The shocking connection between complex numbers and geometry.

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SOURCES and REFERENCES for Further Reading:

This video is a quick-and-dirty introduction to Riemann Surfaces. But as with any quick introduction, there are many details that I gloss over. To learn these details rigorously, I've listed a few resources down below.

(a) Complex Analysis

To learn complex analysis, I really like the book "Visual Complex Functions: An Introduction with Phase Portraits" by Elias Wegert. It explains the whole subject using domain coloring front and center.

Another one of my favorite books is "A Friendly Approach To Complex Analysis" by Amol Sasane and Sara Maad Sasane. I think it motivates all the concepts really well and is very thoroughly explained.

(b) Riemann Surfaces and Algebraic Curves

A beginner-friendly resource to learn this is "A Guide to Plane Algebraic Curves" by Keith Kendig. It starts off elementary with lots of pictures and visual intuition. Later on in the book, it talks about Riemann surfaces.

A more advanced graduate book is "Algebraic Curves and Riemann Surfaces" by Rick Miranda.

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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.

00:00-00:54 Intro
00:55-04:30 Complex Functions
4:31-5:53 Riemann Sphere
5:54-6:50 Sponsored Message
6:51-11:06 Complex Torus
11:07-11:50 Riemann Surfaces
12:11-13:53 Riemann's Existence Theorem

[Aleph]

Thanks for watching! If you have any resources you'd like to recommend, feel free to comment them down below.

[timothypulliam]

The reason exp(1/Z) contains an essential singularity is, if you expand the function as a Taylor series, you will get infinitely many powers of (1/Z). In essence, the singularity can't be removed by multiplying by Z. Therefore, it is "essential"

[dougdimmedome]

One of my favorite things in complex analysis was just seeing that elliptical curve come out of nowhere with the Weierstrass p-function, I felt like I was seeing a fraction of what Wiles saw every day while proving the modularity theorem enough to prove Fermat’s last conjecture.

[SGin]

it’s the main argument of my thesis, I’m so happy to see a video about Riemann Surface ❤️

[mohammedbelgoumri]

No better way to start a day than an aleph0 upload

[StratosFair]

Aleph 0 is back with yet another banger ! Nah but seriously as a grad student in applied and little knowledge of pure maths, i enjoy these videos so much as they give me a glimpse of the beauty of what's on "the other side". Please keep them coming !

[jakobr_]

Riemann’s existence theorem: “Bernhard Riemann exists.”

[dimitriskliros]

i don’t often comment on uploaded videos, but i feel this video is so good that i just wanted to say thank you, and keep up the good work.

[primenumberbuster]

Finally, more Algebraic Geometry content

[omargaber]

When the world needs him he will come back

[jogloran]

I love how you give equal time to "zee" and "zed" 😅

[SultanLaxeby]

This is an extremely good motivation for the elliptic curve equation(s) that I see everwhere, and a very nice explanation why complex tori are elliptic curves (and not just the other way around)!

I'm a bit baffled by your way to write a zeta though...

[gnaistvlogs]

This is one of my favorite results in mathematics. I used this categorical equivalence (along with the equivalence to algebraic function fields) in my master's thesis on prime Galois coverings of the Riemann sphere back in 2007.

[GhostOnTheHalfShell]

my math is such a rust bucket. i need to dust off a bunch of old books, but then recapitulate several semesters just to be sure i had enough of the definitions fixed in my head

[kernel]

Love the channel and the content, no pressure, but I have been eagerly awaiting the course that you talked about developing/releasing.

[headlibrarian]

I don’t know if it’s important, but in the complex torus example the interval is first written as closed [0, 2pi] and later in the example it is written as open [0, 2pi).

[lukiatiyah-singer]

Thanks for the video, very well explained!
On this topic, I found the book by Serge Lang on elliptic functions very helpful, but also Gunning's lectures on Riemann surfaces for every thing beyond genus 1

[magnusre]

Been waiting for a new video from you. Just checked a few days ago. And there it is. I'm already intrigued.

[ianmichael]

Respect. The printed cut outs are beautiful.

[beardymonger]

Great amazing content, I admire the effort that went into making this!!!

I would add a short section about the inversion 1/z (with animation) to explain the essential singularity at infinity.