What is algebraic geometry?
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Published 2023-10-17
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A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!
And another thank you to Davide Radaelli for helpful conversations when making this video.
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CORRECTIONS:
At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.
SOURCES and REFERENCES for Further Reading!
(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.
(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.
(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.
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MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
www.vincentrubinetti.com/
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What is algebraic geometry?: (0:00)
Coordinate Ring: (3:04)
How algebra detects reducibility: (3:54)
How algebra detects a node: (5:15)
Schemes!: (8:00)
All Comments (21)
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Please please please make more algebraic geometry or commutative algebra videos. These are really great!
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Please make more videos on algebraic geometry, please. These videos are treasures.
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Best advanced math education channel on YouTube. I struggled immensely with algebraic geometry in college. The definitions and concepts weren’t properly motivated. So I learned in a painfully mechanical way.
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As an algebraic geometer/commutative algebraist, this video describes exactly how we think about shapes and their corresponding rings. Great job! (for any graduate students reading this: Read Hartshorne's Algebraic Geometry book. It is, IMHO, the end all be all reference for introductory algebraic geometry.)
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4:25 There is a typo I think: should be g(1,-1)=-2. Aside from that, congrats for the really nice explanation.
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An excellent introductory video. I should have watched it before I took algebraic geometry or read Gathmann's.
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Wow, I don't think there is a better introduction to ideals in algebraic geometry.
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Individuals that have spare money, if I were one of you, I would consider donating to this man. He has the most simple yet beautiful way of sharing knowledge I've seen since I discovered 3b1b. Give this man a chance to make more videos like this one more frequently. ❤
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It's really cool that you talked about schemes! For such an advanced topic it's really nice to see a video even mentioning it
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Small error but at 4:27 I believe it should say g(1,-1)=-2 instead of g(1,1)=2
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As a mathematical physicist, the immediate question that popped into my brain is, "How does this relate to differential geometry?" For example, the curve having a self intersection in one of the examples, which corresponds to the ring not being an integral domain, manifests itself in differential geometry as the curve not being a manifold-ie no diffeomorphism with R around the intersection point.
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This is a really nice appetizer, it's so rare for a video on algebraic geometry to actually go far enough to talk about schemes
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04:25 wrong calculation g(x, y) = y - x Okay, but below: g(1, 1) = -2 is wrong g(1, 1) = 1 - 1 = 0 not -2
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You really made my day ❤️, Please make a series out of it, the world will remember you
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Was in the ER this morning, but a new aleph 0 video has made this a good day.
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Such a fascinating video! Your videos tend to ignite a spark of curiousity everytime i watch them, thanks so much!
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The best layman presentation of algebraic geometry I have ever seen. Great!
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Qing Liu’s book is great. I also really like Eisenbud and Harris “The Geometry of Schemes”, and Mumford’s “Red Book” is just a rare jewel, so beautiful, with all those drawings of schemes (some also reproduced in Eisenbud-Harris)
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I was always too intimidated to begin studying this topic I’ve laways been intrested in, but this video has definitely done a good job at helping me croos that threshold. So thanks! Great stuff, as usual
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Appreciable work. Keep on providing introductory videos (+ additional resources) of Advanced Math Courses. As a highly motivated undergrad, it really helped me to study these advanced topics with good intuition and a good introductory recourse (that book you mentioned). Anyway, Thanks and keep on guiding us.☺
