Jacobian Conjecture SOLVED Today?! (LIVE)

[00:00:00]Math is being cooked more and more. And in this live stream, we're going to break down the Jacobian uh conjecture. It is being announced that it is false. And this is a no-nonsense live stream discussing the newest advancements today in AI. I'm going to break it down for you. So, let's dive into it. Mathematician disproves an 85-year-old Jacobian conjecture with AI help. Now, it's my understanding that this was done by an employee over at Claude Anthropics Fable. I mean, this is getting to be a mouthful now.

[00:00:40]But, uh let's read into it, okay?

[00:00:44]Levent Alpoge, I hope I'm pronouncing his name correctly, has announced that here on X that the Jacobian conjecture is false using a specific map in three-dimensional complex space with a constant nonzero Jacobian detriment the determinant of -2. The map collapses three distinct points into the one proving no polynomial inverse exists. A hand-checkable example verified by tools like the Wolfram Alpha, that's a website that you can use to check. Credited Excuse me. Uh AI Claude Fable 5 for the discovery of the math and mathematician Akil Mathieu for the prompt drawing praise from experts like Fields medalist Timothy Gowers as a milestone for AI in math. So, what does this mean? I'm going to tell you right here, right now, no nonsense at all.

[00:01:38]This math problem was created like in in the 1930s and was considered to be true.

[00:01:45]But, here it's proven false, and I'm going to tell you why in simple terms.

[00:01:49]Okay.

[00:01:50]So, imagine that you are a locksmith, and you're creating a key, okay? So, you create a key, just like a regular door locking key, and you say, "For each groove on the key, therefore, the key is made and can unlock a specific door, okay? Or a specific lock." And that was considered true.

[00:02:17]And this is what's known as the Jacobian uh determinant is that actually it is proven false.

[00:02:25]In the map of the 3D world, we we come to the uh conjecture that uh a key can open more than one lock or more than one door. That's a simple way of mathematically looking into it is that this is why the determinant is -2.

[00:02:46]It meaning in this case more than one, more than more than one option.

[00:02:51]It used to be kind of interpreted that only the one key could open the one door and that was it cuz it made sense.

[00:02:59]But as it turns out, one key can actually open multiple doors.

[00:03:05]And that's basically what it boils down to.

[00:03:08]And if you don't believe me, you can check for yourself and then ask that same question. But right now, this is uh a mathematical problem that is was almost 100 years old if I'm not mistaken, and it was finally proven uh false, okay? And we're going to talk about the impli- implications of what this means uh in math and all that stuff as the video progresses.

[00:03:34]So, let's dive into it. This is the man here himself who says, "Hello, everyone.

[00:03:39]Uh the Jacobian conjecture is false. Thanks to my close friend Akil for asking about it and my other close friend Fabel for working during the final cup, okay? This is when I guess it was Spain turned out to be the winner, and he gives this mathematical equation, okay? And right now it has 8.4 million views, okay? That actually went up quite a bit because just a couple of minutes ago it was like 7.5.

[00:04:04]So, this is really uh really jaw-dropping news right now, everybody. If you'd like to live stream, hit me up with a like or a comment. We can talk about it.

[00:04:14]And uh Mr. Levent here, he he does give an application for the website uh wolframalpha.com, and we can click on it, and [clears throat] we can see what the answer is, okay? So, if we click on it, evaluating at two of those points in the Wolfram Alpha. So, if I click on it, it's basically going to give a complicated math equation, and the answer is going to be seen as -2. And thus, this is this is the the real uh like bread and butter of the math equation or whatever.

[00:04:43]So, if we click on it, we're led to a website called Wolfram Alpha.

[00:04:48]And here's the mathematical equation for those who'd like to look at it.

[00:04:52]And as we scroll down, the input interpretation is as follows, okay?

[00:04:57]Which is pretty much the same thing, and the result in this case being 1/4.

[00:05:03]Uh it was my understanding that I thought it would say -2, but I'm not sure I'm not sure uh that I'm seeing it correctly in that sense, but I guess that's what it is. Let's Oh, yes, if we click on number two, the second part of the mathematical website here, here's another uh conjecture point, and there's the results. Okay, it just says 1/4. Oh, okay. I thought it was going to say two.

