DANGIT that "complicated machinery" that you talk about at the very end is *precisely* what I came to this channel to learn about! I've been wanting to actually *apply* homology theory to a certain space or set of spaces, so I'd like to actually see this stuff in action. I have the intuition already, now I'm wanting more. These are some bomb-ass videos you've got here, they just stop exactly short of what I was hoping to learn about. I hope you continue!
I really enjoyed this series and was/am looking forward to how it progresses. The relationship between differential geometry and simplexes has always fascinated me. I assume you are doing more personal and important things, but wanted you to know your work on this channel is appreciated.
fantastic video! Really perfect buildup of the MOTIVATION for studying simplicies - as a high school student, I found your argument extremely easy to follow and appreciate! Keep up the good work
Awesome series!! (Co)homology, and actually (co)homotopy, is the arithmetic our ancestors could only grok the shadows of. The 'algebra' of homotopy types will one day be taught to school children!
Oh well, a well-animated and directed series on a 'advanced' topic such as (co)homology was too good to be true and so this abnormality had to be corrected. At least it's for a good reason, I wish you success in graduate school even if you don't continue this series.
Marvellous videos! Please, keep them up. Btw, I've just finished watching a course on differential forms and by random chance stumbled on this series. Is it just me or these two talk about all the same things but difforms are more of an analysis perspective and homology is what it looks like from the point of view of topology? I feel the same vibes here.
You're seriously on to something here! That something goes by the name of 'de rahms theorem'! For a brief outline I recommend Aleph 0's video 'the derivative isn't what you think'. Basically, homology has a cousin called cohomology, and differential forms turn out to be a special case of cohomology - called de rahm cohomology. There are a number of theorems that tell us that differential forms tell us essentially the same things about a space as homology does!
You wasted an incredibly long time talking about obvious stuff or things that are not even topology. The entire lesson 2 deals with completely irrelevant calculations on convexity that could have been skipped altogether. And then, as soon as you reach the joucy part, you stop making videos. Huge disappointment
@literallyjustayoutubecomme1591 yes, they have lives, and their lifetime should be spent doing meaningful things. When you embark in a project, overdo the irrelevant parts and then quit, you are wasting your time, as well as the time of the people who have faith in you.
While waiting out BBT's hiatus, fellow could-have-invented-homologists might want to check out Aleph 0's related short video titled "The derivative isn't what you think it is". It pretty much picks up from here and even goes into cohomology. (Sorry I can't paste a direct link. My previous similar comment has just been removed by UA-cam, and there's even no notification informing me "Your comment was removed because of some supposed violation". No, Nothing. Silently removed. Coward UA-cam algorithm!)
Depends what you're looking for! Honestly, I don't really have any *introductory* algebraic topology books that I think and particularly good - Hatcher is the standard but it's a bit......... idk I don't like it that much. You might like. Some people like it a lot, and I can see why! But it's not for me. Pierre Albin has some lecture on youtube which are based on hatcher, and they're great, so if you're into recorded lectures, watch those. The other standard book is Munkres, but I haven't read any of it. Any other books I can think of are not introductory. Imo algebraic topology is missing a standard, well written intro book. Maybe one will come in the future. If you have some familiarity with differential stuff, Bott and Tu's 'Differential forms in algebraic topology' is amazing, but not suited for someone who's not already somewhat familiar with AT imo. That's about all I can think of bookwise right now... algebraic topology is a hard thing to learn, and often you kinda have to patch it together from multiple sources. I highly recommend Pierre Albin's lectures for a first course. Good luck in your endeavours!
@@Boarbarktree Thanks a lot for your recommendations. I once read Hatcher's book, and I am not really a fan of this book. In my first topology course I took during my master's degree, the professor gave us an introduction of algebraic topology, but not the way you are doing here. In fact, this introduction was very fast, so what he taught us was fundamental groups, the \pi_1 functor, in order to characterize compact topological space, via van-kampen theorem (if I remember well)... I have never seen this n-simplex you are talking about in your videos, but I am really enjoying it and hope to have an entire course of AT some day... Anyway, thanks again for the recommendations, I'll take a look as soon as I have time. I am looking forward to your next videos!
