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

Hosted by Lester Nare and Krishna Choudhary, this episode moves from astrobiology to science policy to the rapidly changing frontier of artificial intelligence and mathematics. First, researchers analyzing pristine samples returned from asteroid Ryugu report all five canonical nucleobases used by DNA and RNA. We explain what that does—and does not—mean for the origin of life, how JAXA’s Hayabusa2 mission collected uncontaminated asteroid material, and why comparisons with NASA’s Bennu samples strengthen the case that prebiotic chemistry may be widespread across the Solar System. Next, we examine the fight over who controls federal research funding. A proposed overhaul of the rules governing federal grants would give political appointees greater influence over awards, reduce the controlling role of expert peer review, and expand the government’s power to stop grants that no longer align with an administration’s priorities. We break down the roles of Congress, OMB, federal agencies, universities, and the courts—and why this dispute could reshape the American research ecosystem. Finally, we go deep on an AI-assisted counterexample to the Jacobian conjecture, a major open problem in mathematics. Krishna explains coordinate transformations, Jacobian determinants, invertibility, special relativity, and why this result appears fundamentally different from simple brute force. We close with the growing debate over AI-generated mathematics, human verification, open science, attribution, and the future role of mathematicians. Summary All five canonical nucleobases found in pristine asteroid Ryugu samples Hayabusa2, Bennu, and the possibility of widespread prebiotic chemistry The fight over political control of federal research grants Congress, OMB, peer review, and the American science-funding system The Jacobian conjecture and an AI-assisted counterexample Special relativity, coordinate transformations, and invertibility AI-generated mathematics, open science, attribution, and verification Support the showDonate: FFPod.com/donateFollow: @FFPod on X / Instagram / TikTok / Facebook Show Notes A complete set of canonical nucleobases in asteroid Ryugu OMB proposed federal-grant rule Association of American Universities response Levent Alpöge’s Jacobian counterexample announcement Leiden Declaration on Artificial Intelligence and Mathematics Human-verified remarks on the OpenAI-generated Erdős result

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

• 0:00Lester Nare?

  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. 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, this is, you're not getting any more money. He drops this tweet. Hello there. The Jacobian conjecture is false. Thanks. And a T H A N X. Hello, internet. This is your captain speaking, Lester Nare, and I'm joined, as always, by my co-host and our resident PhD, Krishna Choudhary.

• 0:42Lester Nare?

  This week, we have a rundown episode with three very intriguing stories. We're starting off with astrobiology, where new evidence suggests that asteroids possibly could have seeded 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.

• 1:24Lester Nare?

  Who gets to decide who gets the money and how? 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 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 in the near future. And along the way, I think we might

• 2:10Lester Nare?

  Learn a little lesson in 3D calculus. I'm gonna need that one. For the AI story, this is gonna be a little bit more of a deep dive in this rundown because it's pretty important. But as always, we're gonna talk about the science from the ground up today because this is from first principles.

• 2:43Lester Nare?

  So today, we are gonna start off with my favorite subject, aliens. As many of you longtime listeners know, 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.

• 3:06Krishna Choudhary?

  Yeah, this was super interesting to me because the Japanese space agency, JAXA, put out this paper showing that all five of the nucleotides that we know and love here on Earth with all of life, there's four in DNA, A, T, G, and C. And then there's one that RNA uses. It uses the U, uracil, instead of thiamine 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 it's hugely important for origin of life debate, things like that.

• 3:49Krishna Choudhary?

  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 nonliving matter. That's the idea, right? Abiogenesis means from nonliving to living, a bio.

• 4:12Krishna Choudhary?

  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 was liquid water, and then immediately, we have fossils of tiny bacteria-looking thingies. And the question is, how come it happened 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.

• 4:53Krishna Choudhary?

  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 originated somewhere else, but it originates somewhere else, and then an asteroid gets kicked out of that planet, or whatever thing has the life.

• 5:17Krishna Choudhary?

  It roams around space for millions, maybe billions of years, and then it finds its way on Earth.

• 5:22Lester Nare?

  As an example that we've seen recently, that we've covered on the show, is, you know, the interstellar object of 3I Atlas, you know, that came into our neighborhood. That was one of these travelers that was roaming around. Obviously, it did not make impact with Earth, but we've also covered the crater in Arizona?

• 5:40Krishna Choudhary?

  Yes, we've covered the crater in Arizona. We've covered how, like, actual Mars, a Mars meteorite could impact 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 reentry 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'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?

• 6:24Krishna Choudhary?

  But those samples always carried a major asterisk, which is mainly that you can have terrestrial contamination. The most famous of this is in the 1990s, when Bill Clinton, um, he... Got in front of the press.

• 6:38Lester Nare?

  I think it was in the rose garden, maybe?

• 6:39Krishna Choudhary?

  Yeah, he was in the rose garden and he said, I did not have.

• 6:44Krishna Choudhary?

  I always do this because we've covered this so many times. It's like a running joke now. But he was actually announcing a science paper by scientists at JPL that showed that a Martian meteorite that they had found in Antarctica might have had some bacteria 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 because it looks like a bacteria looking thing, doesn't mean that it's actually bacteria from Mars, right? 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.

• 7:29Krishna Choudhary?

  So here, the hero of the story is a satellite slash bombing system by the Japanese space agency called Hayabusa 2. And what it did was go to an asteroid called Ryugu. R Y U G U. 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.

• 8:09Lester Nare?

  This is the alternative plot to Armageddon. Instead of going on to it and drilling, they did it from the atmosphere.

• 8:17Krishna Choudhary?

  Yeah, yeah. And then they did it remotely. Now we can do that, right?

• 8:20Lester Nare?

  Right.

• 8:22Krishna Choudhary?

  So that's the premise of the whole thing. The next video actually shows, this is a video from the JAXA probe and it's showing you're lower down into the part of the asteroid that's been pulverized. And you grab it, you grab that part and they come out. It's pretty cool that we can successfully like chase the spinning top asteroid, blast a crater and then bring back pristine dirt from an asteroid back home.

• 8:49Lester Nare?

  Who was the one who was like, guys, I have an idea? Let's take a bomb.

• 8:53Krishna Choudhary?

  Yeah.

• 8:55Lester Nare?

  Let's explode it just above the surface so that we can then go down.

• 8:59Krishna Choudhary?

  Yeah, and then there's a bunch of dirt instead of rocks.

• 9:02Lester Nare?

  Right, because the idea is the surface is solid, it's a little bit, you can't.

• 9:05Krishna Choudhary?

  Yeah, and the rocks are kind of big.

• 9:06Lester Nare?

  Big, and you don't want to have to have a drilling mechanism on the whatever probe you have.

• 9:10Krishna Choudhary?

  Yeah.

• 9:11Lester Nare?

  It's clever. I'm very curious how that conversation went in the proposal meeting.

• 9:16Krishna Choudhary?

  Yeah, and the other thing I'm actually curious about is if that little bomb changed the orbit of Raigui. Right, because 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.

• 9:43Lester Nare?

  Yes.

• 9:44Krishna Choudhary?

  So, JAXA and the Hayabusa team, they came out with a paper in Nature, in Nature Astro, "A Complete Set of Canonical Nucleobases in the Carbonaceous Asteroid", which means they have found A, G, C, T, and they found U, which is the alphabet of life in some sense, right?

• 10:05Lester Nare?

  It's all the ingredients we need to rat a two air self into the way of life.

• 10:09Krishna Choudhary?

  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.

• 10:35Lester Nare?

  This is, it's a big, it's a big deal for a number of reasons, not only the execution of it, but to your point earlier, we've removed the contamination hypothesis from the list of options.

• 10:45Krishna Choudhary?

  Yeah, 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 nucleotides that we did.

• 11:00Lester Nare?

  So you're saying there's a chance?

• 11:01Krishna Choudhary?

  Yeah. It's kind of crazy, right? And the big picture is, so, this isn't actually the first time, okay? 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.

• 11:30Lester Nare?

  Which is incredible.

• 11:31Krishna Choudhary?

  Which is cool. Yeah.

• 11:32Lester Nare?

  That's incredible.

• 11:32Krishna Choudhary?

  It's not a one off thing. It's not like a special asteroid.

• 11:34Lester Nare?

  We got lucky one time.

• 11:36Krishna Choudhary?

  Yeah, yeah.

• 11:36Lester Nare?

  That's not what we're talking about.

• 11:38Krishna Choudhary?

  Twice and we find nucleotide bases.

• 11:40Lester Nare?

  Not that this, and there's another variable, in a short timeframe as well. It's not this is every hundred years, every 300 years, every 1,000 years. Something happens to come around that has it.

• 11:50Krishna Choudhary?

  Yeah.

• 11:51Lester Nare?

  This was in a.

• 11:52Krishna Choudhary?

  Yeah, this is just local to our solar system, right?

• 11:55Lester Nare?

  Which is also crazy.

• 11:56Krishna Choudhary?

  Yeah, 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 Raigyu composition of nucleotides with the Ben New composition of nucleotides. And they actually have different ratios.

• 12:12Lester Nare?

  Okay, that's interesting.

• 12:12Krishna Choudhary?

  So both asteroids have the same building blocks of adenine, guanine, cytosine, thymine, and uracil. The same nucleotides that we have. But the proportions are completely different. In Ben New, it's heavily skewed towards the pyramidines, which are the cytosine, the thymine, and the uracil. But in Raigyu, 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.

• 12:53Krishna Choudhary?

  It could mean that it has something to do with the water rock interactions that are on the asteroid, the internal temperatures. But the main thing is, it's a lot more complicated up there, right? There's like actual chemistry happening.

• 13:07Lester Nare?

  We have.

• 13:07Krishna Choudhary?

  And it doesn't have to be on a planet.

• 13:09Lester Nare?

  We have a cosmic oven, a cosmic easy bake oven that is our galactic, intergalactic postal service.

• 13:16Krishna Choudhary?

  Yeah.

• 13:16Lester Nare?

  Like you sort of mentioned, which, and I think part of the point, what I'm getting from the point you're bringing up here is, you know, we like to keep our assumptions conservative when it comes to particularly things like.

• 13:34Lester Nare?

  What's happening on Earth is happening everywhere else.

• 13:36Krishna Choudhary?

  Yeah.

• 13:36Lester Nare?

  We like to see it.

• 13:37Krishna Choudhary?

  Yeah.

• 13:38Lester Nare?

  Before we believe it.

• 13:38Krishna Choudhary?

  Exactly.

• 13:39Lester Nare?

  Sure, maybe it's happening other places, but let's actually get not only one use case, Benu, but multiple use cases. And the idea that we're having now, um, a catalog. Now, we now they say there's a catalog.

• 13:55Krishna Choudhary?

  Yeah.

• 13:55Lester Nare?

  Cause it's more than one.

• 13:56Krishna Choudhary?

  Yes, and as we do more and more, I'm sure we're gonna 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.

• 14:09Lester Nare?

  It's really fascinating. This is re- and I, I mean, there's so many, a- again, it feels to 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 replicability, uh, matters a lot.

• 14:26Krishna Choudhary?

  Yeah.

• 14:27Lester Nare?

  These missions are not cheap.

• 14:28Krishna Choudhary?

  No.

