Complex Numbers and Elliptic Curves - Intro 1 (FLT Proof #4.3.0.1)

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It turns out Elliptic Curves over Complex Numbers are central to Wiles's proof of Fermat's Last Theorem. With the first videos on this channel, we managed to keep the math very simple, almost exclusively nothing more advanced than algebra. Now, to complete our brief tour of elliptic curves, we have to suddenly take an ENORMOUS leap forward in the level of the mathematics involved.

I want to make sure we are all on the same page before we begin learning the more advanced stuff in earnest. I am going to assume we are all familiar with high school math and college math through first or second year calculus. I'm going to introduce complex numbers and complex integration in some detail with the next several videos. We need to examine multivariable calculus at all. I am going to include a separate set on that, after we complete this survey of complex analysis.

This series will culminate with the Weierstrass Equation, which is the thing that connects elliptic curves to complex numbers, and thus allows us to connect them to modular forms, which is what Wiles's proof is all about.

This series will take several weeks to complete. The present video is an introduction and summary of what we will be going over in this series on complex numbers. When the introduction is done, we will start at the beginning with the construction of real numbers.

Here is the outline of this series on complex numbers:
0. Introduction
1. Construction of Real Numbers
2. Complex Numbers
3. Complex Functions
4. Exponential and Trigonometric Functions
5. Complex Integration
6. Cauchy's Integral Theorem
7. Cauchy's Integral Formula
8. Laurent Series
9. Complex Residues
10. Lattices and Doubly Periodic Functions
11. Lattices and Tori and Groups
12. The Weierstrass p-Function
13. The Weierstrass Equation: Complex Functions and Elliptic Curves

Please leave any questions, comments, or suggestions in the comments below!

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Credits:
Music: "Christmas Island" by Depeche Mode. I hope the Men from Mode will not mind that I used a few seconds of their music!

Bibliography:
Complex Analysis, by Lars Ahlfors
Complex Analysis, by Bak and Newman
Principles of Mathematical Analysis, by Walter Rudin
Basic Multivariable Calculus, by Marsden, Tromba, and Weinstein
Khan Academy
Many other short, unsigned treatises from the internet

[davidandcookie]

I'm saving this for later! It snuck in under my radar and I don't know why I never got a notification about it. I have to prepare my mind first.
Did you learn all this recently and are now just relaying, or has this been an ongoing life development? Like did you study it at university?

[evariste]

greg, the Y in the W eqn isn't a dependent variable as it is in an elliptic eqn. The W eqn is more like a differential equation over C. So the attempt to connect them via the form of the eqn is a bit misleading?