The derivative isn't what you think it is.
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- čas přidán 31. 05. 2024
- 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.
Homology: This lecture series by Pierre Albin is a beauty. There are a few lectures that cover homology in a slow, accessible way with lots of computational examples. ( • 12. Singular Homology;... )
Cohomology and De Rham’s Theorem: The amazing Fredrich Schuller (who I have raved about in previous videos) has a crystal-clear lecture on Cohomology. ( • Grassmann algebra and ... )
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
The Work of Emmy Noether, one of the great pioneers of Homology: www.britannica.com/biography/...
Thurston’s Paper about the Interpretations of the Derivative: arxiv.org/abs/math/9404236
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MUSIC CREDITS:
Music: www.purple-planet.com
Song: Thinking Ahead
SOFTWARE USED:
Adobe Premiere Elements for Editing
Blender 2.8 for Animations
Follow me!
Twitter: @00aleph00
Instagram: @00aleph00
Intro: (0:00)
Homology: (1:08)
Cohomology: (3:41)
De Rham's Theorem: (7:45)
The Punch Line: (9:02)
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!
What _is_ your familiarity with Geometry? Are you postgrad, grad, an advanced undergrad, or just an obscenely precocious secondary-schooler?
(Or ... ... an autodidact/amateur?)
@@danielsteel5251 Undergrad! Don't know about advanced though :P
@@Aleph0 Very impressive. I'm a math PhD. Didn't specialize in this but studied it (did some work with Hodge decomposition so had to learn this). I think you capture it very nicely here. You have a nice intuition I suspect some professors lack.
anytime you teach something it forces you to master it at a higher level (such that you *can* teach it). i appreciate this video. who knows maybe i'll end up picking up a topology textbook.
@@fcheung2888 This would not be covered in topology. Look into differential geometry. I learned these theorems from Warner's book on the subject, but better introductions likely exist.
Having taken graduate topology and having had some differential geometry, this filled many holes for me.
This comment is brilliant.
^^What that guy said
I read that comment yesterday and it took me until now to understand it....
Same Here. : )
Parth, this is literally me. Did graduate topology (it wasn't algebraic topology, but it gave me a good introduction to topology in general) in my masters course, self studied differential geometry during my phd so I have all these tools and ideas in my head and I've just been missing the link between them all. This video cleared so much up for me.
This dude just summarized two semesters of Diff Geo in a single comprehensible under 10 min vid what a Pro
And Physics II and Calculus III.
Is this the same as Noncommutative Calculus?
Yeah, for me it was 3 semesters. I just watched his video about the Taniyama-Shimura conjecture which summed up the three semesters on modular forms I took. Now I only need to watch two more such videos and my complete advanced math studies are summarized. In less than 45 minutes.
there nothing comprehensible in this video. if you need a math degree to understand it, then it fails.
@@TheZenytram Tell me if this noncommutative music theory as unified reality vid by Fields Medal math professor Alain Connes works for you - I transcribed parts of it in the comments.
You know he is a legend when he can draw symmetrical opening and closing brackets!
Dont do brackets in one go, split them into S and Z shapes and stack them ontop of each other
I don't think there, should he a comma there
My mind has been blown.
i like how you used the verb "draw"
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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.
One might say this channel fills that hole
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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.
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Same lol... I'm in highschool
That's because you shouldn't look at surface-level videos like this imo. Learn differential geometry in a more detailed way
Because there's lots of technical details skipped over that you could say is "boring." Because once the details are filled in step by step, everything becomes "obvious" or "trivial." Because every step is small: a+b=b+a and so on. Every step is like that, small and obvious, it's just that there's lots of steps.
I don't quite understand it either. The video assumes you already know some terminology and how it applies. Boundaries are intuitive enough without a formal definition, but the other stuff is pretty obscure with no definition provided. Linear algebra and abstract algebra cover vector spaces, and I already know what those are, but he still didn't tell me how to add or multiply k-chains by scalars ( which is part of the definition of a vector space ) so I still felt lost.
