@kamik7372 thanks for the nice comments and for letting us know what you are interested in. Manifolds are extremely interesting for Sofia and me, so we will gladly post more often about them and operations on them
@GuillermoSV thanks for the nice comment! We will post another one soon ;) But you can check out this one, it’s technically the next part ruclips.net/video/pNlcQ0Tx4qs/видео.html
@massfikri15 thank you very much for your support!!! This just motives us to produce even better videos, more frequent and to get deeper into each subject!!! 😎👌🏻
@@guscastilloa ¡Muchas gracias por tu apoyo! Me alegra mucho que disfrutes aprendiendo sobre variedades con nosotros. ¡Es un placer compartir este viaje matemático! 😌Déjanos saber qué tipo de contenido te gustaría que publiquemos.😎
You guys deserve more subs! (And yes! more vids on operations on manifolds pls). Also to make it a bit more beginner friendlier, can you add examples or maybe a case base approach so the transition towards abstraction is bit more smooth?
@@RAyLV17 thanks!!! We are glad to know that you enjoy the videos. Thanks for letting us know about what you want us to post about. We are currently working on a video with more examples so that it will be more “tangible” to everyone. Let us know what you think after we post it 😎
Thank you for the video! It’s a nice channel you are running. Just a small comment: I think it would help to have some longer pauses and speak more slowly such that someone seeing this subject for the first time would digest it more efficiently.
@@ליאורפ-ה8כ hi!! Thank you for the constructive criticism, it really helps us. You know, many people are telling us in the comment section exactly this. We are trying to improve the speed and amount of pauses but it seems that the pace is still too fast… we will publish other 2 videos this week with more pauses and slower. Please (if you can) let us know in a future comments if we really fixed the issue 🙏🏻
Pretty good. A few rough spots, but you hit the high points - I was looking for you to mention atlases and smoothness, and you got those. A minor quibble is C^k is a smooth function that is not identically zero. The last bit is why x is C^1, and not C^2; this point is almost never made clear in the definitions used.
The figure 8 is an immersed submanifold of the plane, in particular it is a manifold. It is just not a proper submanifold. Another example the Klein Bottle is a 2d manifold that does self intersect. Also a chart doesn't map to a lower dimensional Euclidean space, it has exactly the same dimension as the manifold. In fact this is how the dimension of a manifold is defined in the first place.
Hmmm...trying to understand what a manifold *is*, but when you first really start talking about manifolds, you just said "let's say we have a manifold" (3:21) and I still don't really know what it is. How is it different from just "a set in R_n" or something? I confess that I didn't really learn what a manifold is from this video.
A manifold is a generalization of a vector space. And since vector spaces can represent abstract mathematical things rather than anything physical, so can manifolds. Manifolds and vector spaces are similar in that locally (around a close neighborhood of a point) the manifold behaves like a vector space. So we can do linear algebra in the context of that point. But they’re different in that the manifold can vary in its local structure as you move from point to point. Think of a straight line function. It’s nice in the sense that local properties like the slope of the line extrapolate globally away from the origin. But if the function were an arbitrary polynomial, then the curviness would mean you can’t know global properties anymore. But you could still know local ones since at any point along the curve, you can approximate it with its own straight line. That’s an example of a 1D manifold. But you can also have more dimensions.
In biology during morphogenesis how does cell differenciate, I suspect the cells knows their location in collective through some kind of cordinate chart which is establish by set of chemical gradient within collective this help them do different thing based on where they are located.
@@nickst2797 I am no expert at this but Cells receive signals from their environment, which can be chemical (morphogens), mechanical (like stiffness), or through direct cell-to-cell communication. These signals trigger intracellular pathways that influence gene expression, leading to differentiation.
