're closed source. So I don't

1:44:19know what's happening inside. It's not really open. I'd have to pay to play. Mhm. >> Um it's unsustainable journal review back reviewer backlogs cuz you can just now like start making all sorts of proofs submit it to the analys of mathematics and then what like >> the humans are going to and then now what we get AI reviewers seems I don't know we don't like that >> I don't know right and then and then there's like 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 theorym what are we going to do? And they made some recommendations. One was mandatory AI disclosure. >> We love disclosure.

1:45:00>> We love disclosure >> here on this podcast. Um enhanced scrutiny and there's proper attribution for human research. Where did these AI models get the inspiration? Right. We'd like to know inner workings. >> Bingo. >> It's going to be tough because a lot of these LLMs are black boxes at the end of the day. Claude uh Anthropic is doing a great job trying to elucidate what is happening in the inner layers of their models, but at the end of the day, it is it is kind of a black box, right? Um they're also trying to go for independent funding from these industries to support mathematicians because mathematical research funding is down. Well,

1:45:41mathematics research is also down. And I don't think it should be one of those things where oh AI can do it. We shouldn't have humans doing it. I mean I think mathematics is one of the most beautiful human disciplines that we can ever take part in. Right? It's something where we can say a statement is true and it will live on forever as long as human civilization is alive. If we can engrave it in stone, right? And it outlasts human civilizations. and aliens came down um Allah Halo and and like saw the forerunners and saw [laughter] you know for last theorem embedded in rock they would be like I kind of understand that right if they saw like squares and Pythagorean theorem like

1:46:21>> that would be so cool >> because they have their own versions of that. >> Yeah. Yeah. >> Um and so like I don't want this to be a death sentence for mathematics. It's really it I really hope that's not what's happening. >> I I do think another kind of aspect to this that you bring up is mathematics as an understanding if we want to retain human oversight over AI systems is fundamentally necessary. >> Yeah. >> Because the the way these systems are built and work requires this as a discipline that is understood very well. >> Yes. Logic, correctness, truth. And you cannot we cannot say oh we'll have AGI

1:47:04and artificial general and artificial super intelligence. And again the argument is well if it's super intelligent humans can't keep up. We need to maintain this cohort of experts for as long as possible to manage whatever this transition is going to be because I think we're now I think deh sir Demisabas has said we are on the uh the shores. Uh he has a great phrasiology for this. Um um oh we are on the the shores of the singularity or we are on basically the the sort of the dawn of the singularity here. And I know Ray Kurtzwhile is known as kind of the modern proponent for the this concept.

1:47:44No one has ever been able to describe in words

Are we nearing a singularity?

1:47:48what it would feel like to be right there >> when the flip was about to switch. And to me, even as a layman, you can start to feel the ground moving. >> Yeah. >> Underneath our feet. It's not to say the killer robot outcome and Skynet, 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 mathematically scientifically engineering wise, socially, politically,

1:48:32economically, religiously, all of the ease. >> Uh, you know, it's like it's someone made the joke. It's like when people say that they're uh why do I care about politics? It doesn't affect me. Oh, that must mean you're not, you know, a public employee, a veteran, a young person, an old person, a disabled. >> You come from wealth. like [laughter] you know but it is I I know I know that everyone is tired of hearing about AI. I I know that I get it. I totally understand it. Um however um if we could have stopped the bankers from doing '08 before it happened. Would people say no to that? >> Yeah. >> And it feels like a similar situation where there's a runaway train here

1:49:14>> and we need to do something. >> Yeah. >> Yeah. >> I don't know what Yeah. And maybe it's not to stop the progress because progress is always good, >> right? >> But we got to we got to talk about it >> and we have to engage in it and we have to have these discussions and >> and we got to have some kind of policy, right? It can't just be, oh, it's like too late, >> right? >> Everyone's unemployed >> cuz it's not >> and now let's try to solve it. >> Solve it because right now it's not too late but very quickly it might be too late for mathematicians for for some of these industries a lot of as oh it's the white collar jobs. Who cares? I I'm blue collar. It's not going to impact me. Well, humanoid robotics are coming. So, it is this is, you know, I I know I'm

1:49:54making this larger philosophical and sociological point. >> Yeah. But the detail of what we walked through today is the canary in the coal mine. 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 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. >> Yeah. Yeah, this was so good, dude. Like, >> yeah, >> this was really, really good. I can't believe I can go around be like, "Oh, yeah, the Jacobian." Yeah, it equals

1:50:35one. >> Yeah, it just [laughter] It's a ratio of uh areas are multi. >> You should say volumes. Yeah, volumes because because to mathematicians, an area is really a 2D volume. [laughter] >> Oh, those mathematicians, they crack me

Final thoughts

1:50:48up. >> We love them. We This was again, we're having a problem staying under two hours. Uh, we're approaching two hours again this episode. One, it's because we love y'all so much and we just we can't stop because it's all so good. We touched on astrobiology. We talked on science funding and we spent a long time 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, Alph Go? Uh the original I can't remember the name. The points in the grid one that OpenAI did >> not that long ago.

1:51:29>> No, that was this year. >> That was >> Yeah, it's all happening this year. >> It's happening right now. We are so blessed to have you all listening. If you are still listening at this point in the pod, uh we need a a quote, a comment for our our deep deep audience. >> Make fun of mathematicians. I love doing it as a physicist. just just say something that's funny about mathematicians. >> Uh I am your host Lester Nari joined as always by my co-host our Princeton undergraduate physics UCLA PhD physics occasional mathematician Christian Chowdery. Thank you all for joining us again. We are coming up on our oneyear anniversary.

1:52:11It's incredible that it's only been a year. We have so many fun things planned for the future. We will see you all next week. >> [music]