A good introduction to abstract algebra would clear some things up. It's not really about numbers or variables like standard high school algebra. It's more about systems of things that behave somewhat like numbers and have similar sets of rules. The systems often apply to things that are nothing like ordinary numbers though, things like shapes or types/classes of shapes. It really isn't as horrible as it sounds, but of course it will make absolutely no sense if you don't start with the most basic definitions then slowly work your way up to more complex definitions that use the basic definitions. If you don't know the definitions its all gonna look like jargon. That's just the way math is.
Most underrated channel I've seen in the past little while.
Right back atcha!! ;)
I don't think it is underrated at all, it just needs more content to explode
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Idk, the content seems a little...derivative.
Totally agree with you
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!❤️
hey, thanks a bunch! that's great to hear that you're learning the subject by self-study; out of curiosity, are you studying point-set topology or algebraic topology?
@@Aleph0 My book chapters in four set-theoretic and primarily algebraic chapters.
@@Aleph0 Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Having gone trough diff. geom. and topology in grad school I can safely say these videos are going to be a great complement that will help you flesh out and visualize the rigorous logical analytical explanations in textbooks.
Ok
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.
arxiv.org/abs/1709.07343 etale cohomology of diamonds
arxiv.org/abs/1905.08229 prisms and prismatic cohomology
arxiv.org/abs/1205.3463 p-adic Hodge theory for rigid analytic varieties
me who hasn’t learned vectors yet: *ah yes the derivative*
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
@@hyperduality2838 I had a generalised Stroke’s theorem reading this
This is pretty much oriented for those that know mathematics well. I was supposed to be a mathematician, but didn't understand much
@@frun I’m glad it’s oriented. I wouldn’t know how to integrate it onto my watching habits otherwise
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Only few channels which put effort into explaining the hardcore maths rather than just beating the bush like explaining basic concepts over and over.
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I'm coming back to this video in a few years after I know what is happening.
Remember me too
*Never comes back
so? do you know whats happening now?
This is how to build a mathematically fluent society, one video at a time. Thank you.
2-chain is a rapper and he may have curves but he ain’t no square
And what about his holes?
@@TindoufandNirriti you mean his zero boundaries that are not the boundaries to anything else?
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@@zeusaurel6714 Don't you mean his form with zero derivative, which isn't the derivative of another form?
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"
Kek
Algebraic holes, or A-holes for short
Cycles
Well either that or a gynecologist, but the career path is way longer.
@@magtovi now that i think of it proctologist will also fit
Anybody else watch these types of videos even though you don't understand it?
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Hell yeah. It's awesome to look at unknown beautiful beasts!
You got me red handed 🙋♂️
@@G3Kappa ah... I thought it would be like people looking into this type of stuff
Yeaaaah, that's me :'D
My hat is forever off to those who create/discover and understand this stuff. Remarkable work! Thanks for sharing 😁
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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!
Thank you, that is really gratifying to hear. Most of the planning for these videos is in figuring out how to cover a lot of material while also not overwhelming the listener. I'm glad to hear that you felt that the video struck that balance. Really means a lot. (Also happy to hear that you liked the paper-ripping and pen-on-paper approach!)
Hey your content is such high quality I actually cannot believe you only have 1k subscribers!! Cannot wait to see this channel blow up ;)
Hey thanks!! Glad you stopped by :)
Now it’s 17k... soon to be a few hundred thousand and eventually a million
@@Brad-qw1te 19k in 5+ hours ... The dude is eventually going to collab with 3b1b
24.5k
24.6. I don't see any holes in this chain.
I hope to see more videos from you again soon. You put so much effort into every part of these videos! I hope the audience knows how much value you stuff into the descriptions too.
Great video ! Well done ! A minor point : one should maybe day that derivative and boundary are ‘dual’ rather than ‘opposite’. This duality is really a categorical duality hence the name
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The big bang is a Janus point/hole (two faces) = duality.
@@hyperduality2838 Dot product is dual to wedge product. Cross product is wedge product multiplied by pseudoscalar, and shouldn't have ever existed.