@@jammasound nice!!! Would you like a video on the Poincaré Conjecture? (I’m just asking you because I want the pleasure of making it, so I need an excuse to do it hahaha)
@@dibeosyes, you guys got me into math as a math lover. The video where the guy explains it and the lady ask all the question somebody not so good at math would ask actually got me to start reading mathematical literature❤ keep doing what yall are doing ❤ love from the Netherlands ❤
@@Nosa344 hi Nosa, it really means the world to us. Thanks for this comment. We hope you will enjoy the future videos as well, because we try as hard as possible to make it as accessible to everyone as possible ❤️
Great video, as always! I always thought it was a bit weird how charts map elements of the surface to the Rn and not the other way around. Wouldn't it be easier to manage a function that maps the other way? Also, I would love a video about connections on manifolds! I learnt differential geometry using Barrett O'Neill's book, and he used connection 1-forms to define the Levi-Civita connection, but it seems that every other author uses Christoffel symbols instead. Could you make a video elaborating this?
For surfaces, regular parameterizations map from subsets downstairs ( the Rn) to the surface. This is how you get parametric surfaces.if you generate the graph of the surface equation, it is quite easy to get these parameterizations. Consider the equation of a sphere 1=x^2+y^2+z^2 we can get the graph of the upper hemisphere as r(x,y)=(x,y,sqrt(1-x^2-y^2)) for x^2+y^2
@@randomdude8171 Thanks for the nice comment! 😄 Regarding your first point: The use of charts mapping elements of a surface to Rn rather than the other way around is an approach that allows us to study the local structure of a manifold in terms of familiar coordinates in Euclidean space, which simplifies calculations. So, reversing the mapping would complicate the analysis, and kind of miss the point, since the goal is to study the manifold itself, not Rn. For your second question on connections on manifolds: The use of connection 1-forms to define the Levi-Civita connection is a beautiful way to express the geometry of a manifold in differential forms. However, most authors prefer Christoffel symbols because they directly represent the components of the Levi-Civita connection in a coordinate basis, so it makes them easier to compute and more clear in the context of tensor calculus. Both approaches describe the same geometric structure, but they do it in different mathematical languages. A video explaining these two methods would be awesome!! We will definitely do it!
@@shutupimlearning Oh, okay. I can see how it can get way more tricky to parameterize the surfaces with more complex shapes and in higher dimensions. Thank you!
@@dibeos I understand. I figured that it was just easier for us to go from Rn to the surface, seeing how most of the time we use the Euclidean space, but I guess I lost sight of the true goal of studying the surface itself. I would greatly appreciate that video! You guys are the best! 😁
Hi Luca and Sofia Di Beo, very nice explanatory video! I really enjoyed! Can you please explain the Calabi-Yau manifolds, as they are my area of interest ?! I did two videos called The Geometry of Physical Space and the next about Einstein, in which i delved into curvatures and Riemannian manifolds. But I lack the mathematical finesse that you guys have!
Wow, thanks for the nice comment. Yeah, it would be a pleasure. I actually had some classes about the mathematics of the Calabi-Yau manifold in my masters. It is really fascinating! We will definetely do it!
Nicely explained. Subbed. Btw, sometimes you cover your mouth fully with the mic and its a bit weird (for me at least) to see the speaker but not their lips moving
Ciao, hai mai letto "Sheaves in Geometry and Logic" di MacLane e Moerdijk? Viene data una definizione di varietà come un particolare tipo di "cofascio", nel senso che se dai la definizione di fascio e ne prendi la opposta in senso categoriale, ottieni uno spazio localmente euclideo (che poi chiedi sia Hausdorrf e abbia base di aperti numerabile...) È bella perché la puoi applicare in altre categorie cocomplete
@@giovannironchi5332 Ciao Giovanni! Grazie per il commento e per il riferimento interessante. Non ho mai letto questo libro, ma la definizione di varietà tramite il concetto di cofasci che descrivi sembra davvero affascinante, soprattutto per la sua applicabilità nelle categorie cocomplete. Devo cercare più informazioni a riguardo, ma sì, sarebbe davvero interessante esplorare questo tema in un video.
