The Road to the Claimed Breakthrough

2:13:52>> Now there's been a lot of work since um the the days of Navier Stokes to try and understand the mathematical properties of this. The first guy I want to talk about is Jean Lay. Um he actually proved that fluids always possess something called weak solutions where the total kinetic energy of the system remains bounded over time. How do you compute kinetic energy? Like kinetic energy is really just um 12 mv^2 right for like a normal pointlike object like you know how Newton's laws can be applied to a point lock object f= ma well mv^2. For a velocity field it's a little bit different right? What we have to do is integrate over a volume because we've got a bunch of velocities in a volume

2:14:34and we say the square of all of the velocities in that volume times 1/2. That's going to tell you sort of the energy density in that volume like the kinetic energy density in that volume >> which would be in this case the velocity field. >> Yeah. Yeah. Exactly. And so um the kinetic energy density can remain finite for these weak solutions. And by weak solutions, these are stuff that don't have this like strong smoothness criteria. And I'm going to be honest at this point, I'm getting into stuff that this is this is the math stuff. This is maybe why I went on a tirade about 5 minutes earlier about how this is all math because maybe there's stuff now that I'm like, okay, I I don't really

2:15:14know, right? But from what I've gathered, what what he's done is loosen the idea of smoothness and show that like, you know, imagine you've got like a violent storm system, >> he showed that the total energy of that storm can't suddenly go to infinity. >> Okay. But what what he he can't rule out is that even though the entire energy density of that storm doesn't go to infinity, you can have small pockets where the velocity is still going to infinity. Okay? you can concentrate into an infinitely fast like tornado somewhere inside and you can't rule that out. >> Mhm. >> So the overall energy is finite but some local small velocity can be infinite. >> Y >> so already it's it's starting to show

2:15:54that maybe there could be ways that the convective term wins. >> Mhm. >> Okay. 1982 Kafarelli Con and Nuremberg they show that if the Navier Stokes equations for a velocity field work then you have partial regularity what that means is if you have singularities if they exist at all they can't be like everywhere >> okay you can't have giant chunks of fluid moving at infinite speed but what you could do is have infinite decimally small pointlike singularities kind of like at the center of a black hole and that works. Okay. And and so the idea

2:16:35the system has some fundamental limitation where it has to it can only happen at particular scales >> particular scales a very small infinite decimal scales. This is why again it's like a mathematical problem right it's not it's it's infinite decimally small scales and it's happening only at particular places. It's not like a continuous singularity right it's like it's at the center of a black hole. It's a single spot. >> Okay. So that's them saying, okay, partial regularity. In other words, you've got these decades of work that is placing a very tight fence around any possible singularity, but we haven't proved that that fence has nothing in it. There could still be a singularity. It just has to be constrained to these things. And so the the the the box in

2:17:18which the convective term wins, >> we've now created a very small box. Yes, >> it can still win. It could still win against viscosity, but it's a very small box. >> Very small box, right? And this brings us to 2000. This is when the Clay Mathematics Institute announces the Millennium Prize problems in mathematics. These are seven very famous mathematics problems that were formerly Hilbert's problems um that you know bunch of them got solved over in the 20th century. Some by John Nash of course um and others. And finally, we have um seven out of those that were picked as like the big ones. The point

2:17:58car conjecture has been solved already by Gregory Pearlman in 2003. >> Um and that was huge. Um he never accepted the mill the million dollars cuz um he said that someone else deserved it. He's kind of like this weird guy. Let's let's I'm going to be honest. He's a weird Russian guy. Brilliant mathematician, but like lives kind of like in in his mom's basement kind of thing. Yeah. Anyways, brilliant mathematician. Um, so here's the problem statement and this is a problem statement by Charles. >> I want to make one quick note here which is uh for those viewing who can see the list, I want you to remember this list, right? Uh because this list is going to

2:18:38come into relevance when we talk about the race between the frontier labs >> and this is the gold medal that they're all chasing after. Yeah. is and infinity stones >> and these are the infinity stones and they are coming after all of these maybe not Yang Mills and maybe not P versus MP yeah but all the other ones all the other ones remon hypothesis they came after it um or at least something that is related >> related >> um Navia Stokes is the one that we're at right now um Yang Mills I mean we we did Yang Mills for an episode last year that's one of my favorite episodes check it out if you haven't and so here is the problem statement so each of the each of these each of these problems have to be well defined, right? Like when the clay