@@user-yb5cn3np5q From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the second law of thermodynamics.
Convex is dual to concave -- mirrors, lenses.
According to the second law of thermodynamics you have a syntropic or hyperbolic perspective hence entropy always increases!
Your mind is syntropic.
Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes or Plato's divided line.
Inclusion (inside, internal) is dual to exclusion (outside, external) -- the Pauli exclusion principle is dual.
Bosons are dual to Fermions -- atomic duality.
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.
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The big bang is a Janus point/hole (two faces) = duality.
That conclusion did blow my mind. I never thought of the derivative as the opposite of the boundary. These are wonderful. Your handwriting is so good, and the style with which this was presented was astounding. Watching all the objects and computer animations was very interesting.
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.
I saw that Visual Complex Analysis book in my student library and thought it was amazing. I can't imagine anyone could do the same with differential forms. I'd love to see it. I never could figure out what the heck differential forms are.
@@theboombody I think the way to visualize it, for 1-forms, is something like "foliations" or "surfaces of constant value of a function", except that for the latter you must remember that the function may only exist locally. The pairing of this with a vector is then proportional to "the number of curves that the vector (drawn as an arrow) crosses". (In the tangent space, where the family of curves looks instead like a bunch of parallel lines.) This is all wishy-washy though.
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I studied differential forms during a one-semester diffgeo course, and have some notions of homology, but I never thought you could link them up together in such a nice geometric way. I was familiar with the De Rham cohomology too, but I never thought about it that way!
I watched this a year ago when I was just starting my Proof Based Differential Multivariable Course course sequence. I'm watching now finishing up my first course in Analysis and Algebra. It's absolutely crazy how it all fits together now compare to then.
That is just the holy grail! This video should be the mandatory introduction of any course on the subject. I tried to study homology and cohomology by myself, and it was a painful experience; basically, I failed, lost in abstractions like exact sequences and functors (I kind of understood the abstract definitions, but I wondered what they were good for, why these notions were introduced in the first place), without even realizing what kind of problem these theories are supposed to solve. The big picture given in this video is just fantastic.
This channel sparked a whole new profound interest in pure mathematics for me. All of these seemingly unconnected ideas in mathematics like derivatives being opposite to boundaries become so elegant when you generalize them. Math is beautiful.
Appreciation for the channel:
Studying physics at the moment, I was simply unaware of the joys of pure mathematics until about two years back, when I first luckily came about such abstractly intriguing stuff on social handles of passionate people in math such as yourself. By now, although I've taken a certain familiarity to these terms-as those used in this video-being thrown around on online math circles, I don't understand them; for firstly, my formal courses never revealed this realm to me, nor did I manage to have enough time to study rigorously pure math from scratch myself. That said, I've tried endeavouring to study LinAlg and Topology from the pure mathematician's perspectives, and from the basics I have tried developing an intuition for (and forgotten other stuff due to halted practice), these videos mean a great deal of precious explanation to me, making for a sort of wormhole opening to these advanced math concepts, that I can grasp with my (limited) lay of the land.
With this small contextualisation I bestow my utmost praise upon this content! Thank you, Aleph0 ^-^
looks like we have ourselves a new 3blue1brown ,definitely subscribing.
Exactly my thought, mate!
@Robin Hack That is true but i didnt mean that from a mathematical perspective I meant that from the perspective of Entertainment and how he presents his content
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More like MinutePhysics and Looking Glass on crack (and I love it)
What a delight! Best quick summary of the connection between Analysis, Topology and Differential Geometry that I know of. Great on Homology, Cohomology, and De Rham's Theorem. Its so beautiful that I have watched three times. Thank you, Aleph Naught.
Holy crap I’m so happy I found this channel. I’ve needed to find a source explaining homology and cohomology in this intuitive way. You my friend are a hero!
Honestly I needed this like 10 years ago when I was struggling through my Masters Analysis course :(
I have a master's degree in algebraic topology and group theory... Things that were kinda off to me at first and required months of study even after my master now seem make more sense than before ! Great job!