• 14:29Lester Nare?

  Right?

• 14:29Krishna Choudhary?

  No.

• 14:30Lester Nare?

  To, to pull this off.

• 14:31Krishna Choudhary?

  Yeah. And this is two, th- this is two different space agencies doing it.

• 14:34Lester Nare?

  Right.

• 14:35Krishna Choudhary?

  Right? So there's no conspiracy unless Japan is in on it.

• 14:38Lester Nare?

  Yeah, right.

• 14:38Krishna Choudhary?

  Somehow.

• 14:39Lester Nare?

  Right, which, look, look, we'll put the, I'm just, I know, 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 seeing their journey.

• 15:08Krishna Choudhary?

  Mm-hmm.

• 15:08Lester Nare?

  And understanding, based on that, that journey, why is that mix different?

• 15:12Krishna Choudhary?

  Yeah.

• 15:13Lester Nare?

  Um, fascinating.

• 15:14Krishna Choudhary?

  Very, very cool.

• 15:16Lester Nare?

  This is fascinating. So, you know, as always, like, you know, I'm the, I'm the alien guy, and this is great, but we're gonna 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 gonna 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 way- 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 watching because each episode, we go through

• 15:59Lester Nare?

  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 would like to support the pod more directly, you can either donate directly at ffpod.com backslash donate, or you can go to our new shop where you can buy some of our great merch. We are going to be dawning that merch more frequently on the pod so we

• 16:41Lester Nare?

  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, it could be genetics, medicine, physics, space, we always talk about the final product, which is the published paper,

• 17:24Lester Nare?

  and usually the history and the buildup to that. 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 funder 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 arisen 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 works. 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.

• 18:09Lester Nare?

  The president proposes a budget every year. At the beginning of the year, they propose a budget. 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. OMB, which is the White House Office of Management and Budget, helps to execute the budget that is fundamentally decided by Congress. It apportions funding to executive agencies and sets government-wide rules about how that money can be spent. So Congress says, here's the money and who can spend it. And then OMB kind of just distributes.

• 18:53Krishna Choudhary?

  Yeah.

• 18:54Lester Nare?

  How that goes. And this is how federal grants, especially in research, but not only in research, are administered.

• 19:00Krishna Choudhary?

  Okay.

• 19:02Lester Nare?

  Normally, in the history of this process, the Office of Management and Budget doesn't select individual science awards or grants. It's not in the process of selecting who gets what money. 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 how that happens. And the process is the agencies will announce a funding opportunity. Researchers will submit proposals. Experts evaluate their merit.

• 19:37Krishna Choudhary?

  Yep.

• 19:38Lester Nare?

  As you know very well in this process.

• 19:41Krishna Choudhary?

  Yeah.

• 19:41Lester Nare?

  And the agency officials ultimately decide which projects get to receive awards.

• 19:46Krishna Choudhary?

  Yeah, and shout out to my dad. He's one of the astrophysicists that's on a lot of these panels that like decides which grants get awarded and things like that.

• 19:55Lester Nare?

  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 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.

• 20:13Krishna Choudhary?

  Yeah.

• 20:14Lester Nare?

  Congress funds it, OMB establishes the rules, agencies select and manage the grants, researchers do the science. That's how the process has worked forever.

• 20:27Lester Nare?

  But now, you know, we're going to get into the current fight.

• 20:30Krishna Choudhary?

  Right.

• 20:31Lester Nare?

  Before we do, I do want to give a lot of credit here to many of the science policy journalists who have been doggedly covering this story closely, particularly Jeffrey Mervis, Jocelyn Kaiser, Lauren Martin Agudelo, Michael Greshko, and David Malakoff at both Science and AAAS. This is sort of in the weeds and they've taken a very complex bureaucratic funding and policy process and have tried to make it connect to the actual science and impact that it has for people. And this is really, again, a great source for us as we try to begin to now incorporate science funding policy into our discussions because we would not be able to, again, have a show without this.

• 21:17Lester Nare?

  So now we get to the current fight. And it's always over who has the power to decide. So I'm going to kind of read through a couple of the details here, which are a little bit in the weeds. But so on May 29th, OMB can only decide the rules, right? OMB released this 412-page proposal to sort of revamp their role in distributing and managing grants and awards.

• 21:45Krishna Choudhary?

  And this is the White House.

• 21:46Lester Nare?

  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. Granted, they have a role in this process because they are the ones who initially derive and drive the national priorities when the budget process starts in January and February of every year. So this new proposal had until July 13th to provide comments.

• 22:14Krishna Choudhary?

  And

• 22:15Lester Nare?

  O M B 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. Number one. Number two, peer review would, quote, remain a part of the process. But the proposal makes clear that scientific reviewers, recommendations, like your dad, as you just mentioned, would be advisory and not controlling.

• 22:59Krishna Choudhary?

  Okay.

• 23:00Lester Nare?

  Currently, they have the controlling. What they say kind of goes. They're saying that's just more of just an opinion and we get to decide.

• 23:06Krishna Choudhary?

  And, and we being these political appointees.

• 23:09Lester Nare?

  Right.

• 23:09Krishna Choudhary?

  Great, nice.

• 23:11Lester Nare?

  Don't really have any scientific.

• 23:13Krishna Choudhary?

  Yeah.

• 23:14Lester Nare?

  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.

• 23:30Krishna Choudhary?

  Okay.

• 23:31Lester Nare?

  And then fourth, the proposal would impose major restrictions on international research collaborations with countries that are deemed to be a national security risk. The fourth one.

• 23:42Krishna Choudhary?

  Fourth one I get.

• 23:44Lester Nare?

  That one has some, some standing and you could argue the details there.

• 23:48Krishna Choudhary?

  Yeah.

• 23:48Lester Nare?

  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 D C, waste, fraud and abuse.

• 24:03Krishna Choudhary?

  Right.

• 24:04Lester Nare?

  I don't think, if you look at the grand scheme of the budget, that scientific funding and grant funding is the biggest source of waste, fraud and abuse. But they argue that the peer review process has become insular and the final responsibility should remain to politically accountable appointees. That's the argument being made for why this proposal's been made in place. And.

• 24:27Lester Nare?

  You know, in a, in a vacuum, you would maybe say, okay, maybe there's an argument here.

• 24:36Lester Nare?

  But obviously, the scientific community has reacted very strongly about this going well beyond financial oversight. This is clearly a battle about power.

• 24:45Krishna Choudhary?

  And culture.

• 24:46Lester Nare?

  And culture.

• 24:47Krishna Choudhary?

  Yeah.

• 24:47Lester Nare?

  Because science, which has always kind of brushed at the edges of culture, has never been quite as part of a zeitgeist, the zeitgeist as it has since COVID happened, in my view, frankly, at least in American culture.

• 24:59Krishna Choudhary?

  Yeah, I agree.

• 25:00Lester Nare?

  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 to get involved in the process. So the, the Association of American Universities submitted a forty-seven-page response, which I might add is the longest public comment in the organization's one-hundred-and-twenty-six-year history. So, clearly, it struck a nerve.

• 25:30Krishna Choudhary?

  Yeah.

• 25:30Lester Nare?

  Because everyone understands what's going on here. 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.

• 25:46Krishna Choudhary?

  That's right.

• 25:46Lester Nare?

  Has not given OMB the authority to decide what categories of scientific research may receive federal funding. That power lies purely within, inside the agencies and their statutory directives.

• 26:00Krishna Choudhary?

  So effectively, the universities are saying,

• 26:05Krishna Choudhary?

  the, the universities are using the Constitution to try and defend the way that we fund science.

• 26:10Lester Nare?

  Which, it is in the Constitution.

• 26:12Krishna Choudhary?

  Yeah.

• 26:13Lester Nare?

  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.

• 26:23Krishna Choudhary?

  Yeah.

• 26:24Lester Nare?

  Period. The executive branch is effectively trying now to insert themselves in the process of how money is spent.

• 26:31Krishna Choudhary?

  Yeah.

• 26:31Lester Nare?

  So there's, to your point, there's a constitutional level issue here beyond even some of the practical matters.

• 26:40Lester Nare?

  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 all the branches of government, they open up it to public comment.

• 26:57Krishna Choudhary?

  Oh, yeah.

• 26:57Lester Nare?

  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 important to cover. So that's what's currently happening in terms of OMB 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.

• 27:31Krishna Choudhary?

  Yeah.

• 27:31Lester Nare?

  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.

• 27:51Krishna Choudhary?

  Yeah.

• 27:52Lester Nare?

  Something you may or may not have some context for.

• 27:55Krishna Choudhary?

  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 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.

• 28:28Lester Nare?

  So as, we have two sides of the story here. There's the proposal that's happening, but there's actions that have already been, been taken since January of 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, you can't do that, right? So that's, you know, been established in federal court.

• 29:09Lester Nare?

  Now, a caveat to that is the government may retain legitimate authority to terminate grants, which will still continue to be challenged. But the judge's central point was that it cannot change the rules after an award has already been made, which is where we get the proposal language, which says they can involve themselves prior to the awards being distributed. 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.

• 29:50Lester Nare?

  Those projects represented approximately $185 million in multi-year commitments. And so they've already received a distribution, and they're saying you're getting no more money. 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 thing, and we're discontinuing the continued funding of what is a multi-year install.

• 30:34Krishna Choudhary?

  That doesn't make any sense. Are you serious?

• 30:38Lester Nare?

  So it's basically termination under a different name.

• 30:41Krishna Choudhary?

  Yeah, it's termination, but...

• 30:42Lester Nare?

  They're trying to just voodoo magic.

• 30:44Krishna Choudhary?

  They're saying they're not terminating, they're just discontinuing.

• 30:47Lester Nare?

  Right, right.

• 30:48Krishna Choudhary?

  Very nice.

• 30:49Lester Nare?

  I'm not the thesaurus, but they may be synonyms in this context.

• 30:52Krishna Choudhary?

  Yeah.

• 30:55Lester Nare?

  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.

• 31:02Krishna Choudhary?

  Right.

• 31:03Lester Nare?

  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 have a program that gets a $50 million allocation in 2025, but it's spread over four years.

• 31:17Krishna Choudhary?

  Yeah, yeah. I mean, a lot of the, like, some of the grants that I was supported during my PhD were these multi-year grants where, like, my advisor or like the collaboration that I was a part of would get money every year, right? You don't get a lump sum.

• 31:33Lester Nare?

  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.

• 31:44Krishna Choudhary?

  Yeah, it's never been that way.

• 31:45Lester Nare?

  It's not a discretionary rule. And this is, this is a very, very contentious issue right now. So the central dispute dispute is whether the administration is simply managing the congressional-funded programs or whether replacing Congress's priorities has, you know, their own right to do, like, if they have their own right to basically displace what Congress has already decided to do. 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 OMB, Office of Management and Budget, is trying to rewrite the government-wide rules

• 32:28Lester Nare?

  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 this same time, and federal courts are weighing in on how much authorities the White House effectively has on canceling existing awards. So all sort of three branches of government, the executive,

• 32:55Krishna Choudhary?

  Wow.

• 32:55Lester Nare?

  the legislative and the judicial, are concurrently engaging in a battle to really define what happens here.

• 33:03Krishna Choudhary?

