Proving Brouwer's Fixed Point Theorem | Infinite Series
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- Опубліковано 17 січ 2018
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There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra.
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Analogous to the relationship between geometry and algebra, there is a mathematical “portal” from a looser version of geometry -- topology -- to a more “sophisticated” version of algebra. This portal can take problems that are very difficult to solve topologically, and recast them in an algebraic light, where the answers may become easier.
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REFERENCES
The functor in today’s episode is called “the fundamental group.” To learn more about the fundamental group and the proof of Brouwer’s Fixed Point Theorem, check out:
Brouwer's Fixed Point Theorem (Proof) on Math3ma:
www.math3ma.com/mathema/2018/1...
Algebraic Topology by Allen Hatcher, page 31:
www.math.cornell.edu/~hatcher...
A Concise Course in Algebraic Topology by Peter May, page 10: www.math.uchicago.edu/~may/CO...
Category Theory in Context by Emily Riehl, page 15: www.math.jhu.edu/~eriehl/conte...
To learn more about algebraic topology, check out:
Algebraic Topology: An Introduction by W. S. Massey www.amazon.com/Algebraic-Topo...
Elements of Algebraic Topology by J. Munkres www.amazon.com/Elements-Algeb...
To learn more about category theory and functors, check out:
“What is category theory, anyway?” www.math3ma.com/mathema/2017/1...
“What is a functor?” www.math3ma.com/mathema/2017/1...
VSauce - Fixed Points
• Fixed Points
Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. • Topology vs "a" Topolo...
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Hey everyone! Today’s episode is a classic illustration of algebraic topology, a branch of math that uses tools of abstract algebra to solve problems in topology. The name of the functor (a.k.a. “portal”) that appears in the proof of Brouwer’s Fixed Point Theorem is “the fundamental group.” We’ve put some links in the description where you can learn more about algebraic topology, the fundamental group, category theory, and today’s proof. Enjoy!
Yes, exactly! Although not mentioned in the video, the same proof works in higher dimensions using the higher homotopy groups. We only showed the case for 2 dimensions.
While the higher homotopy groups are relatively easy to define (instead of loops, aka circles, aka 1-dimensional spheres, you use n-dimensional spheres), they're a lot harder to compute. A real surprise is the fact that π_3(S^2) is not zero; you can map a 3-sphere into a 2-sphere (aka a regular sphere) in such a way that it can't be contracted to a point! This is the Hopf map.
Joseph Van Name. I'm OK with that
For the curious, there's something else called the _homology_ groups that are much harder to define (and motivate) but are much easier to compute. This is why the Brouwer Fixed Point Theorem in higher dimensions is usually proven using homology rather than homotopy.
PBS Infinite Series Umm... Did you guys just explain that if I walk into a room and a molecule of air moves from one side of the room to the other... it's still the same molecule!??!? lol science
My 8th-grade math teacher liked to invite his former students back to give guest lessons. When it was my day to teach, I attempted to explain Brouwer's fixed point theorem using Sperner's 1928 proof (and, admittedly, a lot of simplification and hand-waving at the end). I have fond memories of that day, and this theorem by association, so it made me happy to see this video in my subscription feed :D
Disc is zero and circle is integers? Why? The rest of the proof is obvious, but the most important thing is not explained.
It's about the composition of loops. If you have a loop in the disc, you can always continuously retract that into a point. Now, consider any two loops on the disc, say α and β. Since you can continuously retract any loop on the disc to a point, we can continuously retract α and β in two points. Call the point that we have retracted the loop α into γ, and call the point that we have retracted the loop β into ω. What we can now do is we can create a path that goes from γ to ω, and we can then continuously retract that into a single point, which we can without loss of generality assume to be the origin. Since you can do this procedure for /every loop in the disc/, they are the same up to a concept called "homotopy".