@dylanjayatilaka8533 Manifolds are useful because they allow us to study complex spaces that might not be intuitive by breaking them down into simpler, local charts (atlases). This allows us to apply tools from calculus and linear algebra in local contexts and then relate them back to the entire structure (globally). The kinds of problems we can tackle with manifolds range from understanding the shape of the universe in general relativity to the behavior of phase spaces in classical mechanics. I hope this answers your question, Dylan. We’ll be making videos on this soon!
waiting to learn about lie algebras 😏. Actually sort of understand what manifold is, not on a deep level tho. But it would be amazing to go deeper into the details and operations on the manifolds.
Yes, they are locally homeomorphic. So, if you take a small portion of a circle, it looks like a straight line segment, and you can "zoom in" enough so that this part is indistinguishable from a line segment. This is what it means to be locally homeomorphic - the local structure or "small-scale" geometry is the same. But they are of course not globally homeomorphic, you cannot transform a circle into a line (you cannot glue the ends to make it a circle in topology). Hope this clarifies things!
It has been a while since my last criticism, so I will make a little one. The video is great, the "describe our surroundings" animation is fun, you two are looking beautiful. But ... the question "How to get to Manifolds naturally?" has a well defined answer: - Cartography Manifolds are just the mathematical study of Cartography. That's actually the inspiration. The precise definition is just convention, so not really important. That usually becomes clear when the person studies manifolds with boundary or topological manifolds. The differentiability condition is just to get rid off some cases. For that I imagine a chef frosting a cake. Oh ok, I guess that's not that natural, as the "naturally" in the title would suggest. Haha. Unless it is a vegan cake, I don't know. Hum. It was kind good criticism. I am feeling kind disappointed with myself. Am I losing my dark powers?
I figured out how to embed all n-tori in ℂⁿ, which counts as an equidimensional embedding. Yes, the complex plane _is_ 1D, not 2D. Complex-Riemannian Manifolds are such fun!
Very good. We definitely need more vidéos about manifolds. They are just everywhere....
@@malicksoumare370 yessss we will publish videos on manifolds more often
I really, really appreciate you providing the PDF. It allows for much more careful study of the subject. Merci.
@@goliathsteinbeisser3547 you’re welcome Goliath 😎
Definitely would like to see more stuff on manifolds like operations on manifolds, this is one of the best explanations I've seen on youtube.
@kamik7372 thanks for the nice comments and for letting us know what you are interested in. Manifolds are extremely interesting for Sofia and me, so we will gladly post more often about them and operations on them
A pretty good explanation! I need the second part soon
@GuillermoSV thanks for the nice comment! We will post another one soon ;) But you can check out this one, it’s technically the next part
ruclips.net/video/pNlcQ0Tx4qs/видео.html
So glad this got recommended to me. I'd enjoy more videos like this
@@disnecessaurorex4908 thanks for the nice comment! It really encourages us. Please let us know what kind of content you’d like us to post about 😎
Exceptional audio and language quality.
@massfikri15 thank you very much for your support!!! This just motives us to produce even better videos, more frequent and to get deeper into each subject!!! 😎👌🏻
Thank you very much for your videos, it’s wonderful to learn math with you!! Lovely to address manifolds in a Saturday night 😊
@@guscastilloa ¡Muchas gracias por tu apoyo! Me alegra mucho que disfrutes aprendiendo sobre variedades con nosotros. ¡Es un placer compartir este viaje matemático! 😌Déjanos saber qué tipo de contenido te gustaría que publiquemos.😎
This channel is on a tear fr
@dariuslegacy3406 thanks Darius!!! Let us know what kind of content you’d like to see in the channel 😎
truly an underrated content
@@Yoseph-ph7hh we will get there! Slowly but surely 😎
awesome. I did my graduation project on smooth manifolds, it's a very interesting topic to me.. please continue.. lots of support ❤❤❤❤
@@mahmoudhabib95 thanks for the nice words and for letting us know what you like! We will definitely post more videos on manifolds 😎
You guys deserve more subs! (And yes! more vids on operations on manifolds pls). Also to make it a bit more beginner friendlier, can you add examples or maybe a case base approach so the transition towards abstraction is bit more smooth?
