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Jacobian Conjecture for Baby
- neuralkoi 3mo agoThis was an excellent write-up and taught me more in 20 minutes than my linear algebra teacher did in 6 months. The material however, got fairly dense. After chapter 10, it is no longer "baby" material in my opinion.
- noio 3mo agoI felt the same. In a way the 'baby' language in the first 10 chapters (e.g. "straight" instead of "linear") left me unprepared for the later chapters.
- NooneAtAll3 3mo agobeginning reeked of... maybe not ai, but bad writing but the further into the text, the better the prose by the end it was great --- good to know that conjecture has been false for real numbers for a while now
- samlinnfer 3mo agoIt got more and more AI towards the end.
- CamperBob2 3mo agoAlso more and more use of an extremely annoying effect: text-heavy animations that reset themselves for no reason while you're trying to read them. Step 11, for instance: no value whatsoever is added by erasing and redrawing that figure. Still, overall a very nice piece of pedagogy. If you didn't grok determinants before, this will probably help.
- muchmirulys 3mo agorevised. thanks for the feedback
- CamperBob2 3mo agoGreat work, don't get me wrong. Looking forward to more like it!
- muchmirulys 3mo agorevised for better progression *still ai assisted
- dash2 3mo agoone thing I didn't understand. If the conjecture is false for real numbers, why isn't it false for complex numbers too? Why can't you just take a real number counterexample and use it for complex numbers? I guess the answer is that a polynomial map whose Jacobian determinant is constant non-zero over the reals may not also be constant non-zero over all the complex numbers.
- NooneAtAll3 3mo agoconjecture said "if there is no J(point) = 0, there's no f(point)=f(point)" counterexample for reals moved point=0 outside reals, but kept point=point inside you can see the "jacobian is sum of squares" being mentioned - that is sufficient to say there's no negative values in the reals, but doesn't work for the whole complex field for complex numbers you have to have jacobian be a constant, or you'll get the zero somewhere
- ahofmann 3mo agoI think the entire post was created by an AI, and I had a similar negative reaction to it. Nevertheless, it's an excellent post with clear explanations of mathematical concepts. This just goes to show how impossible it is to tell the difference between something created by a human and something created by an LLM.
- y-curious 3mo agoI followed very loosely and still got it. Very cool, the kind of math that gets close to feeling like "real life". I was really hoping that your mentions of "solving it by hand" and "you can prove it to yourself by hand" at the end were going to come to some clever maneuver I could do with my own hand (like the physics magnetism hand rule). Still, awesome.
- johnfn 3mo agoThis is really good - miles ahead of almost all other math writing I have read. One particular thing I really liked was the explanation of the determinant. When I was in college I spent a very long time trying to get some intuitive explanation for a determinant of a matrix and the best I could get is that you just multiplied and subtracted some numbers and poof there you go. I KNEW there had to be some more comprehensible explanation and it drove me crazy that people just parroted textbooks without any real understanding. This reminds me of that silly video everyone was shown in middle school about inverting a sphere :-) I am very curious how it was written (will there be a write up to the write up?!). It feels likely AI assisted, but this seems to me to be the best possible use of AI -- crunching out a bunch of visualizations of complex ideas so that they are easy to understand just by looking. The writing also feels a little AI written but definitely feels very human-assisted.
- thebiblelover7 3mo agoSeems like it was entirely AI written (claude) from the commits: https://github.com/muchmirul/conjectures/commits/main/ https://github.com/muchmirul/conjectures/commits/main/
- thedevilslawyer 3mo agoShows how well prompted AI can significantly improve human understanding. It's the slop patterns that seem to turn people off.
- muchmirulys 3mo agorevised. *but still ai assisted
- muchmirulys 3mo agorevised. *ai assisted
- muchmirulys 3mo agothanks, revised version up. please do check and hope it helps
- para_parolu 3mo agoAwesome. I was literally asking ai to find blog post like this but couldn’t get anything decent
- muchmirulys 3mo agothanks, hope it helps
- tug2024 3mo ago[dead]
- CGamesPlay 3mo agoChapter 8 is incomplete. It says the local area factor "is zero at one single point... and the map uses it to wrap the plane around twice... so local factor never zero is not enough on its own". "X is false, therefore X implies Y is false". But this is faulty reasoning.
- muchmirulys 3mo agorevised. hope it clear any confusion
- muchmirulys 3mo ago[dead]