  Uh huh.

• 33:04Lester Nare?

  The comment period for the rule proposal changes is over. Uh, OMB wants to set the statutory deadline for October 1st.

• 33:14Lester Nare?

  At which point this would then take play 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 about accountability because there's already systems in place to deal with that. The real kind of issue is, does the executive branch have the authority to make individual funding decisions within programs that Congress has already authorized? And again, this goes back to it's a fundamental constitutional question.

• 33:49Krishna Choudhary?

  Yeah.

• 33:49Lester Nare?

  Because it literally goes counter to the structure of our co-equal branches of government.

• 33:57Krishna Choudhary?

  Yeah.

• 33:57Lester Nare?

  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 four-year cycle or an eight-year cycle does not mean it's then going to flip the other direction. And we cannot afford to have a seesaw problem in the, what has made this country, a huge part of what has made this country, stable and successful and resilient, is the fact that how we won

• 34:41Lester Nare?

  World War II is because we've done this funding in a way that's already been structured.

• 34:47Krishna Choudhary?

  Yeah, I 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.

• 35:10Lester Nare?

  Correct.

• 35:10Krishna Choudhary?

  Okay, because I doubt the, the powers that be are going to understand why it's important to study bacterial immune systems, okay? But then we get like restriction enzymes, genetic engineering, CRISPR.

• 35:25Krishna Choudhary?

  Come on, guys. What are we doing?

• 35:27Lester Nare?

  I don't want the people that did not understand that studies using transgenic mice

• 35:32Krishna Choudhary?

  Yeah.

• 35:33Lester Nare?

  wasn't, uh, talking about the, the gender of mice.

• 35:38Krishna Choudhary?

  Yeah, yeah, yeah.

• 35:39Lester Nare?

  Like, like, you know what I mean? Like, that's, we can't.

• 35:41Krishna Choudhary?

  Yeah, transgen, and then they're like, "Ohhhhh." Guys, like, how, they should not be in charge of, of this kind of power. This is insane.

• 35:52Lester Nare?

  We're gonna keep you all updated on this process, because 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.

• 36:15Krishna Choudhary?

  Yes, yeah.

• 36:16Lester Nare?

  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 sensitive 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 gonna be going to their home districts, they're gonna 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

• 37:01Lester Nare?

  listen to people who reach out. The problem is, no one reaches out.

• 37:04Krishna Choudhary?

  Yeah.

• 37:04Lester Nare?

  And so, all these specialty issues get kind of overlooked, because they just, no one talks 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.

• 37:36Krishna Choudhary?

  No.

• 37:37Lester Nare?

  We don't need the White House involved.

• 37:38Krishna Choudhary?

  We really don't. 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.

• 37:50Lester Nare?

  Right.

• 37:50Krishna Choudhary?

  It's gonna be a hot button issue and we need to make it.

• 37:53Lester Nare?

  A hot button issue.

• 37:54Krishna Choudhary?

  A hot button issue.

• 37:55Lester Nare?

  Hundred percent.

• 37:55Krishna Choudhary?

  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 need to get on it, guys.

• 38:12Lester Nare?

  It matters a lot.

• 38:13Krishna Choudhary?

  Yeah.

• 38:14Lester Nare?

  It matters a lot. And I I know some folks' 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 because it was all over my feed.

• 38:35Krishna Choudhary?

  Yeah.

• 38:36Lester Nare?

  All over my feed.

• 38:36Krishna Choudhary?

  Yeah, this was um, this was kind of huge because it happened in the middle of the World Cup final.

• 38:43Lester Nare?

  You're right.

• 38:44Krishna Choudhary?

  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.

• 39:01Lester Nare?

  Who who won that? I just, I don't know. I can't remember.

• 39:04Krishna Choudhary?

  I wonder, I know it wasn't Argentina.

• 39:06Lester Nare?

  It wasn't Argentina. That's right.

• 39:08Krishna Choudhary?

  That's right.

• 39:08Lester Nare?

  It was not. It was not.

• 39:10Krishna Choudhary?

  It was not Argentina.

• 39:12Lester Nare?

  That's right.

• 39:12Krishna Choudhary?

  You love to see it.

• 39:14Lester Nare?

  I just, you know, for everyone who's suffered through our World Cup coverage, it's over, so we'll stop the sports ball. But I just, I couldn't remember.

• 39:21Krishna Choudhary?

  Yeah, yeah, it's good to, it's good to remind oneself.

• 39:23Lester Nare?

  Just, just, just to set the record straight.

• 39:26Krishna Choudhary?

  That's right. That's right. So this next story is about AI and how it is encroaching on fundamental mathematics.

• 39:35Krishna Choudhary?

  It's very, very cool because I mean, it's completely obvious, I think, to everyone who lives under, who does not live under Iraq, 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 been here since the time of the ancient Greeks with Euclid.

• 40:19Krishna Choudhary?

  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.

• 40:33Lester Nare?

  Very important point.

• 40:34Krishna Choudhary?

  Very important point. It's, you know, we've seen a lot of disciplines fall to computation. Chess has fallen to computation. The game Go has fallen to computation. Protein folding has fallen to computation. And now we are finally seeing the shining city on the hill, which is pure mathematics. It is starting to fall, right? There's crumbles in the infrastructure.

• 41:04Lester Nare?

  I just want to briefly note that, you know, in our time at Princeton, some of the smartest people, we had the pleasure of being able to be around and breathe the same air that they did were the maths folks.

• 41:18Krishna Choudhary?

  Yeah, were the maths folks, yeah.

• 41:20Lester Nare?

  Like, they were a level of smart that is really hard to wrap my own head around.

• 41:27Krishna Choudhary?

  Yeah, yeah, yeah, same for me. I mean, I took some courses with them, right? And I was just like, I'm gonna stick to it. I'll stick to the physics department, which is right next door. I took a few math courses and it was it was extremely fun, but I could just never see myself like being a math major, right? And I mean, even in the physics department, I was like, man, what am I doing 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, which is like the synonymous, the, the, the company of Claude, the, the company that wants AI safety, AI safety.

• 42:07Lester Nare?

  Regulatory capture.

• 42:09Krishna Choudhary?

  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 polynomial 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 a mathematical object that violates whatever the rule is. It's a bit easier to do, albeit, than actually 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

• 42:54Krishna Choudhary?

  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.

• 43:23Lester Nare?

  Yep.

• 43:24Krishna Choudhary?

  Trying to prove.

• 43:25Lester Nare?

  And this is the part of the ongoing debate about whether these AI systems are quote stochastic parrots that don't have any original thought and or anything that they do that's original is just by. Gathering all the information on the internet and or brute forcing. And I understand that sentiment, but I think it is a little bit more nuanced than the way in which it's presented in this black and white way.

• 43:55Krishna Choudhary?

  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 things in a very, in a very real sense, right? AI is restructuring the labor markets. 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 stochastic parrot. Well, in this case, nobody had a, nobody had a proof of this. And it has a novel proof that the smartest people on the planet couldn't come up with.

• 44:32Lester Nare?

  For almost a century.

• 44:33Krishna Choudhary?

  Yeah. It's, it's huge, right? So mathematics, right? As I said, it's been the gold standard of human reasoning. It's the language of physics, the foundations of electrical engineering, every other kind of engineering, economics, medicine. When we say, yeah.

• 44:52Lester Nare?

  Literally everything.

• 44:53Krishna Choudhary?

  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.

• 45:22Lester Nare?

  Would it be fair to say that mathematics is one of, if not the only thing that we have where you can say that there is an objective truth?

• 45:29Krishna Choudhary?

  Yeah.

• 45:29Lester Nare?

  When you come to, when it comes to like a mathematical proof?

• 45:31Krishna Choudhary?

  Yes, it is, it is, it is, it is like 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? Euclid's theorem that there 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 really cool. And when we start thinking about AI starting to do mathematics, not just assisting and not just suggesting, but actually proving and disproving things, right?

• 46:12Krishna Choudhary?

  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 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. We're really going to get into the weeds of it. And I think you're going to be able to understand exactly what it's done.

• 46:53Krishna Choudhary?

  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 the Jacobian conjecture. It's something that was proposed by Heinrich Keller in 1939. So nearly 100 years ago. 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 to understand the Jacobian conjecture, we have to first understand functional maps,

• 47:37Krishna Choudhary?

  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 a hyperplane 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 coordinate and the old Y coordinate, right?

• 48:14Krishna Choudhary?

  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.

• 48:40Lester Nare?

  Which can also scale from 2D.

• 48:42Krishna Choudhary?

  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 kinda rhomboid, whatever, whatever. 3D prism is what it's called, I think.

• 48:57Lester Nare?

  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.

• 49:05Krishna Choudhary?

  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 1. Okay, that's what the Jacobian means. Effectively, it's saying, okay, I have a map and that map is preserving area if the Jacobian is 1. If the Jacobian is greater than 1, 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 really starts from this concept of like maps of the earth, right? You can think about the Mercator projection or these other projections that change the area.

• 49:42Krishna Choudhary?

  And change the shape, but keep the, sorry, they change the shape, but they keep the area the same. For example, with the Mercator projection, right? We all know that areas near the poles get larger. So the Jacobian near the poles is greater than one.

• 49:55Lester Nare?

  Greater than one.

• 49:56Krishna Choudhary?

  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 Mercator 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.

• 50:15Lester Nare?

  So, and people would argue which is better when you talk about geographic maps? Is it better to keep the Jacobian closer to one?

• 50:22Krishna Choudhary?

  Yeah.

• 50:22Lester Nare?

  Uh, where you're going to get some shape distortion? Or is it better to have it have shape, uh, clarity?

• 50:29Krishna Choudhary?

  Yeah.

• 50:30Lester Nare?

  But you have distortions at some extremes.

• 50:33Krishna Choudhary?

  Mm hmm.

• 50:33Lester Nare?

  As a really very, okay.

• 50:35Krishna Choudhary?

  Yeah, so that's what we're talking about.

• 50:36Lester Nare?

  That's what we're talking about.

• 50:37Krishna Choudhary?

  Jacobians, okay? It's really, it's a, it's a ratio of how my transformation is changing little tiny areas, infinitesimal areas. Right? Cause in calculus, we think about like, there's continuous transformation. So the Jacobian here can be different from the Jacobian there and so on and so forth.

• 50:53Lester Nare?

  It's not a family in Game of Thrones, is what you're saying.

• 50:55Krishna Choudhary?

  No, not at all. And so let's, let's do a very simple case, okay? A linear transformation.

• 51:00Lester Nare?

  Yes.

• 51:01Krishna Choudhary?

  Now, in this case, this is a very simple matrix, two, one, one, two. 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 a 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 axis, right? Similarly, the Y unit vector, which is zero, one, that gets transformed to one, comma, two, which means one on the X axis, two on the Y axis. That's why the green arrow is slanted more towards the Y axis. The red arrow is slanted more towards the X axis. And in this case, the determinant of that matrix, if you were to do it, it's four minus one.

• 51:46Krishna Choudhary?

  So that's three. You can see that the square becomes a rhombus that's the size of three. And it has to do with the determinant of that matrix. So when we say a Jacobian determinant, that's what we're talking about.

• 51:59Lester Nare?