I can't even formulate how astonishing this is! Thank you so much for this incredibly clear explanation!!!!
This is probably the reason why I still tolerate google ads. Your channel is giving me goosebumps. I studied fluid dynamics for master and my specific research subject was on solving NS equation using numerical approach that complies with differential geometry. I wish I saw your video when I was doing my literature study, this is definitely a good intuitive explanation.
This is without a doubt the best video I have seen about this topic. Great work!
You have a really unique and original style of presenting complicated topics. Looking forward to your next videos!
Woow, really this channel is growing up on me very quickly, im sure anyone who discovered will love it, it must get far more attention than this, really amazing job dude, keep going ⭐⭐⭐
So lucky this popped up on my recommended; very helpful with my topology course. Please keep making videos. Reading these comments, there's a pretty large audience. Rightly so-there aren't a lot of people making videos on higher level math topics. Even the ones that do seem to cater to non-mathematicians and keep things pretty cursory/basic. Which is fine, but the grad students want something too!
wow I’m floored by how digestible you’ve made this content to someone like me, who only has a peripheral exposure to higher-level mathematics. I don’t know how I’m only just now finding your channel but you are going to be the next rising star in mathematics communication, guaranteed.
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Dude what a timing! I just took manifolds and did not know what to do with these differential forms. Thank you for giving me more intuition into this subject!
Glad to see this channel slowly getting the attention it deserves! Your explanations are great, your diagrams are lovely and your sheer enthusias and passion is amazing! Keep it up! :D
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There was so much effort put into this video that I couldn't not like it!
Very well done 💛
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This was the most amazing thing I've seen! I cannot compliment and thank you enough for this video. Awesome!
This was an awesome video. Understood relationship between homology and cohomology ending with stokes theorem all in 9 minutes
Fantastic video. You obviously put a lot of time into thinking about how to present a topic, and where the audience will get hung up. Great work!
I didn't understand almost anything, but now I have a set of interesting concepts to research into. I prefer not to understand what is precise than understanding what is incomplete. A great video!!!
Hey, great to see you making videos again! Not gonna lie, but this one flew over my head just a little bit. This definitely makes me even more excited to get to the algebric topology stuff (I'm studying point-set topology at the moment), which is rather impressive considering that I was already super super excited for it. What I find interesting on this is that it shows one classic and incredibly powerful trick of a mathematician's trick book, which is taking some key property of an object of interest (In this case the myths you talked about, and the object being the holes) and using it to define the object itself. This can be seen, at least in my experience, particularly in algebra, so I'm not surprised at all that it showed in algebraic topology. I loved too the link at the end with Stoke's Theorem, which to be honest at this point is one of if not my favorite theorem. By the way, did you take that list of the ways of interpreting the derivative from Thurston's letter/article (I really don't know what to call it) "On the Proof and Progress of Mathematics"? Because I just so happen to read it a couple weeks ago, and I have a feeling that I saw it there. Now for some suggestions, considering that you just covered homology and cohomology, I think it might be appropriate to cover ther last one of the "homo" trio in topology; the homotopy. Personally I don't really know that it's all about, but what I've heard about it seems quite interesting. Another topic that might be cool to cover would be Banach and Hilbert spaces. I think there could be an interesting exploration of generalized geometry with Hilbert spaces and more generally with inner product spaces, and of the link that normed spaces and more specifically Banach spaces stablishes between algebra (vector spaces) and analysis (metric spaces). Anyway, lovely video as always, and I'm looking forward to the next ones!
Always love your thoughtful comments! Totally agree with you on Stokes being a favourite theorem -- its the kind of theorem that's so deep that we'd need to meditate on it for hours to understand what it truly means. As per your suggestions, I'm currently learning about Hilbert spaces so I might make a video about that (depending on how well I understand it, that is!). Homotopy is also super fun, so I'll probably find a way to squeeze it in somewhere too 😂
Anything related to Hilbert spaces would be fantastic! I am deeply curious about applications & some conceptual, foundational insight into the idea of Hilbert spaces so I'd love to see a video sharing your thoughts on it.