@@RAyLV17 thanks!!! We are glad to know that you enjoy the videos. Thanks for letting us know about what you want us to post about. We are currently working on a video with more examples so that it will be more “tangible” to everyone. Let us know what you think after we post it 😎
Thanks! Yes, I'm interested in learning more about manifolds!
You guys are making great videos!!!! Please carry on...
@@ayandaripa6192 thanks for the encouraging words, Ayan! It really helps us 😎
Please continue, this was very elegant and accessible.
@@Mahmood42978 thanks Mahmood!! Oh yes, there is definitely more coming (especially on Diff Geom 🤫)
Thankyou bro. It make me clear.
More manifold content please! Maybe you guys could describe sheafs next? Or maybe isometries?
Thank you for the video! It’s a nice channel you are running. Just a small comment: I think it would help to have some longer pauses and speak more slowly such that someone seeing this subject for the first time would digest it more efficiently.
@@ליאורפ-ה8כ hi!! Thank you for the constructive criticism, it really helps us. You know, many people are telling us in the comment section exactly this. We are trying to improve the speed and amount of pauses but it seems that the pace is still too fast… we will publish other 2 videos this week with more pauses and slower. Please (if you can) let us know in a future comments if we really fixed the issue 🙏🏻
Pretty good. A few rough spots, but you hit the high points - I was looking for you to mention atlases and smoothness, and you got those. A minor quibble is C^k is a smooth function that is not identically zero. The last bit is why x is C^1, and not C^2; this point is almost never made clear in the definitions used.
@@scottmiller2591 thanks Scott, for the nice words and the constructive criticism, it really helps us 😁
Take a paper, make one fold, then do another, then another, then another.... that's how you get Mani(y)fold 😁
Underrated comment
@@VCT3333 man, why didn’t you give us this rigorous definition before we posted the video? It is waaaaay better than the way we explained it
The figure 8 is an immersed submanifold of the plane, in particular it is a manifold. It is just not a proper submanifold.
Another example the Klein Bottle is a 2d manifold that does self intersect.
Also a chart doesn't map to a lower dimensional Euclidean space, it has exactly the same dimension as the manifold. In fact this is how the dimension of a manifold is defined in the first place.
Hmmm...trying to understand what a manifold *is*, but when you first really start talking about manifolds, you just said "let's say we have a manifold" (3:21) and I still don't really know what it is. How is it different from just "a set in R_n" or something? I confess that I didn't really learn what a manifold is from this video.
same here
A manifold is a generalization of a vector space. And since vector spaces can represent abstract mathematical things rather than anything physical, so can manifolds. Manifolds and vector spaces are similar in that locally (around a close neighborhood of a point) the manifold behaves like a vector space. So we can do linear algebra in the context of that point. But they’re different in that the manifold can vary in its local structure as you move from point to point.
Think of a straight line function. It’s nice in the sense that local properties like the slope of the line extrapolate globally away from the origin. But if the function were an arbitrary polynomial, then the curviness would mean you can’t know global properties anymore. But you could still know local ones since at any point along the curve, you can approximate it with its own straight line. That’s an example of a 1D manifold. But you can also have more dimensions.
In biology during morphogenesis how does cell differenciate, I suspect the cells knows their location in collective through some kind of cordinate chart which is establish by set of chemical gradient within collective this help them do different thing based on where they are located.
@6ygfddgghhbvdx I don’t know much about biology to say, but it would be interesting if that’s indeed the case
Hey wait! I always had this question. You mean this riddle isn't yet solved by biology?
@@nickst2797
I am no expert at this but Cells receive signals from their environment, which can be chemical (morphogens), mechanical (like stiffness), or through direct cell-to-cell communication. These signals trigger intracellular pathways that influence gene expression, leading to differentiation.
Cool. Its funny because I just started reading "The Poincare Conjecture" and it was discussing this very thing!
@@jammasound nice!!! Would you like a video on the Poincaré Conjecture? (I’m just asking you because I want the pleasure of making it, so I need an excuse to do it hahaha)
@@dibeos LOL Of course I would love it!