2:19:18mathematics institute is like this is the problem, the problem has to be well defined mathematically. So Charles Ferman um field medalist um also a student of Elias Stein at Princeton University the guy who wrote the complex analysis textbook and my professor of complex analysis also. So he's a brother I guess an academic brother of Terrence Tao. >> Okay. >> Because they both had the same PhD adviser also fields medalist. I I just want to note for those who might not know Fields Medalist meaning that they've won this the the most prestigious mathematics award that is available >> for those who are under 40. >> For those who are under 40 >> yes I would say the most prestigious

2:19:58mathematics award that people don't really know about is the Abel Prize. >> We talk I hope >> which is which is I think like the Nobel Prize of mathematics. People keep saying it's the Fields Medal. I really don't think it is. Um >> argue about it in the comments. >> Yeah. Um so Charles Fefferman puts out this statement of what it is that we are trying to prove. Okay. >> Mhm. >> We want the existence and smoothness of Navier Stokes on R3. That means in three dimensions. Two dimensions we've already figured it out. Olga said it's smooth. We're good to go. Okay. A and B say existence and smoothness of Navia Stokes in um three dimensions. C and D say the breakdown of Navier

2:20:41Stokes in three dimensions. And so and just to just to quickly touch on this. So we're saying A and B means viscosity wins. >> Yes. >> And C and D means the convective term wins. >> Exactly. Yes. Okay. >> So A and B say it's always going to be smooth. >> C and D say it's >> going to lose. And there the in the small fence that we've already defined the convective term ultimately wins and we get a a very discreet small singularity somewhere. >> Exactly. And so in A and B like that's a proof of global regularity and that would mean you need to identify some kind of mechanism that prevents this infinite concentration at these very small scales right in practical terms

2:21:23one would like like some kind of quantity or some kind of scale critical quantity that's controlled for all time and it's some structural argument saying that whatever vortex stretching is happening that we were talking about it's going to lose out to viscosity. Okay. >> A proof for the breakdown would be the opposite. You just have to construct a genuine Navier Stokes flow. You have to construct some kind of initial condition in which the concentration reinforces itself. That convective term keeps making smaller and smaller things and all of that pressure and nonlinear cancellations. After all of that, the mathematics becomes demonstrabably something like a singularity. to just not that to make an analogy

2:22:05because this is sort of how people always frame these things when they talk about AI's progress as it relates to these things which a year ago people didn't think it could compete in the math Olympiad and now it's solving C and D of a millennium problem but putting that aside um it almost seems based on this construction we've just described that it's kind of akin to a counter example >> yes 100% >> versus sort of the global solution >> not minimizing in any way cuz it's still a big deal. But like people will nuance and nitpick on that point. It's like great you found like one >> you're looking at this large search space and you found something that has

2:22:46blown up in finite time. >> Well, it's an incredibly small search space, right? And everyone's been looking at it, >> right? But I think I think you rais a very big point and I think we're going to come to that in the next few examples that I'm showing you because in 2000 this is when the this is when the clay mathematics institute came up with it right >> in 2016 big players are involving themselves in this because they smell the blood in the water >> that there is a counter example hanging out somewhere. >> Okay. more and more analysis is going into showing that perhaps there are situations where you get singularities.

2:23:26So Terrence Tao he creates um he creates this averaged version of the Navier Stokes equations and he proved that the standard energy conservation laws are mathematically too weak to prevent a singularity. So he created this like averaged version of the Navier Stokes equations. It's kind of like a blurred out version. Maybe you can think about that. It's like a a stunt double of the fluid equations. And from the outside, this thing obeys all of the exact same conservation laws that the real Naver Stokes equations does. But Tao proved that this stunt devil definitively explodes in finite time. Okay? So this averaged version can create

2:24:07singularities which means that the principles that you would have used to prove that it wasn't possible are also not going to work for the real thing. If if if it was global you wouldn't be able to have an average stunt double that blew up. >> Yes. Exactly. So there's a chance now that at least the original arguments that people were using saying oh energy conservation and things like that that might not work anymore. Okay. So A and B are already not looking great, right? Maybe there is a singularity that's lurking somewhere. >> Okay. And for decades, mathematicians tried to trigger this blowup using something like a fractal shape, like a whirlpool that's shrinking into smaller

2:24:48and smaller whirlpools and so on. Um, this usually failed because it required an infinite amount of energy. And crucially, you can't do that. One of the things the ferman said is you can't just like stir it infinitely fast and be like, "Oh, I got an infinity." >> Right? That's not the interesting question we're trying to ask, >> right? Okay. So, you supplied an infinite amount of energy and you got an infinite amount of velocity. Great. >> Great. Right. >> No, it's supplying a finite amount of energy to get an infinite amount of velocity because the nonlinear convective term is concentrating energy into smaller and smaller structures. So both finite energy and finite time are both relevant in the context >> 100%. And so um Diego Cordova and Luis