[00:05:30]Uh I could have swore, yeah, 0.25. Okay, I I I could have swore there was a two cuz I I did do my homework and check on this website beforehand, and it said two.

[00:05:42]But anyway, you guys can study that in your personal time a little bit. Let's read some comments about what some of the community is talking about.

[00:05:49]Commenter JT says, "Claude Fabel just proved that the Jacobian conjecture is false. Incredible. This changes everything. Do you know what the Jacobian principle is?" And everybody just says no. So, here's here's a meme about it.

[00:06:03]It's a meme about it. But, let's dive into it cuz it's going to get more serious than just jokes aside, everybody.

[00:06:09]I have rarely seen science Twitter so excited and shocked in the scenario. The next few months will be un unforgettable.

[00:06:17]So, nice positive stream here. "Oh my god, this really seems correct," says David. Uh this guy is a really smart mathematician, is my understanding.

[00:06:25]Christian says, "I can't even fathom the next year." Exactly.

[00:06:29]Uh Peter says, "At this point, seeing famous conjectures fall, I would buy at 2% but probably sell at 20." That within the next year language language models will find a set of positive integers with divergent sum of reciprocals. Yeah, that's another math problem.

[00:06:45]But, without arbitrarily long arithmetic progressions. Yeah, as What he's trying to say there is that the mathematical equations that's going to solve some of our hardest mathematical equations are actually going to be relatively short, is what that person is alluding to.

[00:07:05]Let's keep scrolling just a little bit longer before I break it down even more for you guys.

[00:07:09]Uh Claude Fabel 5 has produced a hand checking counter example to the problem to this problem which started in 1939.

[00:07:21]Posted in 1939, the Jacobian conjecture is one of the central open problems in algebraic geometry. But, it was disproved by Al Po-Chi Matthew. Yeah, another complicated way of talking about this problem in geometry terms is like let's say you have uh let's say you're playing bingo and all the balls are scrambling around in the bingo machine, right? And but those balls represent numbers and like mathematical numbers in in the geometry of the balls bouncing around in the box.

[00:07:54]Well, it was basically thought that it would be like a one-for-one in terms of the empty space inside or something like that. It's where my knowledge kind of uh, ends around that type of example, but geometry and the 3D spatial structures is what the Jacobian conjecture is like founded in in terms of the mathematical principle. But, okay, so this is my 32-year-old regular average Joe understanding of this, okay? The Jacobian conjecture roughly says that a multivariable polynomial F has an inverse function made out of polynomials, provided by the Jacobian is non-singular. Matrix of partial derivatives has non-zero determinant. The condition is necessary by the inverse function theorem from multivariable calculus.

[00:08:48]The hard question is whether it is also sufficient. Evidentially, Fabel found such a case.

[00:08:58]How are you guys liking the stream here?

[00:09:00]What math equation do you think should be solved next or tackled next?

[00:09:06]You know what I mean? And this man who was working with Fabel decided to take time out of his day to watch the World Cup of people playing soccer, football, depending where you're living, to work on this.

[00:09:19]So, very appreciative here.

[00:09:22]Uh, Claude Fabel 5 produced a hand-checkable counterexample. The conjecture says that the map is whatever, whatever. Let's get into some more bread and butter here.

[00:09:32]Maybe we'll read some comments.

[00:09:37]Uh, let's back out of here for now.

[00:09:42]We'll scroll down.

[00:09:44]And yeah, what other what other uh news do you guys want to talk about today?

[00:09:49]I mean, it's pretty big news, right?

[00:09:51]It's pretty cool.

[00:09:55]Uh I wish I had like more to offer you guys in that sense.

[00:10:04]Let's check out the latest.

[00:10:14]Uh no, still not seeing anything here.

[00:10:17]A lot of people from the other side of the world are talking about it.

[00:10:25]And yeah.

[00:10:26]Oh, interesting.

[00:10:28]So, that pretty much solves this solves it up there.

[00:10:37]Hm.

[00:10:45]All right.

[00:10:48]It's pretty funny showing large language models Jacobian disproving blowing their minds over and over.

[00:10:56]Hm.