@@dibeosyes, you guys got me into math as a math lover. The video where the guy explains it and the lady ask all the question somebody not so good at math would ask actually got me to start reading mathematical literature❤ keep doing what yall are doing ❤ love from the Netherlands ❤
@@Nosa344 hi Nosa, it really means the world to us. Thanks for this comment. We hope you will enjoy the future videos as well, because we try as hard as possible to make it as accessible to everyone as possible ❤️
Math textbooks be like:
“Ok here is 1 + 1 =2”
(Next page)
“Every module is a direct limit of finitely generated submodules”
Literally 😂
Great video, as always!
I always thought it was a bit weird how charts map elements of the surface to the Rn and not the other way around. Wouldn't it be easier to manage a function that maps the other way?
Also, I would love a video about connections on manifolds! I learnt differential geometry using Barrett O'Neill's book, and he used connection 1-forms to define the Levi-Civita connection, but it seems that every other author uses Christoffel symbols instead. Could you make a video elaborating this?
For surfaces, regular parameterizations map from subsets downstairs ( the Rn) to the surface. This is how you get parametric surfaces.if you generate the graph of the surface equation, it is quite easy to get these parameterizations.
Consider the equation of a sphere 1=x^2+y^2+z^2 we can get the graph of the upper hemisphere as
r(x,y)=(x,y,sqrt(1-x^2-y^2)) for x^2+y^2
@@randomdude8171 Thanks for the nice comment! 😄
Regarding your first point: The use of charts mapping elements of a surface to Rn rather than the other way around is an approach that allows us to study the local structure of a manifold in terms of familiar coordinates in Euclidean space, which simplifies calculations. So, reversing the mapping would complicate the analysis, and kind of miss the point, since the goal is to study the manifold itself, not Rn.
For your second question on connections on manifolds: The use of connection 1-forms to define the Levi-Civita connection is a beautiful way to express the geometry of a manifold in differential forms. However, most authors prefer Christoffel symbols because they directly represent the components of the Levi-Civita connection in a coordinate basis, so it makes them easier to compute and more clear in the context of tensor calculus. Both approaches describe the same geometric structure, but they do it in different mathematical languages.
A video explaining these two methods would be awesome!! We will definitely do it!
@@shutupimlearning Oh, okay. I can see how it can get way more tricky to parameterize the surfaces with more complex shapes and in higher dimensions. Thank you!
@@dibeos I understand. I figured that it was just easier for us to go from Rn to the surface, seeing how most of the time we use the Euclidean space, but I guess I lost sight of the true goal of studying the surface itself.
I would greatly appreciate that video! You guys are the best! 😁
Hi Luca and Sofia Di Beo, very nice explanatory video! I really enjoyed! Can you please explain the Calabi-Yau manifolds, as they are my area of interest ?! I did two videos called The Geometry of Physical Space and the next about Einstein, in which i delved into curvatures and Riemannian manifolds. But I lack the mathematical finesse that you guys have!
Wow, thanks for the nice comment. Yeah, it would be a pleasure. I actually had some classes about the mathematics of the Calabi-Yau manifold in my masters. It is really fascinating! We will definetely do it!
@@dibeos Thank you! I'm not professional, just trying to understanding the basics and the intuition behind them!
Great stuff!
@@MathwithMing thanks Ming! Let us know what kind of content you’d like us to publish videos about 😎
Hey, I learned something today!
@@masterandexpert288 it’s an honor to get a comment like that from a Master and Expert 😎
vcs são incríveis!
@@ramaronin obrigado Ramon!
Yes interested
@@johnborja6518 yes John, thanks!
I hope you can make the next video
@@vincentalejandrogonzalezan9095 us too, Vincent… us too 😎
Nicely explained. Subbed.
Btw, sometimes you cover your mouth fully with the mic and its a bit weird (for me at least) to see the speaker but not their lips moving
Ciao, hai mai letto "Sheaves in Geometry and Logic" di MacLane e Moerdijk? Viene data una definizione di varietà come un particolare tipo di "cofascio", nel senso che se dai la definizione di fascio e ne prendi la opposta in senso categoriale, ottieni uno spazio localmente euclideo (che poi chiedi sia Hausdorrf e abbia base di aperti numerabile...)