Great question! If I were to partition the 2-sphere (write it as a *disjoint* union) then it would be one closed and one open, but here I am using the informal term "breaks up" - the union here isn't disjoint, the balls share a common boundary. That is, you can construct a 2-sphere by gluing two 2-balls along their boundaries. This is in analogy to how the simplices that make up the boundary of a simplex are not disjoint, but rather meet along their boundaries. Does that clear it up for you?
You can construct a map as follows: you choose a point in the interior of the triangle (say, P) to correspond to the center of the disk. Then you scale all the points on the intersection of the triangle and a given ray from P by the distance from P to the boundary of the triangle along the ray. This makes it so that every point on the ray is at most a distance of 1 from P. In particular, the interior points of the triangle are mapped to points at a distance of less than 1 and the boundary points at a distance of 1. I'm not sure how it was animated but this is basically the "blowing up" function described in the video. In fact, this readily generalizes to a homeomorphism between the n-ball and any compact convex set in R^n with nonempty interior.
@@hiltonmarquessantana8202 Yes, that would indeed make the function non injective. For a non convex set it could happen that a ray from P hits the boundary of the set at two places, so scaling both those points by their distance to P would assign them to the same point on the boundary of the sphere.
I assume that in a later episode you will either say that the boundary of a 0-simplex has no faces, or that there *is* a (-1)-simplex and that it is the empty set (and that it has no faces), and that a 0-simplex has one face which is the unique (-1)-simplex ? (Probably the first one. But I think the second one is a nice way to see the, uh, I forgot the name of it. The thing that homology relative to a point is equivalent to. Is it called “Reduced homology”? Idr. Edit: checked, confirmed it is called the reduced homology. I’m not sure why we don’t always use the reduced homology, it seems to me like its definition is more uniform. Edit2: ok, seems like there are some things that work better in reduced homology and some that work better with not reduced homology, and so it is my lack of experience that led me to question why we don’t always use reduced.)
You lost me at 11:55. I don't understand what the problem is, intuitively this should work. If the walls of the cylinder were slightly angled (i.e. if it were a truncated cone), this DOES work, right? So why does it become a problem when the walls are perpendicular to the loop?
A cylinder is a circle stretched through space. It is crucial that opposite sides don't meet - this changes the topology! If they meet at a point, the resulting truncated cone us actually homeomorphic to a disc! The cylinder, however, has a hole through the middle. You can drink through a straw, but you can't drink *through* a bowl! Try the experiment with the balloon and cardboard tube - you won't be able to fit the balloon around the outside (not covering the opening!) without making a hole in the balloon
@@Boarbarktree I had to read your reply 5 times until I understood. You are talking about a cylinder without a top or bottom!!! That makes sense then, of course, but to me a cylinder unambigously consists of wall, top and bottom.
I disagree with the notion that there's no such thing as a (-1) simplex. The (-1) simplex can be seen as the empty set, so a 0 simplex can be seen to have a boundary consisting of 1 face, and that 1 face is homeomorphic to the empty set. Perhaps you intend to clarify this in a future video.
I haven't gone through the details yet but I realized recently that this view is a good way to think about how reduced homology works! I'll include it in a later video, but I'll stick mostly with standard "nonreduced" homology
@@Boarbarktree Very much looking forward to this series. The connection between simplicial complexes and manifolds has been my main area of interest for years now. Specifically, I want to use simplicial complexes to identify both the global topological properties and the local geometric properties of closed Riemann manifolds.
@@zornsllama The final mapping would be 'x' to 'x/(1-||x||)' so the pre image of 0 would be 0, so that is not the problem. We only require continuity in both directions, right? So what do we win using tan?
It DOES NOT REQUIRE ANY MACHINERY TO PROVE THAT RESULT. It's an obvious consequence of just index calculations. The machinery is for more sophisticated constructions.