As for the circle, if you have a loop that goes around the circle, say, 4 times clockwise, there is no way you can continuously deform that loop into a loop that goes around the circle n times clockwise if n is different from 4, or m times anti-clockwise for any m. It's also not possible to continuously deform it into a loop that doesn't go around the circle. So if we have a loop that goes around the circle 4 times, we cannot change the fundamental property that it loops around the circle 4 times without considering discontinuous deformations. As such, when we /compose two loops/ defined on the circle, say λ and μ, we first go around the circle as many times and in the same direction as the loop λ tells us to do, and we then go around the circle as many times and in the same direction as the loop μ tells us to do.
The effective result of the multiplication property of loops λ and μ on the circle is that, if we know how many times and in which direction both loops λ and μ go around the circle, we know how many times and in which direction the loop μλ goes. Furthermore, to find this number of times we go around the circle, and if we write -3 for a loop that goes around the circle 3 times in an anti-clockwise manner, for instance, and we call the number of times we go around the circle for φ(μ) and φ(λ), then we get the remarkable equality that φ(μλ) = φ(μ) + φ(λ). And /THAT IS THE FUNDAMENTAL PROPERTY OF THE INTEGERS/! And that is why, algebraically speaking, a circle is just the integers.
I hope this wasn't too complicated! If it was, I can clarify some things for you.
RandomBurfness Your explanation of topological properties makes perfect sense, but seems to have no connection whatsoever to fixed-points.
وفسّر الماء بعد جهد بالماء
its called fundamental group. there is a video on this channel u could watch, maybe that will make things clearer. If there a still questions, just ask =)
Topology Riddles | Infinite Series watch?v=H8qwqGjOlSE&t=340s
What about the case we go around circle 0 times, it makes no sense to me. (2:43)
Personally, I feel like this should had been a small follow-up after a video about the relation between topology and algebra, or something. It may not be self-contained, but I'm sure I speak for more than myself when I claim that we are not satisfied with a huge part of a proof being "hidden away" as a "portal."
tuchapoltr It is still a valid attention paratoner for the main big subject imo. Fixed point theorem was quite unintuitive at first to me. However your approach makes sense too.
I agree with you that she could have been way more specific. But nevertheless, those "portals" are what I especially love about mathematics as well. (And that such "fantastic" words grip the essence of mathematical situations in so many cases - I love to speak about maths in that way!) Those portals show that in the end everything is connected even if different subcategories of mathematics often seem so far away from each other.
Gonna agree with everyone else here, you're gonna have to explain how the "portal" actually works and WHY it works, otherwise this is magical nothingness and is not useful to people who don't already understand why it works
Don't think she can explain the fundamental group and the basis od algebraic topology in a video like this
@@alexsaiz3741Lol I don't think she could even in 10 videos like this
It's a metaphor
This is what I hear on the video: Let us assume there is a portal and that the circle are the integers and the disk is 0 thefore Brouwer's Fixed Point Theorem is true, and we are not going into any detail to explain it.
That's what a lot of categorical proofs feel like, but it is actually true that any functor between these categories with these properties will be sufficient for this proof, regardless of how it's defined.
Math is math and making up numbers to give value to show proof of a theory isnt math. Its is imagination that isn't solving a problem its creative one.
That's not what I hear people are hung up on the contradiction and assumption it's false to then prove itself
I was excited to see this pop up, but dissapointed in the execution. While the proof presented is valid, the pedagogical value is missing due to one crucial omission. When explaining a proof by contradiction, I find it extremely useful to "unravel" the chain of assumptions and show why the one we're trying to prove false is the only one that can possibly be false. Ex.
We have shown that a pair of functions from the circle to the (closed) disc and back that commute with identity cannot exist due to the functor between topology and abstract algebra. Because we always have a function from the circle to the (closed) disc by inclusion, the thing that cannot exist is the map back that we called h(x). Since we now know h cannot exist, one of its assumptions must be false. Since assigning a point in the disc to a point on the boundary by drawing a ray through some *other* point always works, the assumption that ∀x, g(x) ≠ x is the only thing that can be incorrect. Thus ∃x, g(x)=x, proving the Fixed Point Theorem.
