Fermat's Last Theorem: Elliptic Curves and the Need for Flatness! (13.2, 180)
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- Опубліковано 12 вер 2024
- Chapters 1-4 of My Fermat’s Last Theorem Notes Now Available on My Website! Link Below.
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Don’t stop please !!!! This is legendary !!!!
Next one up tomorrow!
Awesome video! 🎉😊
Just discovered this series today. Keep up the good work!
How many math courses are required to understand this proof? Just curious.
I have a couple of videos on my channel about this exact topic. Do some digging and let me know if you still have any questions.
I was hoping to find some insight into what is meant by "elliptic curve" as opposed to "ellipse" (I know what Elliptic Functions are). Although I have an advanced degree in engineering from a very reputable university, this awe-inspiring document appears to be in a foreign language, using terms I have never even seen on every line. While I don't doubt that those with the required background can understand this work, I do not see the need for showing the pages of your document and reading them (although it does teach one how all these symbols and notations are to be read). I was expecting some kind of diagrams or illustrations or graphics that would justify a video, but no .... Mind you, I am totally in awe of this degree of facility with mathematics at a level wholly inaccessible to me, but I was hoping for some bridge from the land of UA-cam viewers reasonably competent in general mathematics .... Congratulations on this awesome work!
Hi there. Admittedly, by this far into the playlist, the intended audience is advanced graduate students in Number Theory. I recommend doing a quick UA-cam search along the lines "what is an elliptic curve?" and checking out any one of the excellent videos that pop up.
@@Entropize1 Yup, one already came up - haven't watched it yet. I remember how excited I was in high school to see (in my own library research) how a full-strength elliptic integral solves the general pendulum problem. I did browse some of your other vids, particularly the one with the list of hundreds of books. It goes without saying that 95% of that math did not exist in the time of Fermat, no? I can't imagine that even Euler or Gauss had it available either(?) Either Fermat didn't really have a proof (was mistaken), or there seems (naively) something awry in the structure of the universe --- thank you for your answer!
Elliptic curves are related to ellipses by a very long historical line, with the name "elliptic" being passed down until the object no longer had anything to do with ellipses.
First, we had elliptic integrals. These were originally developed to find the arc length of an ellipse, but grew to be much more general. Then, we had elliptic functions, which arose when you took the inverse of elliptic integrals and extended that to the complex plane. The Jacobi elliptic function is still closely related to ellipses, but it is the last link you'll find.
Of course, we finally have elliptic curves, which come from the Weierstrass elliptic function, and that comes from a certain elliptic integral which has nothing to do with ellipses. :)
@@zswu31416 I wouldn't say they have nothing to do with ellipses since they show up in the formula for the arc length of an ellipse. There's also a somewhat recent theorem relating elliptic curves to ellipses, but I can't recall it. I'll have to go digging.
@@Entropize1 The elliptic integral that comes from the arc length of an ellipse and the elliptic integral that leads to elliptic curves are quite different integrals, and only look similar. Correct me if I'm wrong