  Okay, yes.

• 52:00Krishna Choudhary?

  Okay, so this is a very simple case. Now, let's give you kind of a more complicated case, but one of my favorite coordinate transformations as a physicist, which is the Lorentz transformation. This is the transformation that we know and love from relativity.

• 52:14Lester Nare?

  Right.

• 52:15Krishna Choudhary?

  Right, in special relativity, we have the Lorentz transformation. You might have heard that when we move fast, length contracts and time dilates. So the clock runs slower and length gets shorter. That's two different things, right? 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 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.

• 52:59Krishna Choudhary?

  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 show you by considering this plot. So let's consider this, this plot. This is a plot that 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 1 is not 1 meter. The 1 is 1 light second. So it's really like 300,000 kilometers, 600,000 kilometers. So it's, it's the speed of light multiplied by 1 second, 2 seconds, 3 seconds and 4 seconds.

• 53:44Krishna Choudhary?

  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.

• 53:52Lester Nare?

  Yeah.

• 53:52Krishna Choudhary?

  Right? Because it's like 1 to 1. It's like, okay, in 1 second, it travels 1 light second. In 2 seconds, it travels 2 light seconds. So you've got a nice, the dotted green line is what light would do.

• 54:03Lester Nare?

  Yes.

• 54:03Krishna Choudhary?

  In this coordinate system, okay? So that's, that's usually how we represent things in relativity.

• 54:11Krishna Choudhary?

  All right.

• 54:11Lester Nare?

  For many folks who math may not be something you do very often. The closest thing you may have seen that touches this concept is interstellar.

• 54:20Krishna Choudhary?

  Yes.

• 54:20Lester Nare?

  When they're on the wave planet, the guy is on the ship still. And they have that time dilation issue because of exactly what we're talking about.

• 54:28Krishna Choudhary?

  Because of exactly what we're talking about, exactly. And so we've considered that plot. Now let's consider, for example, we're in this plot, right? It's the two of us.

• 54:40Lester Nare?

  Yes.

• 54:40Krishna Choudhary?

  Okay, we're both stationary. We're both in the same reference frame.

• 54:44Lester Nare?

  Mate, I got a lot of motion. I don't know what you're talking about.

• 54:46Krishna Choudhary?

  But as of now, I'm gonna, I'm gonna have you move. I'm gonna have you move in the next plot. But in this plot.

• 54:52Lester Nare?

  I don't have motion right now, but I will in the future.

• 54:54Krishna Choudhary?

  I wanna, I wanna give you a little sense of how relativity is so cool. Okay? So let's consider, we've got, we're, we're sitting here. This is x equals zero, this table.

• 55:04Lester Nare?

  Okay.

• 55:05Krishna Choudhary?

  And we set the time, we, we have a stopwatch. I've got a stopwatch on my iPhone. I set it to time equals zero. And let's say there are three explosions that happen, okay? Or like three light switches. The, the, in the physics textbooks, it's always like light switches. Let's do, let's do explosions.

• 55:23Lester Nare?

  I like explosions.

• 55:24Krishna Choudhary?

  But, but we can survive the explosions, okay? There's an explosion that happens here, where we're sitting, but in two seconds time. Okay? There's an explosions that happens in front of us, two light seconds away. So 600,000 kilometers away.

• 55:41Lester Nare?

  Real close.

• 55:42Krishna Choudhary?

  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 same time. Okay? So we're like, okay, cool. The coordinates of these explosions in spacetime. Now we're thinking about coordinates in spacetime, not just space. The coordinates in spacetime are, the explosion that happens here is at space equals zero.

• 56:14Lester Nare?

  Yes.

• 56:14Krishna Choudhary?

  Because it's right here and the time equals two seconds. The explosion in front of us is space equals two.

• 56:20Lester Nare?

  Yes.

• 56:20Krishna Choudhary?

  And the time equals two because it happened at the same time. And back there, it's negative two, time equals two.

• 56:26Lester Nare?

  When you say spacetime in this example, we're talking about two axes.

• 56:29Krishna Choudhary?

  Yeah.

• 56:30Lester Nare?

  One is space and one is time. And so you can represent spacetime on a, on a two dimensional X Y graph in this example. And it's not this.

• 56:40Krishna Choudhary?

  No, it's literal now.

• 56:41Lester Nare?

  It's yeah.

• 56:42Krishna Choudhary?

  It's literal on a coordinate axis. This is the genius of Einstein.

• 56:45Lester Nare?

  Right. Okay.

• 56:46Krishna Choudhary?

  Right? As he's like putting this stuff on a coordinate plane and he's asking, right? And this is actually, Lorentz came up with this stuff even before Einstein. He was, he was thinking about Lorentz transformations and Einstein's the real guy to be like, let's just take it seriously, guys. 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 that way.

• 57:15Lester Nare?

  Yes.

• 57:15Krishna Choudhary?

  To the front.

• 57:16Lester Nare?

  Yes.

• 57:16Krishna Choudhary?

  At, at very close to the speed of light. Okay?

• 57:21Krishna Choudhary?

  The 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, but before you, before you, yeah, so, so this is me, right? This is, this is my.

• 57:36Lester Nare?

  Yes.

• 57:37Krishna Choudhary?

  So, so to me, everything's happening at the same time, which is why everyone's in the same time axis.

• 57:41Lester Nare?

  Yes.

• 57:42Krishna Choudhary?

  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 the speed of light.

• 57:51Lester Nare?

  Which is at two, two.

• 57:52Krishna Choudhary?

  Yep, which is at two, two for me.

• 57:54Lester Nare?

  For you.

• 57:54Krishna Choudhary?

  For me.

• 57:55Lester Nare?

  But I'm moving towards it.

• 57:56Krishna Choudhary?

  But you're moving towards it, right?

• 57:57Lester Nare?

  At the speed of light.

• 57:58Krishna Choudhary?

  At the, at, near the speed of light, near the speed of light.

• 58:01Lester Nare?

  Sue me.

• 58:02Krishna Choudhary?

  Yeah, okay. 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. 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.

• 58:22Lester Nare?

  Yeah.

• 58:23Krishna Choudhary?

  So the light from that explosion is gonna come meet you.

• 58:25Lester Nare?

  Mm hmm. Because, because we're basically, it's like two trains coming at each other.

• 58:29Krishna Choudhary?

  Yes.

• 58:30Lester Nare?

  But because of the speed at which I'm moving towards it, from your perspective, it's gonna take time for that.

• 58:37Krishna Choudhary?

  It's gonna take two seconds.

• 58:39Lester Nare?

  But because I'm basically meeting the light before it gets to you at that speed, it's, it's going to happen.

• 58:44Krishna Choudhary?

  Yeah. And so for you, that event is gonna happen sooner than two seconds, right? And it's gonna happen closer than two light seconds because the length is contracted. Similarly, the event that happens here is gonna take a little bit longer than two seconds because this thing happens for me at two seconds, but then the light has to go to wherever you are. So it's gonna happen a little bit longer than two seconds. And the guy, and the explosion that was way back there, that's gonna 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 and Y.

• 59:27Lester Nare?

  Yes.

• 59:27Krishna Choudhary?

  My X and Y is, is, is stationary.

• 59:30Lester Nare?

  And, and when we say down on the Y-axis, because our Y.

• 59:33Krishna Choudhary?

  So it's sooner in time.

• 59:35Lester Nare?

  Sooner in time.

• 59:35Krishna Choudhary?

  And it's sooner in space.

• 59:37Lester Nare?

  Right. And, and getting closer to the origin. Yep.

• 59:39Krishna Choudhary?

  Yep, yep. Okay.

• 59:41Lester Nare?

  Got it.

• 59:41Krishna Choudhary?

  So now, let's, 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 giving you some analysis of, of what we're talking about. The, as I said, the diagonals are light, right? So the diagonals are what light would travel, like the path that light would travel in my spacetime.

• 1:00:05Lester Nare?

  Because again, our, our X-axis is light, uh, light seconds.

• 1:00:09Krishna Choudhary?

  Mm hmm.

• 1:00:10Lester Nare?

  And our Y-axis.

• 1:00:11Krishna Choudhary?

  Is just time.

• 1:00:12Lester Nare?

  Time.

• 1:00:12Krishna Choudhary?

  Seconds.

• 1:00:13Lester Nare?

  Right.

• 1:00:13Krishna Choudhary?

  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 equals one.

• 1:00:19Lester Nare?

  Right.

• 1:00:20Krishna Choudhary?

  Then it would travel like at the, it would be the dotted line that's just above the t equals zero.

• 1:00:25Lester Nare?

  This is why we want to decrement the X-axis in this way so that it creates.

• 1:00:29Krishna Choudhary?

  Yes.

• 1:00:29Lester Nare?

  This simplicity to understand.

• 1:00:31Krishna Choudhary?

  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, a line, a black line, right on the. Right on the 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 x equals zero.

• 1:00:50Lester Nare?

  Yes.

• 1:00:51Krishna Choudhary?

  For all of time.

• 1:00:52Lester Nare?

  Yes.

• 1:00:52Krishna Choudhary?

  Right? So no matter what time it is, I'm always at x equals zero. Okay? So now, 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 you'd see me go backwards.

• 1:01:06Lester Nare?

  Yes.

• 1:01:06Krishna Choudhary?

  Because to you, I'm coming from up there and then going backwards. Okay? Now, let's go ahead and do the Lorentz transformation to see what you would see.

• 1:01:15Lester Nare?

  Okay.

• 1:01:16Krishna Choudhary?

  Okay? What would you see? What you would see is the following. You'd see the coordinate axes change.

• 1:01:23Lester Nare?

  Yeah, yeah, yeah.

• 1:01:24Krishna Choudhary?

  Okay? And notice, first thing we want to notice is the three red dots are doing exactly what we 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 two seconds, it's a little bit more, because the light has to catch up to you for you to be like, "Ah, there was an explosion." And the event that happened way back there is happening way farther at two seconds, right? But notice two things. And for one, the line that is me, which is usually time t equals zero, x equals, time, time equals whatever, x equals zero, that thing is shifting to where at negative time,

• 1:02:07Krishna Choudhary?

  so before time t equals zero, I'm ahead of you. And then I go behind you. Right? That's the black line.

• 1:02:13Lester Nare?

  Yep.

• 1:02:14Krishna Choudhary?

  Here's the key thing about this coordinate transformation, which is the genius of the Lorentz transformation. All of the diagonals are still diagonal. 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 getting faster, but they're happening at the same rate, such that the diagonals are always diagonal.

• 1:02:45Lester Nare?

  Yeah, they're just shifting on the same ratio and proportionality to each other.

• 1:02:50Krishna Choudhary?

  Yes, yes, by the Lorentz factor, actually, is what it is. And so, it preserves relativity, because one of the tenets of relativity is that the speed of light is the same in any reference frame.

• 1:03:03Lester Nare?

  It's constant.

• 1:03:03Krishna Choudhary?

  So even when we go into your reference frame, the diagonals are still diagonal, but...

• 1:03:10Krishna Choudhary?

  The space has changed and the time has changed. The other cool thing about this is the volume of spacetime is always the same. The area of the square becomes a rectangle, but the area of each rectangle is exactly the same.