Everyone learns homology first, of course. With homology, for interesting cases, you can in theory calculate cohomology as in Poincare duality etc. and vice versa. Beyond the first course, it is all cohomology, especially uses in algebraic geometry, number theory, logic because of extensibility of cohomology to sheaves. Characteristic classes are all about cohomology ring of the Grassmannian manifolds and classifying spaces such as B(U(n)) made possible by the Chern-Weil map. The ring structure of cohomology is what makes algebraic topology interesting. Homotopy is way too hard to work with. Homology is too simple mechanically. However, I do not disagree with you as De Rham theory is hard to do without the De Rham Theorem. BTW, K Theory is also a generalized cohomology theory which enable us to calculate Bott Periodicity of the compact Lie groups of the GL(n) genre. You can introduce yourself to cohomology theory with the De Rham stuff. Cobordism is too a generalized cohomology theory too, with which I am somewhat familiar. It was a favorite topic of Grothendiek and Jean Pierre Serre.
You wasted an incredibly long time talking about obvious stuff or things that are not even topology. The entire lesson 2 deals with completely irrelevant calculations on convexity that could have been skipped altogether. And then, as soon as you reach the joucy part, you stop making videos. Huge disappointment
![Imaginary Numbers Are Real [Part 13: Riemann Surfaces]](http://i.ytimg.com/vi/4MmSZrAlqKc/mqdefault.jpg)
One year passed, hoping for part 4!!! Thank you so much for these!
looks like he is a lost UA-camr
1:28 sound design is clearly your true passion
Haha, appreciate the shout out!
AH YES BABY, THIS IS WHAT I'VE BEEN WAITING FOR, THIS IS WHAT IT'S ALL ABOUT. KEEP MAKING EM!
DANGIT
that "complicated machinery" that you talk about at the very end is *precisely* what I came to this channel to learn about!
I've been wanting to actually *apply* homology theory to a certain space or set of spaces, so I'd like to actually see this stuff in action. I have the intuition already, now I'm wanting more.
These are some bomb-ass videos you've got here, they just stop exactly short of what I was hoping to learn about. I hope you continue!
it seems to me he disappeared for unknown reason
if I want to learn about it? how should I go about it? any playlists? what is it called?
@@wargreymon2024 in previous video he said hes sarting his phd, and so he would be busy.
I really like your style! Keep up the good work :)
This channel is about to explode!
And by the way, whenever I hear the intro I am reminded of that chopin piece. Cool idea.
The minor 6th of Chopin Nocturne op.9 no.2
@@mmoose3673 I know
You must be fun at parties lol
Thanks 😁 I'm a big Chopin fan. P.S be nice lol
@UCFeIEAkqvS4fJMTwUtF4OFw Yeah sorry I was joking. Your videos are great!!!
Excellent series, still spreadin joy and understanding. Hope you are doing well out there.
I really enjoyed this series and was/am looking forward to how it progresses. The relationship between differential geometry and simplexes has always fascinated me. I assume you are doing more personal and important things, but wanted you to know your work on this channel is appreciated.
Ooooooh..... Bourbaki - Boarbarktree! 😂
The only and the best tutorial for homology has ended in part 3((( hope the 4th part will come soon.
Mathematics: making choices to avoid making choices
PLEASE MAKE MORE BEFORE MY FINAL! I REALLY NEED THIS!
Can’t wait for the next video!!! Thank you for the time and care you put into these. Good luck with your phd program!
Looks like you can wait a long time. Just when the Oracle was going to reveal the secrets of the universe it was swallowed by a black hole.
Please don’t stop, this series is marvellous
Will you continue this someday? It is too good to end here..
This series is just pefect, I've been following since the first episode. Each one that comes out is more polished than the previous one
fantastic video! Really perfect buildup of the MOTIVATION for studying simplicies - as a high school student, I found your argument extremely easy to follow and appreciate! Keep up the good work
This Channel will make a fine addition to my math channel collection. Good Work, keep it up!
This is REALLY well done and I hope he continues it.
Awesome series!! (Co)homology, and actually (co)homotopy, is the arithmetic our ancestors could only grok the shadows of. The 'algebra' of homotopy types will one day be taught to school children!
Oh well, a well-animated and directed series on a 'advanced' topic such as (co)homology was too good to be true and so this abnormality had to be corrected.
At least it's for a good reason, I wish you success in graduate school even if you don't continue this series.
Number 4 would be great. Excellent description
Boarbarktree's channel needs and deserves much more love!