Matt Olkowski It's exactly the same as cleaning up the call stack when evaluating a functional program. Proper proofs have a symmetry to them in which we start somewhere with an idea of where we're going, get there through a chain of modus ponens, then return with our nugget of truth, wrapping it in contex and meaning as we move back to the place from which we came. By skipping the "second half" a lot of people (as can be seen from the other comments) are left with a "wait, why does that prove the theorem?" feeling.
Yes, but neither you nor the presenter proved that h(x) is actually a continuous function, you've only shown how to construct if from g(x). This should be part of the proof since the continuity of h(x) is part of the contradiction
Actually your extension of the proof in the video, makes it a lot clearer. I understood anyway, but you're right in saying that this should have be done
DarkChaos110 Fair enough, but the continuity of h is rather straightforward from the continuity of g, which is assumed. For any given ε of distance from h(x) on the circle, there's very obviously a δ within which we can adjust x to get a g(x) so that the ray from x through g(x) intersects the circle within ε of h(x). The exact calculation depends on the disc we're working in, but it should be evident that h(x)'s construction cannot introduce discontinuity.
I didn't really get this video, but your explanation here made it click for me.
One thing that was keeping my from understanding the video is that the functor wasn't really explained too well, so my first instinct when it was shown that the functor wouldn't work properly was to say that the functor doesn't necessarily hold true all the time as a model, not that ∀x, g(x) ≠ x could be wrong.
As an intuitionist mathematician, Brouwer would turn in his grave if he saw this proof using contradiction.
Nice introduction to functors though :)
is it true, that the original proof was via calculus (for smooth functions) and then there was employed some sort of smooth approximation of continuous maps? Do you know about some modern account on the original proof?
+
Did he proove it in another way? I thought he turned away from this theorem and other results afterwards, just because it was not intuitive enough for him :D
@@semikolonne I believe this is correct
@@semikolonne - I read he turned away from topology and gave only lecture about intuitionism, because the latter interested him much more. I doubt that he had only non-constructive proof of his fixpoint theorem.
This video seems to be attracting quite a bit of negative feedback, scrolling through the comments. I... like what they're trying to do with it, but I can also see how it can fall down in practice. The videos under the new hosts are trying to squish more complicated ideas into the same time slot, and something, unfortunately, has to give. This explanation of Brouwer's Fixed Point Theorem, for example, skims over several important details, leading to the explanation feeling unsatisfying. Under the old videos, you had simpler concepts, but explained thoroughly, and, I feel, allowed you to take the ideas presented and use the ideas more generally as well - the knowledge being imparted also felt like deeper knowledge rather than the shallow knowledge that is hard to replicate elsewhere that's presented in videos like this one. (Whether or not that's actually the case, though, is likely an entirely different story, but that's what it feels like to a viewer.)
I think this is a part of the growing pains of the new hosts, and I'm sure this is something that will get better with time - while there are problems, I think some of the comments are perhaps overly dramatic.
I felt the same exact way when Matthew O'Dowd replaced Gabe in PBS SpaceTime. And when Jay Leno replaced Johnny whats-his-name on the Late Night Show.
So of course, if a disc is represented by 0, it will be a 'black hole," i.e. any function that maps to it will only map to one thing. But you didn't convince me that the disc should be mapped to 0. If a circle maps to the integers, then I would think the disc should map to some bigger set, like the real numbers. Mapping the disc to 0 is the heart of your proof by contradiction, and I would love to understand that choice better.
The mapping from topological spaces that is referenced in the video is called The Fundamental Group Functor. As the name suggests, it always outputs a mathematical object known as a group. The basic idea is that a group is created from the collection of fundamentally different loops that can be formed in the topological space. Because all of the loops in the disk are contactable to a single point, no two loops are fundamentally distinct. Hence, the group assigned to the disk must have a single element. Since all groups contain an identity element, a one element group must be simply the identity element, namely {0}.