• 1:03:24Lester Nare?

  Yes, again, because everything is proportion. It's a shift that's happening. It goes back to this mapping point that you bring up earlier. We are making this transformation in our map, but fundamentally, our Jacobian remains as one the entire time.

• 1:03:38Krishna Choudhary?

  Yes, that's exactly right. And what that means is if like the volume of spacetime is always preserved.

• 1:03:45Lester Nare?

  Yes, yes.

• 1:03:46Krishna Choudhary?

  In the Lorentz transformation, okay? That's the central tenet of relativity.

• 1:03:50Lester Nare?

  Okay.

• 1:03:51Krishna Choudhary?

  That ensures that the speed of light is always the same for the both of us. And really, 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.

• 1:04:07Lester Nare?

  Correct.

• 1:04:07Krishna Choudhary?

  Right? Your perspective is the same as my perspective. So the speed of light better be the same. And the universe is doing all this nonsense of contracting length and making time go slower just so that the diagonals remain diagonal.

• 1:04:23Lester Nare?

  I'm not saying it's a simulation, but if you wanted to make it a simulation.

• 1:04:27Krishna Choudhary?

  Yeah, this would be a rule that I would code up.

• 1:04:29Lester Nare?

  It would make sense.

• 1:04:30Krishna Choudhary?

  Yeah.

• 1:04:31Lester Nare?

  And I'm being facetious here, but I think the point you're bringing up is important, which is that the constant of the speed of light requires the universe to do all this other stuff.

• 1:04:40Krishna Choudhary?

  Yeah.

• 1:04:41Lester Nare?

  So that regardless of your speed or your position in the universe, that constant remains the same.

• 1:04:50Krishna Choudhary?

  Exactly.

• 1:04:51Lester Nare?

  Which requires it to do all this other stuff.

• 1:04:52Krishna Choudhary?

  All those other things.

• 1:04:53Lester Nare?

  So it's almost like the speed of light is a golden rule.

• 1:04:56Krishna Choudhary?

  Yeah.

• 1:04:57Lester Nare?

  It always must be followed.

• 1:04:58Krishna Choudhary?

  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?

• 1:05:03Lester Nare?

  We'll figure out the other stuff to make it happen.

• 1:05:05Krishna Choudhary?

  Yeah, exactly. And so that's a central tenet of Einstein's special relativity, okay? And that's the Lorentz transformation that gives rise to all of the rich behavior that is a consequence of special relativity. Relativity. So that again is just a, that's a linear transformation, as I've said, right? It's um, it's not too complicated.

• 1:05:27Lester Nare?

  I was able, I was able to get it.

• 1:05:28Krishna Choudhary?

  Yeah, yeah, exactly. Special relativity is like actually not that bad. When you like really just think about it, it's like, yeah, okay, kind of makes sense. It's when we get to general when it's like, I don't, whoa, whoa, whoa, whoa, slow down. I just got here. Like, so, so that's, that's special relativity, right? And the Jacobian there is one, meaning that the volume in spacetime is preserved no matter what reference frame you're in. Okay?

• 1:05:55Lester Nare?

  And when you say reference frame, you mean both position and speed?

• 1:06:00Krishna Choudhary?

  And time.

• 1:06:00Lester Nare?

  And time. So both position and time.

• 1:06:02Krishna Choudhary?

  Yeah, yeah. Whatever volume in your position and time versus my position in time are going to be exactly the same. It's just that the, each individual thing is going to be changed, but the product of the two are going to remain the same.

• 1:06:14Lester Nare?

  Yep.

• 1:06:15Krishna Choudhary?

  Right? Because the Lorentz can't, the Lorentz factor is going to cancel out.

• 1:06:18Lester Nare?

  Makes sense.

• 1:06:18Krishna Choudhary?

  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 Minkowski tensor and all, and all sorts of stuff. But 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 spacetime 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 x plus sine of y over 2, and the y-axis goes to y plus sine of x over 2. This is what I mean by now you can start mixing stuff, right?

• 1:07:02Krishna Choudhary?

  In the other one, you were also mixing, but here you're doing it in a nonlinear way. You're not just like adding some Lorentz factor times the constant. This is like sine, which is an oscillatory 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 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 that sector that's right in the middle of the 1-1 box.

• 1:07:47Krishna Choudhary?

  Okay, now

• 1:07:50Krishna Choudhary?

  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 gonna be a line. That's all calculus is. Like, the essence of calculus is weird stuff is happening, but if I take a small enough element, it's mostly just linear.

• 1:08:13Lester Nare?

  Calculus, just zoom in.

• 1:08:15Krishna Choudhary?

  Just zoom, yeah, no, exactly. Just zoom in and the square becomes a parallelogram. It's not becoming a weird curved shape.

• 1:08:23Lester Nare?

  It goes back to the example we just talked about previously, but it's all about what is your perspective of the or context by which you're making your calculation.

• 1:08:34Krishna Choudhary?

  Yeah, exactly. And in this case, we can understand what the Jacobian is, because the Jacobian at that point is gonna be, what is the ratio of the square area to the new parallelogram area? Okay, in this case, it looks like 1, because like it's getting squished and this, like the rhombus looks about the same. So in that locality, the Jacobian is 1, but somewhere else, it could be 2, somewhere else, it could be less than 1, and so on and so forth. But that's the beauty of calculus is that you can zoom in and you can figure out a well-defined Jacobian at every given point.

• 1:09:03Lester Nare?

  We have a larger complex system, but at sufficient zoom in, that complex system still comes back to the fundamental that we just talked about.

• 1:09:11Krishna Choudhary?

  Yeah, 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 2 means that the ratio went up, less than 1 means the ratio went down. There's also negative Jacobians, which means that the axes got flipped. Okay, so it's like, I mean, it's just flipped. Yeah, it's like 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?

• 1:09:56Krishna Choudhary?

  And it doesn't collapse. Now here, we've got An example that is at the center of the Jacobian conjecture, okay? We gotta consider situations where the determinant is a non-zero constant. It's a constant everywhere, meaning no matter where I am in my map, 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 I came up with in Desmos. There's a lot of really cool Desmos pre-made applications where you can plug in your own custom linear map

• 1:10:40Krishna Choudhary?

  and you can visualize how the map is gonna change the axis. So in this case, the new x-axis is x plus 1/4 y squared and the y-axis is just the y-axis. So every y gets plotted to its own y, but the x gets shifted by a parabola. That's why these things are becoming a parabola, but like the horizontal lines are remaining horizontal lines.

• 1:11:04Lester Nare?

  It's going from lanes on a highway to a track, the elliptical of a track.

• 1:11:08Krishna Choudhary?

  Yeah, and so and the lanes on the highway is the x, right? So that's why that's getting shifted into a parabola. But 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 squished, 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,

• 1:11:49Krishna Choudhary?

  meaning that the transformation of area everywhere to all of infinity in x and y, the area is preserved, okay?

• 1:12:00Krishna Choudhary?

  Now, given this, one can ask, suppose I was, suppose I was given the end product.

• 1:12:08Krishna Choudhary?

  Okay, if I was given the end product of the, of the last, like, the, the pair, the, the, the points in that, um, after the transformation.

• 1:12:18Lester Nare?

  Yes.

• 1:12:18Krishna Choudhary?

  Suppose I were given the points after the transformation. Can I go backwards?

• 1:12:23Lester Nare?

  Right.

• 1:12:23Krishna Choudhary?

  Meaning, is the map invertible?

• 1:12:25Lester Nare?

  Yes.

• 1:12:26Krishna Choudhary?

  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 gonna call it U and V, because the U is gonna be my new X axis, V is gonna be my new Y axis. If I wanna recover the old X and Y axes that, like, gave me that point, I can just, like, do the mathematics and plug in, it's actually U minus 1/4 V squared. So the, the plus 1/4 becomes a minus. Okay? And then I can recover the old map.

• 1:12:55Lester Nare?

  Is it, is, as a concept to help me grok this, is, is it similar to when you talk about encryption and decryption? When you have the key in the middle?

• 1:13:02Krishna Choudhary?

  Very good.

• 1:13:02Lester Nare?

  Um, you're, you know, you're able to go back, even though it's jumbled.

• 1:13:05Krishna Choudhary?

  Yeah.

• 1:13:06Lester Nare?

  Because you have the key that is, allows you to, okay?

• 1:13:08Krishna Choudhary?

  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 points

• 1:13:27Krishna Choudhary?

  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.

• 1:13:37Lester Nare?

  Right.

• 1:13:38Krishna Choudhary?

  Right?

• 1:13:38Lester Nare?

  Right.

• 1:13:39Krishna Choudhary?

  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

• 1:13:47Lester Nare?

  Right.

• 1:13:47Krishna Choudhary?

  the thing came from.

• 1:13:48Lester Nare?

  You, you need both, you need, you need both.

• 1:13:50Krishna Choudhary?

  Yeah.

• 1:13:51Lester Nare?

  You can't have only one and then zero on the other side.

• 1:13:53Krishna Choudhary?

  Exactly, right? And that happens when the Jacobian is zero. So when the Jacobian is zero, I can't go backwards.

• 1:13:58Lester Nare?

  It's not, it's not invertible.

• 1:13:59Krishna Choudhary?

  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 not zero.

• 1:14:07Lester Nare?

  Right.

• 1:14:07Krishna Choudhary?

  Then I can go backwards.

• 1:14:08Lester Nare?

  That makes sense.

• 1:14:09Krishna Choudhary?

  Okay? Now, we can finally understand the conjecture.

• 1:14:12Lester Nare?

  Okay.

• 1:14:13Krishna Choudhary?

  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?

• 1:14:25Lester Nare?

  Right.

• 1:14:25Krishna Choudhary?

  Everywhere, okay? If the constant is, 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 means like numbers, okay? Mathematicians have other types of rings, but for us, let's just say they're numbers, okay? They have a characteristic of zero. A characteristic of zero, a ring really means something that has addition and multiplication, okay? 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,

• 1:15:09Krishna Choudhary?

  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 nonzero, where like, I add the number 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. In any case, we're talking about normal things here, okay?

• 1:15:39Lester Nare?

  Yep.

• 1:15:39Krishna Choudhary?

  Let K have a characteristic of zero. So this is just real numbers, complex numbers, things like that. And if J F, which is the Jacobian determinant, is a nonzero constant of some functional map, then F, which is the functional map, has an inverse function G that plots my n-dimensional space

• 1:16:00Lester Nare?

  Yep.

• 1:16:00Krishna Choudhary?

  to, back to the n-dimensional space.

• 1:16:02Lester Nare?

  Yeah, yeah, yeah.

• 1:16:03Krishna Choudhary?

  Okay?

• 1:16:03Lester Nare?

  Yeah, yeah.

• 1:16:04Krishna Choudhary?

  And this thing is regular, meaning that its components are polynomials.

• 1:16:08Lester Nare?

  Okay.

• 1:16:09Krishna Choudhary?

  Okay? So if I have a polynomial function that's going in, polynomial meaning things like x cubed plus x squared y, like powers of stuff.

• 1:16:17Lester Nare?

  Yep.

• 1:16:18Krishna Choudhary?

  Then the inverse map is also a polynomial, okay? That is the Jacobian conjecture.