Thank you so much!!
I hope you return to making content
your animations style is very pretty, and the music choice is quite satifying, I'm really glad that I found this channel!
No part 4! That was a coitus interruptus.
¯\_(ツ)_/¯
@@Boarbarktree will there be a part 4?
truly poetic phrasing
Great work! Please make more videos. Thank you so much!!🧡
Hi. You're the best!! Hope your plans for grad school are moving along well and that you know life is good.
Outstanding design and explanation! Will there be a part 4?
Hello, I'm here since the start. Was hoping to get more videos.
Marvellous videos! Please, keep them up. Btw, I've just finished watching a course on differential forms and by random chance stumbled on this series. Is it just me or these two talk about all the same things but difforms are more of an analysis perspective and homology is what it looks like from the point of view of topology? I feel the same vibes here.
You're seriously on to something here! That something goes by the name of 'de rahms theorem'! For a brief outline I recommend Aleph 0's video 'the derivative isn't what you think'. Basically, homology has a cousin called cohomology, and differential forms turn out to be a special case of cohomology - called de rahm cohomology. There are a number of theorems that tell us that differential forms tell us essentially the same things about a space as homology does!
@@Boarbarktree Oh, thanks! Now that makes sense. Cool! When is the next video gonna be?
You wasted an incredibly long time talking about obvious stuff or things that are not even topology. The entire lesson 2 deals with completely irrelevant calculations on convexity that could have been skipped altogether. And then, as soon as you reach the joucy part, you stop making videos. Huge disappointment
@@aaaab384Relax my man, people have lives
@literallyjustayoutubecomme1591 yes, they have lives, and their lifetime should be spent doing meaningful things. When you embark in a project, overdo the irrelevant parts and then quit, you are wasting your time, as well as the time of the people who have faith in you.
3blue1brown reminded about your channel. But there are no new videos here. :( Please, continue the series, it was sooo great!
so this is what real tragedy looks like, no more video uploaded to this channel in over 2 years
I was very excited to see the notification for this video and not disappointed!!! Super interesting
The music is a perfect fit.
I can’t wait to see you make more!
Still waiting for part 4 😭😭
Outstanding work as usual. These videos are a sheer delight.
While waiting out BBT's hiatus, fellow could-have-invented-homologists might want to check out Aleph 0's related short video titled "The derivative isn't what you think it is". It pretty much picks up from here and even goes into cohomology.
(Sorry I can't paste a direct link. My previous similar comment has just been removed by UA-cam, and there's even no notification informing me "Your comment was removed because of some supposed violation". No, Nothing. Silently removed. Coward UA-cam algorithm!)
Algebraic topology 🐐🐐 🐐
Part 4 please!!
"space with a missing boundary is boundless" (:
Amazing videos keep doing this you’re going to blow up your animation style is very pleasing and your explanations are perfect
Please make more. More
Such a calming music
I want to know more!!! 😀😀😀
I'm here waiting for part 4
💀
Can't wait to see you do a video on De Rham's theorem. :)
Waiting for part 4 :)
Dang, did he stop making these?
No boundary= kind of boundless. Kind a revelation for me.
Wonderful video!
please please please come back!!!!!!
I need moooooooooore
Wonderful
mustve taken quite a bit of restraint to animate the stretching of a balloon onto a cylinder, and not make any jokes
Is there relationship betwwen homology and free modules?
This was EPIC!!!!
Where’s the next video? Is this project discontinued?? 😔
Amazing video!
really nice videos
1:28 i bursted laughing lmao genious
wheres pt 4 ? :(
I miss u
wait what? how long is your hiatus? I am here in 2022. By the way how is your Ph.D. going?
"That can't be it. Where's the rest of it??"
awesome!
12:08 as an enthusiastic category theory beginner I scream in ecstasy whenever I see a commutative diagram
Please, video 4
Could you suggest some references books about Algebric Topology??
Depends what you're looking for! Honestly, I don't really have any *introductory* algebraic topology books that I think and particularly good - Hatcher is the standard but it's a bit......... idk I don't like it that much. You might like. Some people like it a lot, and I can see why! But it's not for me. Pierre Albin has some lecture on youtube which are based on hatcher, and they're great, so if you're into recorded lectures, watch those. The other standard book is Munkres, but I haven't read any of it. Any other books I can think of are not introductory. Imo algebraic topology is missing a standard, well written intro book. Maybe one will come in the future.
