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Turn a number system into a geometric space. Intro to algebraic geometry, Spec(R) and schemes.#SoME4

Audience:

Tags: algebrageometryalgebraic-geometry

I describe a concept that is central to modern algebraic geometry - schemes. It allows us to do geometry over pretty much any number system you can think of, including the integers, p-adics, and so on. It's used in the proof of Fermat's Last Theorem by Andrew Wiles. I first define locally ringed spaces by generalizing properties of continuous functions on a topological space. Then, I reverse this process and construct a locally ringed space called Spec(R) from any commutative ring R. I achieve the goal in the video by defining a scheme as a locally ringed space that is locally isomorphic to Spec(R). Finally, I touch on some advanced topics such as Proj of a graded ring, quasicoherent sheaves, invertible sheaves, sheaf cohomology, relative Spec and Proj, and blowing up. And I make a connection with number theory via the ideal class group. This is a submission to 3Blue1Brown’s #SoME4 contest. Works Cited Foote, Richard, and David Dummit. Abstract Algebra. Danvers, John Wiley & Sons, 1991. Hartshorne, Robin. Algebraic Geometry. New York, Springer, 1977. Jacobs, Konrad . “File:Alexander Grothendieck - Face.jpg - Wikimedia Commons.” Wikimedia.org, 21 Oct. 2024, commons.wikimedia.org/wiki/File:Alexander_Grothendieck_-_face.jpg. Accessed 1 Sept. 2025. Munkres, James R. Topology. New York, Ny, Pearson, 1974. Vakil, Ravi. The Rising Sea: Foundations of Algebraic Geometry. 8 Sept. 2024. Wiles, Andrew. “Modular Elliptic Curves and Fermat’s Last Theorem.” The Annals of Mathematics, vol. 141, no. 3, 1995, p. 443, https://doi.org/10.2307/2118559.

Analytics

4.88 Overall score*

124 Rank

8 Votes

5 Comments

Comments

The introduction is great. It clearly establishes the goal of the video, and it’s an exciting goal. Schemes sound powerful, and I want to know more. The ladder to climb up, however, is tall and steep. I was following for a while, as long I was already familiar with the definitions introduced, but then I had trouble when I got to the new definitions. Rather than assuming little background and trying to walk the viewer all the way up, an alternative approach would be to assume some fair bit background (say, undergraduate abstract algebra and topology courses) and spend more time building the other concepts on top, with plenty of slow examples. (Also, adding chapters to the video would help orient the viewer to where we are; you could do that now.) I really would like to understand schemes, though, so a followup video would be welcome!

5.2

This is a good video with a lot of thought put into it. Admittedly, the topic is above my knowledge so I don’t have much helpful feedback. I can tell it was well thought out. It was a bit wordy and abstract but I suppose that is the topic. Nice work!

An overwhelming amount of lengthy technical definitions and syntax that doesn’t feel organically motivated, with a large amount of text on the screen and few visualizations.

4.8

Motivation: 4/9 - I felt like the motivation is there, but it was delivered in broad strokes. The question of representing number systems as geometric spaces is catchy and I liked it, but I would explore that topic and make this question have sence intuitively for the viewer before attempting to explain it rigorously.

Clarity: 3/9 - Throwing definitions without any context or visualization didn’t do it for me, sorry :/ I think the topic is so vast and you wanted to explain so many ideas before the main problem that the viewer might lose motivation before even trying to understand it. I think, sadly, you chose a topic that was too broad for a single youtube video, especially if you’re choosing to explain it so rigorously.

Novelty: 6/9 - The topic is something I’ve only vaguely heard of before and it turned out to be very interesting! Nice!

Memorability: 2/9 - The presentation was a slide show with only a few visualizations. Wasn’t very memorable in my opinion. Of course, I know it’s hard to visualize such topic, but still, all I see is white text on black screen. I think the video lacked the most in this area and even small improvements like highlighting certain elements of the text, maybe a slower pace in certain moments and more examples would make it a much better video.

Nonetheless it isn’t a bad entry! Easier to follow (and visualize) topic would be great! Overall: 6,75/9

2.5

I thought you did a great job of motivating the idea of a scheme. “How can we do geometry on arbitrary number systems” is a great motivating question, and you gave lots of examples of the uses of schemes in the first few moments.

However, I found that the jargon and notation got out of hand very quickly with this topic; there were lots of definitions being shown which may not be internalized fully by an audience with no background in the material. Certainly your effort to give some intuition was clear, but these snippets did not always suffice for the depth of the material covered.

Also, there were some large jumps in logic that were not easy to follow. For instance, moving from continuous functions being functions “where a small change in the input results in a small change in the output” to functions which maintain that “if U is an open set in the output space, then its preimage is open in the input space”. This may be clear to someone who has taken a first course in topology, but is not obvious necessarily for someone who hasn’t. There are more similar jumps throughout the rest of the video.

On the other hand, it’s worth noting that there may not be a great solution to this problem. The material you want to cover is of course at a really high level, and so I understand that in order to make the concepts approachable, you need to start at a low level and work your way up. But you don’t have the time to do the justice to the building blocks without making an egregiously long video. Perhaps this topic would be better as the culmination of a short series of two/three/four videos? Or could be presented with some assumptions about the viewers knowledge beforehand, i.e. a cursory knowledge of topology/rings.