• 1:16:24Lester Nare?

  Okay, so this, this is actually interesting because now, now I can understand the formulation here of what we, like what this open problem

• 1:16:33Krishna Choudhary?

  Yeah.

• 1:16:33Lester Nare?

  is trying to.

• 1:16:36Lester Nare?

  Suggest.

• 1:16:36Krishna Choudhary?

  Yes, it's trying to suggest that anytime.

• 1:16:39Lester Nare?

  You're right.

• 1:16:39Krishna Choudhary?

  I have a function where the Jacobian determinant is a constant everywhere, meaning the the function preserves area, preserves area, no, I shouldn't say preserves area. It scales area, but it scales it in the same way everywhere.

• 1:16:54Lester Nare?

  Yeah.

• 1:16:55Krishna Choudhary?

  Okay, if this function scales it in the same way everywhere, then I should always be able to go backwards.

• 1:17:01Lester Nare?

  I should always be able to decrypt my encryption.

• 1:17:04Krishna Choudhary?

  Yes, okay, as long as it preserves 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 Stephen Smalley's 1998 list of important mathematical challenges, and it has very famously defeated top, top mathematicians, which we're gonna get into.

• 1:17:32Lester Nare?

  Top 20, baby.

• 1:17:34Krishna Choudhary?

  Yeah.

• 1:17:34Lester Nare?

  That's top 20.

• 1:17:35Krishna Choudhary?

  Top 20, top 20.

• 1:17:37Lester Nare?

  That's quite nice.

• 1:17:37Krishna Choudhary?

  Open problems.

• 1:17:38Lester Nare?

  Yeah, that's quite nice.

• 1:17:39Krishna Choudhary?

  Now, July 19th, 2026, in the middle of the World Cup final, okay, Levant Alpugue, he is part of the Harvard Society of Fellows and also one of the researchers at Anthropic, he drops this tweet.

• 1:17:57Lester Nare?

  I saw it within 30 minutes of it going up.

• 1:17:59Krishna Choudhary?

  Yeah, because it was insane.

• 1:18:01Lester Nare?

  It literally 30 minutes.

• 1:18:03Krishna Choudhary?

  Yeah, first of all, it's all lowercase, it's, he's, it looks like he's trolling.

• 1:18:07Lester Nare?

  Yeah, yeah, yeah, yeah.

• 1:18:08Krishna Choudhary?

  You know, hello there, the Jacobian conjecture is false, thanks, and T H A N X.

• 1:18:14Lester Nare?

  This is very.

• 1:18:15Krishna Choudhary?

  Thanks to my close friend Akil, Akil, we're gonna 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 shows the counter example, he just writes it out.

• 1:18:31Lester Nare?

  There's so many things, but I'm not gonna bring it up now.

• 1:18:33Krishna Choudhary?

  And it could fit in a tweet. You know, there's like the hundred character, whatever limit thingy. He just fit the entire proof in a tweet. That's all you need.

• 1:18:42Lester Nare?

  Arguably, if NFTs were still a thing, this NFT would be very valuable.

• 1:18:46Krishna Choudhary?

  Bro, that is, I would buy this NFT.

• 1:18:48Lester Nare?

  This would be so valuable.

• 1:18:49Krishna Choudhary?

  Yeah, I would. We would get this NFT for the MFP vault.

• 1:18:53Lester Nare?

  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 apparently also were not watching the World Cup final at the time. The the immediate

• 1:19:17Lester Nare?

  conversation that arose

• 1:19:21Lester Nare?

  literally within an hour

• 1:19:22Krishna Choudhary?

  Yeah.

• 1:19:23Lester Nare?

  of this being out there was as a bystander, as like an as an innocent bystander, was one of the most fascinating things.

• 1:19:30Krishna Choudhary?

  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 algebra and they like dropped hints of it about how it was unsolved. But it's like fairly easy to understand.

• 1:19:43Lester Nare?

  Yeah.

• 1:19:43Krishna Choudhary?

  Right?

• 1:19:44Lester Nare?

  Yeah.

• 1:19:44Krishna Choudhary?

  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 about 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's talk about that exact solution that it has. Okay? And in the in the next, in the next overlay, we have it typed formatted because within hours it was on Wikipedia.

• 1:20:11Lester Nare?

  Which is.

• 1:20:12Krishna Choudhary?

  Right? You go into the Jacobian conjecture Wikipedia article, it says counter example. And it shows that it's not true.

• 1:20:18Lester Nare?

  You know that meme where the we got him?

• 1:20:20Krishna Choudhary?

  We got him.

• 1:20:20Lester Nare?

  With with what's his name and the balloons and confetti?

• 1:20:24Krishna Choudhary?

  Yeah.

• 1:20:24Lester Nare?

  They did it immediately.

• 1:20:25Krishna Choudhary?

  Yeah, yeah, immediately, right? And this is a counter 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 negative 2, meaning one of the axes is flipped. So it's like getting flipped, but and if you've got a cube of the size 1, it becomes a cube of size 2. 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,

• 1:21:07Krishna Choudhary?

  but two different points are being plotted to the same resulting point.

• 1:21:15Lester Nare?

  Uh huh.

• 1:21:17Krishna Choudhary?

  Meaning if you gave me that resulting point, I couldn't tell you which of the two it came from.

• 1:21:23Lester Nare?

  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 because two points point to the same place.

• 1:21:36Krishna Choudhary?

  It could be one of two things. So it's 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.

• 1:21:47Lester Nare?

  Right.

• 1:21:47Krishna Choudhary?

  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 I could show you my work. It's in the next one.

• 1:22:00Lester Nare?

  Yes.

• 1:22:01Krishna Choudhary?

  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 coordinate is some seven degree polynomial that has, that mixes X, Y, and Z. The new Y coordinate is another, I think, seven degree polynomial. And then the, the, the Z coordinate is not a seven degree polynomial. It's a little bit simpler, but you, I've got this map where X, X, Y, and Z goes to a new X, new Y, new Z. I calculated the derivatives, which are on the left. So that becomes a matrix, which you see on the top right.

• 1:22:34Lester Nare?

  Right.

• 1:22:34Krishna Choudhary?

  And then I did a little bit of cheating because I decided I didn't want to multiply all those polynomials. 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, I still got it. So then, so then I went on Wolfram Alpha and I just multiplied all of those polynomials to calculate the determinant. And the determinant is negative 2. And then I went ahead and plugged in those two numbers. And they went to the same number, negative one fourth, zero, zero. There you go. I did it on a single piece of paper, plus, okay, a little bit of Wolfram 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.

• 1:23:16Lester Nare?

  This is so interesting because, you know, and, 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. And the nature of the disproof.

• 1:23:36Krishna Choudhary?

  Yeah.

• 1:23:37Lester Nare?

  If that's the right naming.

• 1:23:38Krishna Choudhary?

  The counter example.

• 1:23:39Lester Nare?

  The count, sorry, the counter example.

• 1:23:42Lester Nare?

  You, it does not, if anyone can do it and see that it is true, going back to our point about mathematics.

• 1:23:49Krishna Choudhary?

  Yeah.

• 1:23:49Lester Nare?

  And the objective nature of it.

• 1:23:51Krishna Choudhary?

  Yeah, exactly. I, I did it. Anyone can do it. Right? This is something that algorithmically, there's no AI here.

• 1:23:57Lester Nare?

  Right.

• 1:23:58Krishna Choudhary?

  Right? It's just, it's an algorithmic process of taking derivatives, taking the determinant of the Jacobian. There you go. Boom. There's a constant determinant and two points point to the same thing. So I can't go backwards.

• 1:24:09Lester Nare?

  I can't go backwards and it's not invertible, which means the Jacobian conjecture fundamentally not a real thing.

• 1:24:15Krishna Choudhary?

  Is not a real thing, right?

• 1:24:17Lester Nare?

  It's not a real thing.

• 1:24:18Krishna Choudhary?

  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.

• 1:24:28Lester Nare?

  Yeah.

• 1:24:28Krishna Choudhary?

  Because I can just take, like, uh, consider the trivial map where I take a function of four dimensions. Um, let's say X, Y, Z and T. T is our fourth dimension in Allah relativity. X, Y and Z do what that function was doing and T just points to T.

• 1:24:45Lester Nare?

  Mm-hmm.

• 1:24:45Krishna Choudhary?

  Right? So the time doesn't change. That is going to have the, the same Jacobian determinant and the things 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 five, N equals six, any dimension greater than three now. Which, so this is, this proved all dimensions greater than three.

• 1:25:05Lester Nare?

  Right. So the, it's only true in two.

• 1:25:08Krishna Choudhary?

  Yeah. Now it's like, doubt's an open question.

• 1:25:10Lester Nare?

  Uh, right. Cause we don't know.

• 1:25:11Krishna Choudhary?

  We don't know.

• 1:25:11Lester Nare?

  Ah, ha, ha.

• 1:25:13Krishna Choudhary?

  Right?

• 1:25:13Lester Nare?

  Ha. So now the Jacobian conjecture is, yes, it's just only in two.

• 1:25:17Krishna Choudhary?

  Only in two dimensions. Now it's still an open problem.

• 1:25:20Lester Nare?

  Open problem, not, not yet solved. Okay. But the, but that's interesting.

• 1:25:24Krishna Choudhary?

  But like for, for all the infinity of dimensions above.

• 1:25:27Lester Nare?

  Which is like, really. Much more meaningful.

• 1:25:30Krishna Choudhary?

  Yeah, yeah, it's crazy.

• 1:25:31Lester Nare?

  No, this, and I think what's so interesting is that, again, and 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, it's just, oh, was, you just ask Fable, the latest frontier model from Anthropic, to just brute force it. And if you look at the amount of time required to brute force, it's just, it's like, to the power of.

• 1:26:06Krishna Choudhary?

  To the power of crazy. And that's actually my next point.

• 1:26:10Lester Nare?

  Okay.

• 1:26:10Krishna Choudhary?

  We can actually, we can actually ask, did Fable just brute force?

• 1:26:13Lester Nare?

  Right, right.

• 1:26:14Krishna Choudhary?

  Did it just do like supercomputer nonsense?

• 1:26:17Lester Nare?

  Because they have a lot of compute.

• 1:26:18Krishna Choudhary?

  Yeah, and, and there's history behind supercomputer nonsense, okay? So let's talk about, like, this is a counter example.

• 1:26:23Lester Nare?

  Okay.

• 1:26:23Krishna Choudhary?

  Okay, and historically, computation has been very good at producing counter examples, okay? Way back in the 1960s, actually. So let's talk about Euler's conjecture. Euler's conjecture is the following, that, you remember, we were talking about like Fermat's Last 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. Trivial example, three squared plus four squared equals five squared, because nine plus sixteen equals twenty-five. Plato, way back in the day, had found another kind of Pythagorean-looking thing, where he showed that three cubed plus four cubed plus five cubed is actually six cubed, which

• 1:27:08Krishna Choudhary?

  is kind of cool.

• 1:27:09Lester Nare?

  Yeah.

• 1:27:09Krishna Choudhary?

  It's called, it's called Plato's Number.

• 1:27:13Krishna Choudhary?