[00:10:59]Yeah, let's read the the uh the simplified Wikipedia version.

[00:11:04]Mathematics, the Jacobian conjecture is a famous problem concerning polynomials in several variables.

[00:11:10]Uh it states that if a polynomial function forms an end-dimensional space to itself has a Jacobian determinant, which is a non-zero constant. Yeah, it's basically what I was saying about the lock and the key function.

[00:11:23]Then the function has a polynomial inverse. The conjecture was first stated for two variables by Ludwig Gordan 1844.

[00:11:32]Wow.

[00:11:33]And then stated full in generality in 1939. Wow.

[00:11:40]As an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus.

[00:11:47]Yeah, because these AI models, they don't get tired and they can kind of like find a I'm not saying that they're creative by themselves, but they can find creative solutions to a problem that otherwise people would burn out of trying to think about.

[00:12:04]You know what I mean? Or even work on for a couple of weeks. I'm assuming that this probably got solved in like 5 minutes.

[00:12:10]You know, and I'm assuming that the man who solved it with the help of Fabel, I'm assuming he's going to have about a 1-hour interview or a 20-minute interview coming live.

[00:12:20]And uh yeah.

[00:12:23]Can't wait to see it.

[00:12:25]Is notorious for the large number of published and unpublished proofs that turned out to contain stumble errors.

[00:12:32]Uh today or yesterday Anthropic employee using a large language model found a concrete counterexample to the conjecture proving it false.

[00:12:41]Da da da da, sometimes too sometimes the best way to move ahead in math is not to create something new, but to like uh how do I say this? Not to create something new, but to like disprove something with a counterexample. That's one of the surefire ways to also increase your intelligence and to assess the intelligence of someone you're talking to. If you can if you cannot not disprove something in the world of science, then it is possibly true. So, instead of trying to find out if something is true, then you have instead of doing that, you can go from a backwards principle to say, can it not be can it not not be disproven?

[00:13:26]It's not just can it not be disproven, it's can it not not be disproven with a like a counter example.

[00:13:34]And I think if you were to watch this video and just heard what I said and explicated that into an AI model of your own and asked it to go out there in the world and solve certain mathematical equations that can't not not be falsifiably disproven or whatever then that would in turn solve some of the unsolved math equations actually.

[00:14:02]So, we're going to end it there you guys.

[00:14:05]Uh hope you guys like the video. Hope you guys enjoy hanging out talking as much as I did.

[00:14:10]Uh Vadim, I see you in the stream. Let's read some comments guys. Two is the determinant one over four is the projection point. Okay, thank you, sir.

[00:14:17]Thank you. Thank you.

[00:14:19]Uh be sure to like and subscribe if you guys want to talk in the future cuz I do do live streams where I just hang out with people too sometimes.

[00:14:26]And uh yeah, sounds good. Okay.

[00:14:29]So, I salute you guys for doing your research, studying hard, and trying to make a difference in the world. I know it's not easy but just keep at it doing what you're doing. Take care of yourself.

[00:14:39]And uh as the days pass, Dr. Alex Wester Gross recently said on a podcast that a new AI a new AI a new AI model will be dropping every single day potentially in the month of January, which is kind of strange to wrap your head around because we used to get an AI model pretty much every month and now we get a new AI AI model drop every 10 days but with the rise of China dropping the free models and the open source models, if it like uh you can see where the complications arise from dropping a new model every single day.

[00:15:18]Like in terms of uh the recursive self-improvement of the old model and then creating a new model but then the next day after that recursively self-improving the model that was just dropped the day before.

[00:15:31]Which is kind of strange because like just when you download the open weights onto your Mac minis or whatever and whatever or just uh or like or in the big data centers just when you got the latest model you pretty much have to shut it down and update the next model the next time around like I don't comment below where you think that's heading because I don't really understand uh like if you have a like say for example open AI has the soul Luna and Terra model but if they drop a new model tomorrow will it be like the Milky Way galaxy the universe and beyond or something the next day and then the next day after that after that after that you know what I'm saying so why would you drop three models in one day if you're already dropping one model a day?

[00:16:18]You know what I mean like so I I I don't know comment below what your thoughts are on that if you even made it this far into the video and we'll see you guys later okay?