2:25:30Martinez Zora and Fan Jen in 2023 and 2025. This is very recent. They changed the paradigm by abandoning this whole single shape thing and relying on something called a layer cascade. It's kind of like a bunch of nested gears or um you know those Russian dolls where you have like a big doll and then smaller doll and smaller doll and smaller dolls. That's what they were constructing. Instead of a single vortex that they do this vortex stretching in, they've got slowmoving outer vortex and then a medium inner vortex and then a smaller inner vortex that spins faster because of the medium vortex. And the medium vortex is kind of like a parlay or like not a parlay like an

2:26:11intermediary between the outer and the inner and it acts upon an even smaller layer and so on and so forth. And you can design this geometry so that the feedback between these gears doesn't disrupt the big gears >> and the feedback between the smaller gears goes up up and up and you can get unbounded velocity. >> Mhm. >> Okay. And if you have an infinite number of gears then perhaps you can have an infinite velocity. And so the mechanism they called a dynamical amplification across scales. And they used this to prove finite time blow up for the unforced 3D oiler equations. Remember the oiler equations without viscosity

2:26:51>> without viscosity. >> They showed that without the smoothing term you could actually have finite blowup. the pressure won't do as much because the pressure also does a little bit of smoothing >> cuz it was both the pressure and the external force were the only two of the three on the force part of the equation. >> Yeah. And so this is unforced. So no force. You just set up an initial >> ah okay. >> Yeah. There's no stirring. There's no force. You just set up an initial condition and the scales converge in such a way that you get an infinite blow up. >> Uh they they >> Now this is huge. They they're chipping away at each of the variables on the the right side of the equation. >> Yes. And so >> good >> the problem was so so then so that was

2:27:32for the unforced 3D oiler equations, right? But obviously we're interested in the ones that have viscosity. >> So there's this one type of equation called the hypo dissivative dissipative Navier Stokes equations. It's a variant that utilize utilizes um this thing called a fraction leloian instead of the actual leloian. the the dell squared u that we had for viscosity the triangle squared u that's a leloian this thing has a fractional >> fractional leloian >> and they showed that in this case we can use the same sort of paradigm and get blowup >> mhm >> the only problem was that they required um a forcing term that was rough meaning

2:28:15it's not smooth like the the forcing term itself has tiny little bits of infinity >> mm like the way you're stirring it has like little jolts of infinite stuff. >> You're pumping infinity into your stirring. >> And so >> we're getting closer though because the infinity is like small enough that you could like maybe figure out a way to smooth it and still preserve the dynamics. >> Okay. And and so I just want to pause really quick because what we're sort of saying is human mathematicians have now taken this uh Navier Stokes Millennium Prize looking at uh uh finite

2:28:56blowup. >> Mhm. >> Um and then they started with looking at you know taking out the smoothness and taking out all the force. We got some solutions which are effectively the oil the oiler equations without the f and then we've now started adding in back in the viscosity but that but the viscosity wasn't smooth it was rough. And so when people see this smoothness and roughness, part of what we're saying is the viscosity term um has some special sauce >> that doesn't quite get you to the the natural smoothness that arises in the

2:29:36equations with viscosity on >> Yeah. the raw viscosity raw viscosity. >> Yeah. And and >> but it's still important because these these fun these are fundamental. >> No. And and this is what even open AI's thing is built on, okay? Is like the there's a reason why the Navier Stokes is the first of the six infinity stones to fall, >> which we just saw. >> It's because we got so close as human beings. >> It's like a racetrack and someone was on the la the fourth lap of whatever the 800 meter and uh you know, open eye just shows up, grabs a baton and finishes. >> Yeah. Yeah. >> Not minimizing. Not minimizing of course but I'm just saying a lot of human input

2:30:18has gone into this right so Cordova and uh Martinez Zora they establish this layer bylayer scale amplification it's a huge deal >> because everyone can sense that we're getting close right okay and shortly before the open AI announcement mathematicians Tristan Buckmaster and Levant Alpog they published preprints that elevated this methodology to accommodate for smooth forcing of oiler meaning you've got the oiler equations but now with a smooth force the boyesque equation which is something that's like kind of similar to the Navier Stokes equations >> and this is where the sort of >> drama starts but I'm going to save that for later because a few days after that openai announces that they have solved

2:31:00it. >> Here we go. >> Here we go on the Navier Stokes Millennium Prize problem. They announce it on Twitter. They have it on their website and the actual paper is something like 165 pages long. The