If you have some familiarity with differential stuff, Bott and Tu's 'Differential forms in algebraic topology' is amazing, but not suited for someone who's not already somewhat familiar with AT imo.
That's about all I can think of bookwise right now... algebraic topology is a hard thing to learn, and often you kinda have to patch it together from multiple sources. I highly recommend Pierre Albin's lectures for a first course. Good luck in your endeavours!
@@Boarbarktree Thanks a lot for your recommendations. I once read Hatcher's book, and I am not really a fan of this book.
In my first topology course I took during my master's degree, the professor gave us an introduction of algebraic topology, but not the way you are doing here. In fact, this introduction was very fast, so what he taught us was fundamental groups, the \pi_1 functor, in order to characterize compact topological space, via van-kampen theorem (if I remember well)... I have never seen this n-simplex you are talking about in your videos, but I am really enjoying it and hope to have an entire course of AT some day...
Anyway, thanks again for the recommendations, I'll take a look as soon as I have time. I am looking forward to your next videos!
I love this videos (from Perú)
Might be a bit off-topic:
7:06 Why does the 2-sphere breaks up into two closed 2-balls, (instead of one closed and one open 2-balls)?
Great question! If I were to partition the 2-sphere (write it as a *disjoint* union) then it would be one closed and one open, but here I am using the informal term "breaks up" - the union here isn't disjoint, the balls share a common boundary. That is, you can construct a 2-sphere by gluing two 2-balls along their boundaries. This is in analogy to how the simplices that make up the boundary of a simplex are not disjoint, but rather meet along their boundaries. Does that clear it up for you?
@@Boarbarktree Ah, I see. Thank you BBT! Good luck with your school starting affairs~
Nice work, Bro! One question, there is a explicit map between the triangle and disk? If not, how did you animate this transformation?
You can construct a map as follows: you choose a point in the interior of the triangle (say, P) to correspond to the center of the disk. Then you scale all the points on the intersection of the triangle and a given ray from P by the distance from P to the boundary of the triangle along the ray.
This makes it so that every point on the ray is at most a distance of 1 from P. In particular, the interior points of the triangle are mapped to points at a distance of less than 1 and the boundary points at a distance of 1.
I'm not sure how it was animated but this is basically the "blowing up" function described in the video. In fact, this readily generalizes to a homeomorphism between the n-ball and any compact convex set in R^n with nonempty interior.
@@HilbertXVI Nice, I had something similar to this in mind. But this idea would work for concave sets, right? Or this function could be non-injective?
@@hiltonmarquessantana8202 Yes, that would indeed make the function non injective. For a non convex set it could happen that a ray from P hits the boundary of the set at two places, so scaling both those points by their distance to P would assign them to the same point on the boundary of the sphere.
when will your 'hiatus' be over?
I assume that in a later episode you will either say that the boundary of a 0-simplex has no faces, or that there *is* a (-1)-simplex and that it is the empty set (and that it has no faces), and that a 0-simplex has one face which is the unique (-1)-simplex ?
(Probably the first one. But I think the second one is a nice way to see the, uh, I forgot the name of it. The thing that homology relative to a point is equivalent to. Is it called “Reduced homology”? Idr. Edit: checked, confirmed it is called the reduced homology.
I’m not sure why we don’t always use the reduced homology, it seems to me like its definition is more uniform.
Edit2: ok, seems like there are some things that work better in reduced homology and some that work better with not reduced homology, and so it is my lack of experience that led me to question why we don’t always use reduced.)
Subscribed and liked
How does homology relate to homotopies?
NICE love the videos!
5:18 "the boundary of an n-simplex consists of n-1 simplices"
The boundary of an n-simplex consists of n (n-1)-simplices 😉 the terminology is a bit tricky when said out loud, I could have been clearer
Correction: you need n+1 (n-1)-simplices - as many of them as there are vertices of your n-simplex, of which there are n+1
3D tetrahedron’s surface made of 4 2D triangles
You lost me at 11:55. I don't understand what the problem is, intuitively this should work. If the walls of the cylinder were slightly angled (i.e. if it were a truncated cone), this DOES work, right? So why does it become a problem when the walls are perpendicular to the loop?
