How the counterexample works

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

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

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