I love what you're doing with this channel, my friend. Making complex, mind-blowing mathematics approachable is a difficult task to say the very least. Have a good day!
Right now, I'm going through the John M. Lee's "Introduction to Smooth Manifolds" textbook. This video is such a nice introductory to the intuition behind it :)
@Kha Vu Chan Great to hear!
When I first took calculus I stumbled upon this video while trying to wrap my head around the concept of the derivative for the first time. It made absolutely no sense to me but it was also strangely beautiful so I added to my Watch Later playlist until I had learned enough to understand. Now I'm watching it again while procrastinating on a paper I'm writing on homology. This dude planted a seed inside my brain over two years ago and it changed the course of my academic life.
I'm studying industrial engineering, I'm doing some of the IT on my own and now I must say just found a new hobby, self-studying topology.
Thank you very much gentleman.
Greetings from Mexico 🌎🇲🇽
I didn't understand anything because two reasons:
- I don't know English
- I don't know mathematics
X2
XD
F
Awesome video, always so nice to watch videos on subjects that are usually very vague
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This video deserves a published article. Beautiful!
wow!, this was great! Most of it was over my head but I couldn't stop watching. Great content, man! it's condensed and beautifully explained, and illustrated - thanks a lot for posting this
This videos are so enlightening, thank you a lot, keep the good content!
Thanks! Glad you had fun watching them :)
You deserve my professor's salary.
Hats Off man.
Wonderful video ⚡
I really want to say thank you to this video. I took two math undergrad course: differrential geometry and smooth manifold. And both of them studies cohomology, but I had no motivations at all and I could not even have any intuitions from what I learn. This really blowed up my mind and explains most of things that I have learned so far. Thank you.
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Every video on this channel makes me more excited about and gives me a deeper understanding of mathematics, and I have an undergraduate degree in it. Great work shedding a bit of light on the abstract and opaque but quite beautiful field of algebraic topology.
That comment just made my day -- thanks, and I'm glad you enjoyed the vid :)
@@Aleph0 Glad to hear it! Keep it up and this channel will grow I'm sure of it.
This was an *amazing* video on these concepts. Good work, my friend.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
The big bang is a Janus point/hole (two faces) = duality.
I've studied this stuff and the video isnt perfect, but its very impressive you managed to talk about the topic in such an approachable way at all.
I'm glad you point out the video isn't perfect. With such heaps of unqualified praise in the comments, there needs to be some warning to take it with grains of salt.
When you say the video's not perfect, what do you mean? It has gaps? It glosses over details? It oversimplifies? It gets things wrong?
Woah! That was awesome. The piece of abstract math put so intuitively. You've cleared the fog for once and for all, thanks.
You are producing quality explanations, showing a deeper-than-average understanding of mathematics.
What the hell, this might be one of the best maths video I've ever watched
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Wish we had teachers...like u at IITs...a lot of Indian talent would have bloomed.
I hardly understand anything about what's going on here, but the little bit that I do get is wonderful on its own, and I can definitely appreciate the depth of this subject, as well as how the fundamental qualities of derivatives and boundaries are related. It's something you might hear in a conversation and just go "ah, sure, I believe it", and not think about it again. But being shown in detail how it works, even if I'm not fully able to understand, makes it all the more amazing.
The best introductory explanation of what cohomology is that I've come across. I could never get my head around this stuff.
Amazing vid! (literally gave lectures about singular homology and de Rham cohomology only last week, lol)
Few comments though:
- I think your part about de Rham's theorem wasn't entirely correct. Yes, H_k(M) and H^k(M) are isomorphic but just because they have the same dimension. As you explained correctly, the more technical statement is that integration induces a dual pairing between H_k and H^k, but this then induces an iso of H^k and the *dual* of H_k (which is not naturally isomorphic to H_k), so the construction of an isomorphism H^k to H_k would be to fix a basis \omega_1, ..., \omega_n of H^k and then for each i to find a chain such that its integral of \omega_j is 1 if i=j and 0 otherwise.