  So, all right, the Euler looked at this, looked at Plato's Number, and he said, well, look, if. If if I need, in that case, I needed three cubes.

• 1:27:24Lester Nare?

  Yes.

• 1:27:25Krishna Choudhary?

  To get to another cube.

• 1:27:26Lester Nare?

  Yeah, right.

• 1:27:26Krishna Choudhary?

  Okay.

• 1:27:27Lester Nare?

  Right.

• 1:27:27Krishna Choudhary?

  So Euler is like, what if I need N to the Nth power?

• 1:27:30Lester Nare?

  Right.

• 1:27:31Krishna Choudhary?

  To get to another Nth power. I need at least N things to the Nth power to get to another Nth power. So if I wanna, if I want something like A to the fourth plus B to the fourth plus C to the fourth, I would need another D to the fourth in order to get to an E to the fourth.

• 1:27:46Lester Nare?

  He was trying to generalize.

• 1:27:47Krishna Choudhary?

  Yes, he's trying to.

• 1:27:48Lester Nare?

  Play this number.

• 1:27:49Krishna Choudhary?

  Exactly. He's trying to generalize, because Pythagorean theorem already happened, right? Three squared plus four squared equals five squared. 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 Euler's conjecture, remains untrue for a very long time. Lo and behold, Lander and Parkin, in a very, very famous mathematical paper, one of the shortest math papers of all time, right? They're like, direct search with the CDC 6000, 66,000, which is one of the first successful supercomputers. They just brute forced it.

• 1:28:25Lester Nare?

  Yep.

• 1:28:26Krishna Choudhary?

  And they found a counter example.

• 1:28:30Krishna Choudhary?

  27 to the fifth power plus 84 to the fifth power plus 110 to the fifth power plus 133 to the fifth power equals 144 to the fifth power. So I only needed four fifth power thingies to get to a fifth.

• 1:28:42Lester Nare?

  We don't like that.

• 1:28:43Krishna Choudhary?

  Yeah, and there was just like, it's the smallest instance in which five fifth powers. There you go.

• 1:28:49Lester Nare?

  Sorry.

• 1:28:50Krishna Choudhary?

  Sorry, Euler, you're wrong. Which is crazy to say.

• 1:28:52Lester Nare?

  Right, right.

• 1:28:53Krishna Choudhary?

  Euler is wrong.

• 1:28:56Lester Nare?

  But, but this, this is a perfect point. This is why people immediately go to, again, you know, the.

• 1:29:01Krishna Choudhary?

  Oh, they just brute forced it.

• 1:29:02Lester Nare?

  Yeah, right.

• 1:29:03Krishna Choudhary?

  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.

• 1:29:16Lester Nare?

  And it happened to be that.

• 1:29:17Krishna Choudhary?

  And the search space is small enough where with a big enough supercomputer, I can do it.

• 1:29:22Lester Nare?

  And this was some, some time ago.

• 1:29:24Krishna Choudhary?

  Yeah, this was in the 60s. And now we've found more and more exceptions of Euler'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.

• 1:29:36Lester Nare?

  Yes.

• 1:29:37Krishna Choudhary?

  But let's consider, could Fable have 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 of polynomials, right? There's x, y, and z. So let's just do like, okay, it found like the seven degree polynomial, right? There was an x to the seven there.

• 1:30:03Krishna Choudhary?

  If you just think about seven degree polynomials and seven degree maps. So you can have seven degrees in x, seven degrees in y, seven degrees in z, right? And each of those can have different numbers, right? So that's, first of all, that's 343 different polynomial terms. Each of those polynomial terms can then make up 1,300 cubic equations. And then the coefficients of that equation, right? Is it 1 x squared y? Is it 2 x squared 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 energy range of, let's say, negative 10 to 10, and try all of the different coefficients,

• 1:30:45Krishna Choudhary?

  that would be something like 10 to the 475 candidates, right?

• 1:30:49Lester Nare?

  Right.

• 1:30:49Krishna Choudhary?

  Even like a million qubit quantum computer, if we were to create a quantum algorithm that could somehow do this, which, I'm just, I'm just saying, we don't even have one.

• 1:30:59Lester Nare?

  For another day.

• 1:31:00Krishna Choudhary?

  Yeah, that's for another day. We don't even have one. But like, you can't, you can't do that. This is not a brute force thing, right? Like, Fabel had to understand something about the structure of polynomial maps, understand something about the properties of polynomial maps where the Jacobian is constant, and then look for places where things would overlap. Because if the Jacobian is constant, that means that any given, like, infinitesimal area is, is being put into the same thing. So it's not just trivially that like,

• 1:31:33Lester Nare?

  Right.

• 1:31:33Krishna Choudhary?

  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.

• 1:31:46Lester Nare?

  I think what's been interesting to hear about the discussion of open problems in AI systems, and this is, I think, is a perfect example. There was a discussion, it's gonna escape me who it was with, but there were sort of three, there are three people in the field who were coming from three different perspectives about

• 1:32:05Lester Nare?

  AI's potential in particularly this idea of novelty. Um, or, or scientific discovery as a general not well-defined term. And one of the arguments was that even if

• 1:32:22Lester Nare?

  a system like Fable has scooped up every math paper ever.

• 1:32:29Krishna Choudhary?

  Yeah.

• 1:32:29Lester Nare?

  Right? It also has the benefit of having every physics paper, every biology paper. And because of the way structurally, um, 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.

• 1:33:05Krishna Choudhary?

  Yeah.

• 1:33:06Lester Nare?

  And simply by just having that same level of expertise in a cross-functional, in a cross, a cross-field way.

• 1:33:16Lester Nare?

  Cause a lot of science is making these connections that are non-obvious. The ability for these systems to make connections that are non-obvious is just exponential.

• 1:33:24Krishna Choudhary?

  Yeah.

• 1:33:24Lester Nare?

  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.

• 1:33:43Krishna Choudhary?

  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 some insight into how it got to this particular functional map.

• 1:34:01Lester Nare?

  Right.

• 1:34:02Krishna Choudhary?

  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 Feynman 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.

• 1:34:31Lester Nare?

  And so in

• 1:34:33Krishna Choudhary?

  And that's why I'm like freaked out kind of.

• 1:34:35Lester Nare?

  So, and this, I think this is an important point because I wanted to lay that out there.

• 1:34:39Krishna Choudhary?

  Yeah.

• 1:34:40Lester Nare?

  To this exact note, which is like, even if that's true, what I just walked through.

• 1:34:45Krishna Choudhary?

  Yeah.

• 1:34:46Lester Nare?

  What we're talking about with this solution to the Jacobian conjecture seems to be outside of that, potentially outside of that sandbox.

• 1:34:53Krishna Choudhary?

  Yeah, yeah.

• 1:34:54Lester Nare?

  Which

• 1:34:54Krishna Choudhary?

  It seems.

• 1:34:55Lester Nare?

  It seems.

• 1:34:55Krishna Choudhary?

  I don't know, I don't know enough. And I'd love to talk to Levant.

• 1:34:59Lester Nare?

  Even the potential.

• 1:35:00Krishna Choudhary?

  Yeah.

• 1:35:01Lester Nare?

  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.

• 1:35:11Krishna Choudhary?

  Yeah.

• 1:35:11Lester Nare?

  Is, um, you know, and people, part of it's the, you know, the Levant who discovered it is obviously one of the world's greatest mathematicians, has been so for a long time.

• 1:35:20Krishna Choudhary?

  Well, but he's not.

• 1:35:21Lester Nare?

  Oh, he's not?

• 1:35:21Krishna Choudhary?

  No.

• 1:35:22Lester Nare?

  Oh.

• 1:35:22Krishna Choudhary?

  I mean, look, no offense to Levant.

• 1:35:24Lester Nare?

  Oh, look, I, I don't want that heat.

• 1:35:26Krishna Choudhary?

  Look, look, I love this Levant. 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?

• 1:35:38Lester Nare?

  That's fair.

• 1:35:38Krishna Choudhary?

  Um, and, but he, he knows clearly, I mean, he's, he's a researcher in anthropic and he's very good at understanding how to use and leverage anthropic, right?

• 1:35:48Lester Nare?

  Right.

• 1:35:48Krishna Choudhary?

  And clearly he knows a lot about mathematics.

• 1:35:50Lester Nare?

  Yes.

• 1:35:50Krishna Choudhary?

  Right? But this isn't like someone like Peter Sarnak, that's like a number theorist using this stuff, right? And That's why I really want to know what the, what the, um, what the prompt was.

• 1:36:02Lester Nare?

  Which, which almost makes it again even more.

• 1:36:05Krishna Choudhary?

  Yeah, um, like he's clearly gifted, but he's not at, at that top level that's getting fields medals.

• 1:36:10Lester Nare?

  Fascinating. I just, there's so many angles to this story that I think are, you know, again, a lot of these CEOs and people around who are at the frontier of these systems who have unlimited compute with them, by the way.

• 1:36:24Krishna Choudhary?

  Yeah.

• 1:36:24Lester Nare?

  Which is part of, part of the value out 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.

• 1:36:33Krishna Choudhary?

  It is.

• 1:36:34Lester Nare?

  And everyone's kind of been like, yeah, yeah, show me.

• 1:36:37Krishna Choudhary?

  Yeah. Well, yeah. And there's several examples, right? Now, before we get into the other examples, one thing I wanted to talk about was like how humans have been approaching this problem.

• 1:36:47Lester Nare?

  Right.

• 1:36:47Krishna Choudhary?

  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 polynomial of degree like 100, like X to the 100 plus to X to the 99, da, da, da, da, da, right?

• 1:37:26Lester Nare?

  Right.

• 1:37:26Krishna Choudhary?

  And so we've been focusing on this like highly complicated thing. But here Claude Fable 5 proves that a low degree, degree seven, it's just in a higher dimension.

• 1:37:36Lester Nare?

  Yeah.

• 1:37:37Krishna Choudhary?

  Dimension three.

• 1:37:38Lester Nare?

  Right.

• 1:37:38Krishna Choudhary?

  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 three, sorry, in dimension three.

• 1:37:50Lester Nare?

  Yes. Yeah. Which, which goes back to then what is the source material for the insight?

• 1:37:54Krishna Choudhary?

  And, and historically, there's, there's kind of an irony. If you look at, um, Yiteng Zhang, he's a very famous number theorist. He's one of the first guys to bring down the prime gaps to something 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 was like, ah, right? But his story is insane because like during his PhD at Purdue University, he was under the advice of Tsong Sing Mo.

• 1:38:40Krishna Choudhary?

  And his work was on this very conjecture. It led to years of hardship. He couldn't get anywhere.

• 1:38:47Krishna Choudhary?

  It's a really kind of a sad story because his dissertation relied on this auxiliary corollary about the conjecture that was proved by his advisor. Later discovered to be false under peer review. 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, his advisor was kind of pissed off about that.

• 1:39:10Lester Nare?

  Yeah.

• 1:39:10Krishna Choudhary?

  He couldn't get a recommendation letter and he was, he was a subway worker, minimum wage, and he lived out of his car.

• 1:39:18Lester Nare?

  Jesus Christ.

• 1:39:19Krishna Choudhary?