È bella perché la puoi applicare in altre categorie cocomplete
@@giovannironchi5332 Ciao Giovanni! Grazie per il commento e per il riferimento interessante. Non ho mai letto questo libro, ma la definizione di varietà tramite il concetto di cofasci che descrivi sembra davvero affascinante, soprattutto per la sua applicabilità nelle categorie cocomplete. Devo cercare più informazioni a riguardo, ma sì, sarebbe davvero interessante esplorare questo tema in un video.
More please
@@censoredamerican3331 yessssss 😎👍🏻
❤❤❤
@@omargaber3122 thanks Omar. Your comment means more than a thousand words 😎❤️
So manifolds are things that can by related to atlases. Clear. Then? What problem can we see the solution to, with that?
@dylanjayatilaka8533 Manifolds are useful because they allow us to study complex spaces that might not be intuitive by breaking them down into simpler, local charts (atlases). This allows us to apply tools from calculus and linear algebra in local contexts and then relate them back to the entire structure (globally). The kinds of problems we can tackle with manifolds range from understanding the shape of the universe in general relativity to the behavior of phase spaces in classical mechanics. I hope this answers your question, Dylan. We’ll be making videos on this soon!
waiting to learn about lie algebras 😏.
Actually sort of understand what manifold is, not on a deep level tho. But it would be amazing to go deeper into the details and operations on the manifolds.
@@victork8708 hi Victor, yes our next few videos will be mainly on diff geom and topology, but right after that we will make the one on Lie Algebras
very nice
@@benjaminleonpape thanks Benjamin ;)
@@dibeosi really like that you link some book sources and pdfs summarizing the video's topic
@@benjaminleonpape thanks for letting us know that, really. We will keep doing that 😌👌🏻
Commenting for the algorithm
@@Jop_pop Replying for the algorithm. (Thanks though, appreciate it)
A line segment is ABSOLUTELY not homeomorphic to a closed loop? What do you mean??
Yes, they are locally homeomorphic. So, if you take a small portion of a circle, it looks like a straight line segment, and you can "zoom in" enough so that this part is indistinguishable from a line segment. This is what it means to be locally homeomorphic - the local structure or "small-scale" geometry is the same. But they are of course not globally homeomorphic, you cannot transform a circle into a line (you cannot glue the ends to make it a circle in topology). Hope this clarifies things!
@@dibeos Yes that is right of course. I just objected to what you said (and wrote) in the video. I have no doubt that you actually know it better ;)
I`ll go google if there are a non integer dementional manifolds
It has been a while since my last criticism, so I will make a little one. The video is great, the "describe our surroundings" animation is fun, you two are looking beautiful. But ...
the question "How to get to Manifolds naturally?" has a well defined answer:
- Cartography
Manifolds are just the mathematical study of Cartography. That's actually the inspiration. The precise definition is just convention, so not really important. That usually becomes clear when the person studies manifolds with boundary or topological manifolds. The differentiability condition is just to get rid off some cases. For that I imagine a chef frosting a cake. Oh ok, I guess that's not that natural, as the "naturally" in the title would suggest. Haha. Unless it is a vegan cake, I don't know.
Hum. It was kind good criticism. I am feeling kind disappointed with myself. Am I losing my dark powers?
@@samueldeandrade8535yeah, you are losing it a little bit hahaha but I like you way better this way, to be honest 😂
This is making my Brane hurt.
10k views is criminal.
@neoieo5832 couldn’t agree more 😎 little by little the algorithm will catch up
hi
@@ucngominh3354 what’s up
I figured out how to embed all n-tori in ℂⁿ, which counts as an equidimensional embedding. Yes, the complex plane _is_ 1D, not 2D. Complex-Riemannian Manifolds are such fun!
@@PerpetualScience that’s awesome, really!