- You're using some weird cube homology... I haven't really heard about it, does it yield the same homology groups as singular homology? Because you make it seem much more intuitive than triangulating everything ;)
Keep it up! This channel is bound to blow up if you stay dedicated.
Thanks so much!!
@@Aleph0 Absolutely correct, just in case you don't have entire and utter resolve that this channel will reach hundred k's of subs in a few years (maybe even months), this is exactly the point where a YT channel blows up. Beyond exponential how it behaves. Grateful to be a part of the early community here!
Brilliant content! The clarity of the presentation, the drawing and the pace are so enjoyable! Hope your channel grows soon :)
First time here, I subscribed at second 27, without reading the comments. Something you said on the intro made a deep impression on me.
can someone link me to the exact article he displays at the start of video? I am into learning this. I check the description that was something else or it might have changed the underlying content. plz someone get me this. thanks a lot!
1:03 is that taken from Lee? If so is it intro to smooth manifolds or intro to topological manifolds?
Yes! Smooth.
you make this very digestable for people with the right background education. I applaud your successful efforts
First video I see from this channel, already subscribed. The punchline totally got me, I will need some time to bring together all pieces of my blown mind
Hi, I just really really wanted to thank you. So basically, I watched your video 'How to learn maths on your own" and it got me really interested in abstract algebra. I got my hands on Gallian's modern abstract algebra through moderately legal means and I started reading it about a week ago. For a bit of background, in French high-schools we have two maths classes. We have what we call "math speciality"(6 hours a week) and what we call Expert math(3 hours a week). My speciality math teacher doesn't really like me(and he's not particularily good at maths either) so since the beginning of the year I've not been as good in school maths as I was last year. My Expert maths teacher, on the other hand, is a really good teacher and seems really knowledgeable. We began learning arithmetic in Expert maths on monday and I was naturally really good at it since I got a grasp of the main concepts thanks to the book. So my teacher asked me to stay at the end of the lesson and he told me that I should consider going in a preparatory class in Paris(not uncommon for good students in France, but not what i considered either), and that he even could get me recommended. This means that I may become an advanced maths student thanks to you. I really hope that you read this comment, and I wish that you continue making your videos because advanced maths vulgarisation is too rare on the internet :)
Hey Kmeleon, thanks a lot for taking the time to write this. It's really gratifying to hear that you found the self-study video useful, and that you've started reading Gallian's book. Your story is really inspiring! It's also super cool that you're getting recommended for an advanced math class; I'd love to hear how the class goes. Best of luck!
@@Aleph0 Thank you very much, right now we're in partial lockdown in France so it's not going as smoothly as I hoped it would but I think that maths class should be much more enjoyable for me from now on :)
@Kmeleon I'm curious, so the standard high school in France gives two math classes every year? and how many years is the High school over there and in which ages you are expected to enroll to it? 15 to 18?
@@zy9662 It's 3 years and it's from 15 to 18 as you guessed. Basically, at the end of the first year, we are supposed to chose whether or not we'll continue studying maths, if we do, we have a course called "speciality math" which lasts for 4 hours a week. At the end of the second year, we can either discard or keep this course, except that it lasts for 6 hours a week during the final year. Aditionnally, we can chose an extra math course which is called "expert math". In this one, we study complex numbers, matrices and modular arithmetics for example.
@@zy9662 One thing to note is that before university we don't have classes called "algebra", "geometry", "analysis" or whatever. It's just a single course called "mathematics" where the whole curriculum of the year is taught, by the same teacher, to the same group of students. (Well this is not totaly true as there are also more advanced classes you can take in High-School, but its still all mixed in, not separated into different fields of maths). If you say "I have taken Algebra 2", a typical French student (myself included) would not know exactly what that means, if its a very basic or very advanced class.
