Invertibility and the conjecture

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

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

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

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

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

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

1:16:08>> Okay. >> Okay. So if I have a polomial function that's going in polinomial meaning things like x cub plus x^2 y like powers of stuff y >> then the inverse map is also a polomial >> okay >> that is the jacobian conjecture. >> Okay so this is this is actually interesting because now now I can understand the formulation here of what we like what this open problem yeah >> is trying to suggest. Yes, it's trying to suggest that any time >> right >> I have a function where the Jacobian determinant is a constant everywhere meaning the the function preserves area >> preserves area I shouldn't say preserves

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

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