A cylinder is a circle stretched through space. It is crucial that opposite sides don't meet - this changes the topology! If they meet at a point, the resulting truncated cone us actually homeomorphic to a disc! The cylinder, however, has a hole through the middle. You can drink through a straw, but you can't drink *through* a bowl! Try the experiment with the balloon and cardboard tube - you won't be able to fit the balloon around the outside (not covering the opening!) without making a hole in the balloon
@@Boarbarktree I had to read your reply 5 times until I understood. You are talking about a cylinder without a top or bottom!!! That makes sense then, of course, but to me a cylinder unambigously consists of wall, top and bottom.
Wouldn't the cylinder have to have infinite length? If the cylinder had finite length, it would simply be homeomorphic to the sphere, right?
It doesn't include the end caps! It is like a straw
which software do you use to make those animation?
He stated in the other video that he uses Adobe Animate
Are you okay? You disappeared???
I disagree with the notion that there's no such thing as a (-1) simplex. The (-1) simplex can be seen as the empty set, so a 0 simplex can be seen to have a boundary consisting of 1 face, and that 1 face is homeomorphic to the empty set. Perhaps you intend to clarify this in a future video.
I haven't gone through the details yet but I realized recently that this view is a good way to think about how reduced homology works! I'll include it in a later video, but I'll stick mostly with standard "nonreduced" homology
@@Boarbarktree Very much looking forward to this series. The connection between simplicial complexes and manifolds has been my main area of interest for years now. Specifically, I want to use simplicial complexes to identify both the global topological properties and the local geometric properties of closed Riemann manifolds.
Why tan(||x||)? Why not something simpler like 1/(1-||X||)?
Riddle me this: what is the pre-image of 0 under your proposed map? :)
(A more direct answer: tan(x) is the easiest “standard function” that has all the right properties.)
@@zornsllama The final mapping would be 'x' to 'x/(1-||x||)' so the pre image of 0 would be 0, so that is not the problem. We only require continuity in both directions, right? So what do we win using tan?
@@gijsb4708 ah, I see! In that case it’s really a question of personal preference.
@@zornsllama Alright :)
It DOES NOT REQUIRE ANY MACHINERY TO PROVE THAT RESULT. It's an obvious consequence of just index calculations. The machinery is for more sophisticated constructions.
pity that the author has stopped broadcasting
cohomology is more interesting, has product structure, and generalizable as in generalized cohomology. homology is more provincial.
Well maybe but not for everyone. Homology is more simple to compute and to understand with less prerequisite.
Everyone learns homology first, of course. With homology, for interesting cases, you can in theory calculate cohomology as in Poincare duality etc. and vice versa. Beyond the first course, it is all cohomology, especially uses in algebraic geometry, number theory, logic because of extensibility of cohomology to sheaves. Characteristic classes are all about cohomology ring of the Grassmannian manifolds and classifying spaces such as B(U(n)) made possible by the Chern-Weil map. The ring structure of cohomology is what makes algebraic topology interesting. Homotopy is way too hard to work with. Homology is too simple mechanically. However, I do not disagree with you as De Rham theory is hard to do without the De Rham Theorem. BTW, K Theory is also a generalized cohomology theory which enable us to calculate Bott Periodicity of the compact Lie groups of the GL(n) genre. You can introduce yourself to cohomology theory with the De Rham stuff. Cobordism is too a generalized cohomology theory too, with which I am somewhat familiar. It was a favorite topic of Grothendiek and Jean Pierre Serre.
So I'm assuming you're trying to do research in topology for your Ph.D.?
Uh……..weird Satanic messaging at the beginning?
You wasted an incredibly long time talking about obvious stuff or things that are not even topology. The entire lesson 2 deals with completely irrelevant calculations on convexity that could have been skipped altogether. And then, as soon as you reach the joucy part, you stop making videos. Huge disappointment
"That can't be it. Where's the rest of it??"