"the derivative is the opposite of a boundary", isn't that also seen in Gauss's divergence theorem where you take the surface integral of a vector field leaving a volume and equal that to the gradient of the field in the volume over the space?
Yes, it's exactly the same! Such theorems are all fundamental theorems of calculus and they have the same overall "shape"; this is just the generalization to all kinds of derivative (the divergence being one of them)
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Holographic principle anyone?
@@kiharapata how does this cover the fractional derivative in linear algebra 🤔
"And now you can go to school and back going downhill both ways" is such a perfect description for this! It really links with the geometric intuition for a gradient vector field. Excellent video!
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I find your videos the day after I finish my semester in differential forms and manifold calculus, wish it had been a bit sooner! The conciseness and motivation for construction you provide is excellent.
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Best channel on YT according to me alongside 3B1B
High praise! Thanks for stopping by :)
@@Aleph0 just stating facts, also- your video about self study is letting me study mathematics despite the various academic and institutional difficulties in my country. Thank you may Lord Euler make your subs go up like e^x goes up
This was good. Like, really good.
Love from India.
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Blew my mind totally. I knew the fact and involvement of stoke's theorem. But they way you presented it was my entire course on algebraic topology. Thanks for this video. The best thing i have seen today, this video. This never showed off in the course on differential forms either.
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Nice video! Suddenly differential geometry sounds more interesting. I love this kind of thing where you take a mathematical concept and translate it to think about it from several perspectives. This is why Representation Theory is so great.
CZcams algorithms just brought me a very awesome channel!
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I'd clarify that this is the exterior derivative, which is not the same as the usual derivative lol
They are the same, if you're doing multivariable calculus, no? Exterior derivative is also dual to boundary operator of a complex (known in undergrad courses as simply the "differential") with regards to operation of integration, it follows from DeRham theorem.
@@hrsmp Exterior derivative is a morphism of abelian groups/vector spaces, whereas usual differential is a morphism of vector bundles on different spaces, so it really depends on what you mean by ''the same".
@@robertcrumplin1232 can you clarify this without using as many big words?
@@PhilfreezeCH If you have smooth manifolds X,Y (i.e. objects locally looking like R^n) and a smooth map f: X -> Y, the derivative is meant to be some kind of linear approximation to f. So it should be a linear map between the tangent spaces at any point (tangent space you can think of as a tangent plane made into a vector space). This is usually written (df)_p : T_p X -> T_f(p) Y for each point p of X. You can then glue these together to get map df : TX -> TY where TX, TY are called the tangent bundles. Anyway the exterior derivative d is the differential of the de Rham complex, which is dependent only on X say.
Homology is dual to co-homology.
Integration is dual to differentiation -- the generalized Stoke's theorem.
Convergence is dual to divergence.
Union is dual to intersection.
The initial value theorem (IVT) is dual to the final value theorem (FVT) -- optimized control theory.
The dot product is dual to the cross product.
Points are dual to lines -- the principle of duality.
"Always two there are" -- Yoda.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
I do self study math. Your videos for me are a sneak peak ahead. But of course I understand that modern math knowledge builds thoroughly ground up, not vice versa. Though thank you for a huge piece of motivation!
Hey man, I just wanted to say usually your videos feel a little too abstract for the content but this one felt perfect. I actually feel like I got a good understanding of homology groups and cohomology groups which is a subject I haven't had much experience with. The link to the Generalised Stokes' Theorem did indeed blow my mind. Thanks for the content.
Great video! Differential forms do have a beautiful visual interpretation. Sadly, this is not very well known, and almost never taught.
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which book talks about that beautiful visual interpretation of differential forms lol please tell me, All I know is that they're like elements of the cotangent spaces or something lol I'm not even close to understanding them...yet :D
There is no such book unfortunately.....it's hidden knowledge
@@julesjacobs1 lol wdym? really? is there at least a blog post or a stack exchange answer or something?
why does the sphere have a hole?
I watched this at the beginning of differential topology and could not get it. Just came back because I’m taking a course on equivariant cohomology and THANK YOU! This was so helpful, and very interesting too!