  For a while before he proved the twin, the 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's maybe can save mathematicians' time. But at the same time, it's like, how much of this is AI going to do to the point where our mathematicians going to be left with a 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 OpenAI model disproved the ESD planar unit distance conjecture. That was, that was conjectured by Paul ESD in 1946. This is the proof for it.

• 1:40:03Krishna Choudhary?

  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? Ersch showed that there's like an upper bound on how many points I can pack with this amount of distance. And the OpenAI 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 human verified version of that counter example. Again, it's a counter

• 1:40:45Krishna Choudhary?

  example. But here, the authors are a who's who of mathematicians. We've got Timothy Gowers from Trinity College at the University of Cambridge in the UK. 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 Polymarket, it's all collapsed to 99% for people. That's like, okay, somehow it got leaked and now no one's like betting anymore. Jacob Zimmerman, actually, I think he was my, he was my TA in freshman year.

• 1:41:18Lester Nare?

  Oh, we love that.

• 1:41:18Krishna Choudhary?

  Yeah.

• 1:41:19Lester Nare?

  We love to see it.

• 1:41:20Krishna Choudhary?

  Because he was a PhD student of Peter Sarnak.

• 1:41:21Lester Nare?

  Hey, come on the show. We'd love to talk to you about it.

• 1:41:25Krishna Choudhary?

  Yeah, we'd love to talk to you about how you, how you gave me a, never mind. Let's not talk about how you graded my midterm. But, you know, so this is a who's who and what they're doing is taking the work of OpenAI and trying to make it human digestible. So is that what mathematicians are gonna just end up doing?

• 1:41:42Lester Nare?

  Right, right, right. Being translators of the machine genius.

• 1:41:45Krishna Choudhary?

  Yeah. And funny thing about this, right? So OpenAI solved that Ersch conjecture. Levant, our hero from this story at Anthropic, he tweeted, again in all lower caps, he's like, "Over the weekend, I checked the obvious thing, which is whether Mythos is able to solve the Ersch unit distance problem, aka Ersch problem. The answer is yeah."

• 1:42:11Lester Nare?

  I love this.

• 1:42:13Krishna Choudhary?

  I think he's just trolling. It's just all lowercase.

• 1:42:18Lester Nare?

  Gen Z.

• 1:42:19Krishna Choudhary?

  Yeah.

• 1:42:20Lester Nare?

  Capitalization language.

• 1:42:20Krishna Choudhary?

  And then his colleague is like, you know, huge, huge credit to the OpenAI team. And but he's like retweeting, but he there are colleagues and he's like trying to make it a little bit more.

• 1:42:31Lester Nare?

  Yeah, yeah, yeah. Corpse speak.

• 1:42:32Krishna Choudhary?

  Yeah, yeah, yeah. But but it's just so funny. I think the guy's just trolling.

• 1:42:37Lester Nare?

  This is so, there's so much here. Sorry, continue though.

• 1:42:39Krishna Choudhary?

  Yeah. Um, and the final thing I want to end with is this is clearly now becoming an existential crisis for mathematics. Right? 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 gonna lie. It makes me happy that we're we're seeing such obvious counterexamples for things and like stuff that I can understand. And, you know, who knows, we'll maybe find a proof for the Riemann zeta. Or maybe we'll find a counterexample for the Riemann zeta hypothesis where they it's gonna just come up with a zero for the Riemann function. That's like, I don't know, two billion on the one half real line. Who knows, right?

• 1:43:21Krishna Choudhary?

  Probably not, because every mathematician that I know says that the Riemann hypothesis is true. But who knows, right? At this point, it would be kind of cool to find it because that I could also check.

• 1:43:31Lester Nare?

  Yeah.

• 1:43:32Krishna Choudhary?

  Like, not by hand, maybe, because I'd have to do it a bunch of times, but I could probably check on Wolfram.

• 1:43:37Lester Nare?

  Yeah.

• 1:43:37Krishna Choudhary?

  Right? So it's it's it's concerning. It's concerning, to say the least, for pure mathematics. And in June 2026, there was a statement from the Laurent Center regarding AI-generated proof. It was called the Light and 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.

• 1:44:15Lester Nare?

  Yes. Because they're closed source.

• 1:44:17Krishna Choudhary?

  Yeah, they're closed source, so I don't know what's happening inside. It's not really open. I'd have to pay to play. Um, it's unsustainable journal.

• 1:44:27Krishna Choudhary?

  Review backlog reviewer backlogs because you can just now like start making all sorts of proofs, submit it to the analysis of mathematics and then what like the humans are going to and then now what we get AI reviewers.

• 1:44:39Lester Nare?

  Yeah.

• 1:44:39Krishna Choudhary?

  Seems I don't know.

• 1:44:41Lester Nare?

  We don't like that.

• 1:44:41Krishna Choudhary?

  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 theory. What are we going to do? And they made some recommendations. One was mandatory AI disclosure. We love disclosure.

• 1:45:00Lester Nare?

  We love disclosure.

• 1:45:01Krishna Choudhary?

  Here on this podcast. Enhance scrutiny and there's proper attribution for human research. Where did these AI models get the inspiration, right? We'd like to know in our workings.

• 1:45:12Lester Nare?

  Bingo.

• 1:45:13Krishna Choudhary?

  It's going to be tough because a lot of these LLMs are black boxes at the end of the day. Claude 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? They're also trying to go for independent funding from these industries to support mathematicians because mathematics, like you just covered a story about how fundamental research funding is down. Well, mathematics 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?

• 1:45:53Krishna Choudhary?

  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. Allah Halo.

• 1:46:11Krishna Choudhary?

  Saw the forerunners and saw, you know, Fermat's Last Theorem embedded in rock. They would be like, I kind of understand that, right? If they saw like squares and Pythagorean theorem like.

• 1:46:21Lester Nare?

  Yeah.

• 1:46:21Krishna Choudhary?

  That would be so cool.

• 1:46:23Lester Nare?

  Yeah.

• 1:46:23Krishna Choudhary?

  Because they have their own versions of that.

• 1:46:25Lester Nare?

  Yeah.

• 1:46:25Krishna Choudhary?

  Yeah. And so like, I don't want this to be a death sentence for mathematics. It's really.

• 1:46:35Krishna Choudhary?

  I really hope that's not what's happening.

• 1:46:37Lester Nare?

  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.

• 1:46:50Krishna Choudhary?

  Yeah.

• 1:46:51Lester Nare?

  Because the way these systems are built and work requires this as a discipline that is understood very well.

• 1:46:56Krishna Choudhary?

  Yes, logic, correctness.

• 1:46:58Lester Nare?

  Yes.

• 1:46:58Krishna Choudhary?

  Truth.

• 1:46:59Lester Nare?

  And you cannot, we cannot say, oh, we'll have AGI and artificial general and artificial superintelligence. 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 Sir Demis Hassabis has said, we are on the shores, he has a great phraseology for this. Oh, we are on the shores of the singularity, or we are on basically the sort of the dawn of the singularity here. And I know Ray Kurzweil is known as kind of the modern proponent for this concept.

• 1:47:44Lester Nare?

  No one has ever been able to describe in words what it would feel like to be right there when the flip was about to switch. And to me, even as a layman, you can start to feel the ground moving.

• 1:48:03Krishna Choudhary?

  Yeah.

• 1:48:03Lester Nare?

  Underneath our feet. It's not to say the killer robot outcome in 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 here now, what does that mean? And there are so many ways in which mathematically, scientifically, engineering-wise, socially, politically, economically, religiously, all of the E's.

• 1:48:36Lester Nare?

  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.

• 1:48:49Krishna Choudhary?

  Yeah.

• 1:48:49Lester Nare?

  A disabled.

• 1:48:49Krishna Choudhary?

  You come from wealth.

• 1:48:50Lester Nare?

  Yeah, like, you know, but it is, I know, I know that everyone is tired of hearing about AI. I, I know that, I get it, I totally understand it. However, um, if we could have stopped the bankers from doing oh eight before it happened, would people say no to that?

• 1:49:09Krishna Choudhary?

  Yeah.

• 1:49:10Lester Nare?

  And it feels like a similar situation where there's a runaway train here and we need to do something.

• 1:49:17Krishna Choudhary?

  Yeah. Yeah.

• 1:49:18Lester Nare?

  I don't know what the answer is.

• 1:49:19Krishna Choudhary?

  Yeah, and maybe it's not to stop the progress, because progress is always good.

• 1:49:23Lester Nare?

  Right.

• 1:49:23Krishna Choudhary?

  But we gotta, we gotta talk about it.

• 1:49:26Lester Nare?

  And we have to engage in it, and we have to have these discussions, and...

• 1:49:30Krishna Choudhary?

  And we gotta have some kind of policy, right? It can't just be, oh, it's like too late.

• 1:49:34Lester Nare?

  Right.

• 1:49:34Krishna Choudhary?

  Because everyone's unemployed.

• 1:49:36Lester Nare?

  Because it's not.

• 1:49:36Krishna Choudhary?

  And now let's try to solve it.

• 1:49:38Lester Nare?

  Because right now, it's not too late, but very quickly, it might be too late. For mathematicians, or for some of these industries, a lot of, 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 making this a larger philosophical and sociological point.

• 1:49:57Krishna Choudhary?

  Yeah.

• 1:49:57Lester Nare?

  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.

• 1:50:23Krishna Choudhary?

  Yeah.

• 1:50:24Lester Nare?

  Especially with taxpayer-funded money.

• 1:50:26Krishna Choudhary?

  Yeah.

• 1:50:27Lester Nare?

  This was so good, dude, like.

• 1:50:29Krishna Choudhary?

  Yeah.

• 1:50:30Lester Nare?

  This was really, really good. I can't believe I can go around being like, oh yeah, the Jacobian, yeah.

• 1:50:34Krishna Choudhary?

  Yeah.

• 1:50:34Lester Nare?

  It equals one.

• 1:50:35Krishna Choudhary?

  Yeah, it's just, it's a ratio of, uh, of areas.

• 1:50:39Lester Nare?

  Multi-dimensional.

• 1:50:40Krishna Choudhary?

  You should say volumes, yeah, volumes. Because, because to mathematicians, an area is really a 2D volume. Oh, those mathematicians, they crack me up.

• 1:50:48Lester Nare?

  We love them. We, this was, again, we're having a problem staying under two hours. We're approaching two hours again this episode. One, it's because we love you 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, Alpha Go,

• 1:51:23Lester Nare?

  the original, I can't remember the name, the points in the grid one that OpenAI did, not that long ago.

• 1:51:29Krishna Choudhary?

  No, that was this year.

• 1:51:31Lester Nare?

  That was.

• 1:51:32Krishna Choudhary?

  Yeah. It's all happening this year.

• 1:51:34Lester Nare?

  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,

• 1:51:41Lester Nare?

  we need a quote, a comment for our, our deep, deep audience.

• 1:51:45Krishna Choudhary?

  Make fun of mathematicians. I love doing it as a physicist. Just, just say something that's funny about mathematicians.

• 1:51:53Lester Nare?

  I am your host, Lester Nare, joined as always by my co-host, our Princeton undergraduate physics, UCLA PhD physics, occasional mathematician, Krishna Choudhary. Thank you all for joining us again. We are coming up on our one-year anniversary. It'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.