The derivative isn't what you think it is.

The derivative's true nature lies in its connection with topology. In this video, we'll explore what this connection is through two fields of algebraic topology: homology and cohomology.

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SOURCES and REFERENCES for Further Reading!

In this video, I give a quick-and-dirty introduction to differential forms and cohomology. But as with any quick introduction, there are details that I gloss over for the sake of brevity. To learn these details rigorously, I've listed a few resources down below that I found helpful.

Differential Forms: The book “A Geometric Approach to Differential Forms” by David Bachman is a treasure. Instead of leading with the formalism, it gives a nice intuitive picture of what forms do, and then provides the precise definitions.

More on Cohomology: I also came across the book “Differential Forms in Algebraic Topology” by Bott and Tu, which starts off with De Rham’s Theorem and goes into much more depth about the relationship between the boundary and the exterior derivative. This is quite advanced (read: I only got through the first few chapters before I stopped understanding what all the words meant ...), but if you’re up for it, read along!
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LINKS

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MUSIC CREDITS:

Song: Thinking Ahead

SOFTWARE USED:
Adobe Premiere Elements for Editing
Blender 2.8 for Animations

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Twitter: @00aleph00
Instagram: @00aleph00

Intro: (0:00)
Homology: (1:08)
Cohomology: (3:41)
De Rham's Theorem: (7:45)
The Punch Line: (9:02)

[Aleph]

Thanks for watching! I’ve been meaning to make a video about cohomology for a while, so I'm glad I finally got around to making it. I am by no means a master on this topic, and I’m sure that you have insights into the material that I don’t – so share them in the comments below. Feel free to ask questions and recommend learning material as well. I’ve learned a ton just by hearing people’s thoughts and questions in the comments section, so let’s learn some math together! <3

[darkdevil]

This dude just summarized two semesters of Diff Geo in a single comprehensible under 10 min vid what a Pro

[parthdatar]

Having taken graduate topology and having had some differential geometry, this filled many holes for me.

[mohammedfaizan]

You know he is a legend when he can draw symmetrical opening and closing brackets!

[sambaird]

This is above what I can comprehend right now, but I’m happy there’s someone making concise, beautiful videos for more advanced mathematics topics. I notice people with higher background knowledge on other channels like Numberphile commenting they want to see more equations and logic, and I think this channel fills that void.

[TheoriesofEverything]

Most underrated channel I've seen in the past little while.

[x_gosie]

I’m actively learning topology by self-study, I don’t entire understand some of this right now. But it help me to appreciate how beautiful this subject is!❤️

[kerrychou]

I didn't understand a single thing, and I'm kinda okay with that. Makes me happy that content like this proves the endless capacity for human specialization.

[kartikkalia]

Only few channels which put effort into explaining the hardcore maths rather than just beating the bush like explaining basic concepts over and over.

[MaximQuantum]

me who hasn’t learned vectors yet: *ah yes the derivative*

[dhickey]

This is how to build a mathematically fluent society, one video at a time. Thank you.

[theflaggeddragon]

Beautiful content Aleph 0, I especially loved your explanation of de Rham cohomology. A minor nit pick, however, is that de Rham's theorem doesn't really state that H^k_{dR} = H_{k, sing}. This is really a consequence of the symmetry of de Rham cohomology (namely H^k_{dR} = H^{n-k}_{dR}) and Poincare duality. It's a bit subtle so I'll try to explain carefully.

That H_{dR}^k = H_{sing}, k happens to follow from the "duality" of differential forms, namely that Omega^k(X) and Omega^{n-k}(X) have the same dimension, but may not be canonically isomorphic, but if you endow your manifold a Riemannian metric, there is a canonical isomorphism called the Hodge star. de Rham's theorem, however, is the statement that singular _cohomology_ and de Rham cohomology yield the same answer. This is so-called a _comparison theorem_ between cohomology theories. More precisely, de Rham's theorem states that the integral over k-cycles (modulo k-boundaries) of closed differential k-forms (modulo exact forms whose integral always vanishes) is a perfect pairing. Symbolically, the map ∫:H^k_{dR}(X) x H_{k, sing.}(X) --> R which is a perfect pairing, hence, by linear algebra, we have

de Rham cohomology = H^k_{dR}(X) ~ Hom_R(H_{k, sing}(X), R) = H^k_{sing} = singular cohomology.

The fact then that H^k_{dR} = H_{k, sing.} is an "accident" that follows from the symmetry H^k_{dR} = H^{n-k}_{dR}, namely

H^k_{dR} = H^{n-k}_{dR} = H^{n-k}_{sing.} = H_{k, sing.}

where the last equality is Poincare duality.

I'll say the spirit of your claim is correct, namely that de Rham cohomology gives another way of measuring holes using calculus which coincides with the classical way of doing so, and your video does drive that main point home, especially for the layperson (say a math major), and for that you've done a superb job! It's just that the precise statement is subtly incorrect.

BTW, comparison theorems in cohomology are a very active subject of modern research. For some spectacular examples, look up "p-adic Hodge theory", whose inception in large part was devoted to showing that various l-adic and p-adic cohomology theories (crystalline, algebraic de Rham, etale, etc.) give the same answer. The reason these problems are so extraordinarily difficult is formulating their statements and carriying out their proofs requires a really deep dive into the cohomology of algebraic varieties (first one was etale cohomology from Grothendieck), schemes, stacks, rigid analytic spaces, etc. For some recent progress, check out some of the following papers by Bhargav Bhatt and Peter Scholze.

[anarchistalhazen]

Man, I am not a math major, but I am tempted to change to it now that I know I would be able to say "I study holes"

[oxman]

Hey your content is such high quality I actually cannot believe you only have 1k subscribers!! Cannot wait to see this channel blow up ;)

[OwenMcKinley]

My hat is forever off to those who create/discover and understand this stuff. Remarkable work! Thanks for sharing 😁

[halvard]

I'm coming back to this video in a few years after I know what is happening.

[tau.neutrino]

Despite learning about this years ago it is still the one duality I think about more than any other. Your passion really shows, thank you for sharing it.

[xfigury]

I just saw on Amazon that Tristan Needham is coming out with a visual differential geometry and forms book. I've been waiting for him to write a new book and here it is. If it's anything like his Visual Complex Analysis it will be great.

[happynightmare]

Anybody else watch these types of videos even though you don't understand it?

[arongil]

This video is so satisfying and beautiful and perfect! At first I had no idea how you would fit everything into the nine minutes, but you did by presenting concepts both fast and with intuition. The punchline indeed blew my mind -- I feel I need to look into cohomology more now! Best of all, the presentation did not rely on the viewer thinking at lightning speed. Somehow, even in the nine minute timeframe, you leave time to pause and ponder. And the paper and marker style felt fresh. I liked when you ripped your myths in half! Simply perfection. Inspiring. I don't have time to enumerate the other choices you made that set the video apart, but I'm every second was planned, and it paid off. Keep up the good work!