You immediately earned a like and a subscription after about a minute and a half of this video. Such have a wonderful approach to explaining - one can immediately tell you're really fascinated by this stuff. So am I... and very glad to have found this channel to help me learn about it.
After 10 years out of academia (with an MPhil in philosophy, formal logic and philosophy of science), I just began a course of studies in computational mathematics in order to get a better chance at grokking Homotopy Type Theory/Univalent Foundations - I think it might actually manage to unify the Logicist, Formalist and "Geometric/Topological" approaches from the Whitehead/Russell/Ramsey-school, the Hilbert-program and the Erlangen-program. The philosophical import is immense (especially in philosophy of science, I think) - and I feel how I imagine a young Russell or Quine must have felt - a new mathematical/logical tool in front of us, barely explored - but obviously incredibly powerful.
But to understand that, you don't just have to understand a good deal of formal logic and proof-theory, but also model-theory, category-theory, linear algebra, topology, algebraic geometry... this channel seems excellent for diving in deeper! - Thank you so much :)
Just came here to check I wasn't the only one watching a video on higher dimensional calculus before heading to bed...
Calculus, topology, group theory. I don't know what's going on
That...blew my mind
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absolutely amazing. thank you so much for making this video and I am so glad I found your channel. I absolutely love this and I really hope your channel becomes popular. I am going to share it with my friends
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OMG this is easily the best explanation on cohomology and it's profound conection with stoke's theorem, wow
please, suggest us some good books for calculus as in the self-study video!
thanks a lot, this channel is great!
Hey Racky, thanks! for 1st year calculus, "Understanding Analysis" by Stephen Abbott is amazing, and I've added links to CZcams videos that follow the book in the self-study video. For differential forms, check out the description of this video for some links for books and lecture videos. Happy learning :)
@@Aleph0 Thanks a lot! I hope this channel grows more and more to help people like me, I already shared it with my friends though, thank you again.
Great video! You said that k-forms aree very abstract and i agree, but there is a way to explain tem as some kind of flow. I dont remember it exactly, but it isnt just abstract (even though i dont understand them peferctly too ;) ) :D
That's really cool, I've never heard of that interpretation. Any video / book / article you know that talks about it?
@@Aleph0 Search for Bixler, Visualizing Exterior Calculus
As with everything, the journey is more important than the destination.
However, I feel like the answer in the end is the most natural intuition you can have about derivatives after having spent sufficient time working with discontinuous functions under integrals.
Great video
So glad I found this channel. Wonderful video!
If k-chains are a vector space, how do you add two maps from [0,1]^k to X together? Obviously you can't just add the respective points in X since it doesn't come with a form of addition... right?
Also, what field is the vector space over? I assumed it was reals because X was related to reals.
That's a really good question. It's a real vector space, and you take *formal* linear combinations of chains. You then define the integral over a formal linear combination of chains to be the linear combination of the integrals over the chains. You're not actually adding functions, because as you pointed out, the target space doesn't have an addition.
(Spivak's book Calculus on Manifolds has a section explaining this is in more detail, but this is basically it.)
@@Aleph0 that sounds like a cheat, how would anyone come up with "ok, let's just sum these two things even if it doesn't make any sense and maybe we'll find holes" ??
@@dottormaelstrom It needed to be done because some primitive chains don't have a boundary that is also a primitive chain. The boundary of a cylinder is 2 circles which doesn't exist in the set of maps from [0,1] to X but it is a linear combination of two maps from [0,1] to X.
@@Aleph0 I may have done some research on the topic and found that generally, the "vector space" is over integers. This would mean it isn't actually a vector space in the end but actually a module that's formed when creating the set of k-chains. I think there's also an equivalency relation between the chains that two primitive chains are equal if one can be continuously morphed into the other.
This seems to line up with the fact that all the homology groups I saw were discrete things like Z^2 for H_1(T) (where T is the torus) which wouldn't make sense if it was the quotient space between two real vector spaces (since such a space would itself have to be a real vector space).