Jeffrey Bernath exactly! Although I am not demanding a rigorous proof of why the disc can only be mapped to zero it is necessary to have constructed a better appreciation of what we are even doing here! "All loops are points, therefore zero!" doesn't cut it!
Hi Jeffrey, I'm glad you asked. The functor ("portal") assigns a very particular group to each topological space, called the fundamental group. I didn't mention this in the video, but this group is constructed by taking "homotopy classes of continuous functions of the circle into the space." In other words, a group element is an equivalence class of loop in the space (with a fixed base point), where we consider two loops to be equivalent if one can be "wiggled" (i.e. homotoped) to another. Once you unwind all this, it turns out that the fundamental group of the disc is 0, i.e.the trivial group. In other words, there is only *one* equivalence class since (and here's the key) *every loop in a disc can be "wiggled" (i.e. homotoped) to a point.* Therefore the functor assigns the disc to 0. Interestingly enough, the proof that the fundamental group of the circle is the integers is high non-trivial, so I'm glad it made sense intuitively! If you're interested in more details, be sure to check out the links in the description!
-Tai-Danae
See also "Who (else) cares about topology? Stolen necklaces and Borsuk-Ulam" by 3Blue1Brown
I am very disappointed with this channel since the hosts changed. I am not familiar with algebraic topology but I have a master's degree in computer science, so I do know some maths. This video did not provide any insights into algebraic topogogy, since there was no mathematical argument to follow. The "proof" was based on the claim that there is a "bridge" between topology and algebra but instead of properly definig it or illustrating why this "bridge" would make sense with examples, they show an example where it does not make sense and state that this is a contradiction.
While the hosts appear as they know what they are talking about, they lack some very fundamental didactic skills.
This comment expresses most clearly why I found the video unsatisfying. We need some indication of why we should believe that mapping the disk to 0 and the boundary to the integers is meaningful in the context of continuous functions, and the best way to do that (for an audience who aren't already topologists) would be to show some examples of how various intuitively obvious properties of the disk and the boundary map naturally to 0 and the integers.
So... why exactly do these scenarios correspond? Why does a circle get "assigned" Z? Why does the disk get "assigned" 0? What does this have to do with loops? Why this assignment in particular, and not something else? What's the motivation behind it, if any? The video left me more confused than I was about BFPT.
Vsauce video about BFPT provides some really good intuition and insight, you may want to see it. This one was terrible.
Good question. I gave an bird's-eye-view explanation in the video, but you're right to ask for more! Here's the answer in full detail: This functor assigns a very particular group to each topological space, called the fundamental group. This group is constructed by taking "homotopy classes of continuous functions of the circle into the space" - a.k.a. equivalence classes of loops - and the group operation is determined by loop concatenation. The two topological spaces we see in the video - a disc and a circle - each have their own fundamental group. And once you unwind the definition, you discover that the fundamental group of the disc is isomorphic to 0, the trivial group, and the fundamental group of the circle is isomorphic to the integers. But the proof of this is highly non trivial - it involve some very heavy machinery. (If you're interested though, definitely check out the links in the description!)
The motivation for using the fundamental group is that it allows us to *probe* topological spaces with loops. Loosely speaking, it is a "hole-detector" - it's how we can tell topological spaces apart from each other. Formally: if the fundamental groups of two spaces are different, then the two spaces must different (i.e. they are not homeomorphic). This is a key theorem in algebraic topology.
-Tai-Danae
You should explain the analogies you are using before using them, or it gets very hard to understand you. Also, you have not explained how it works.
I used to love this channel, but I have not seen any of her videos and left with the feeling I really understood her.
The nice thing about Space Time and Infinite Series is that it simplifies the topic they are explaining without turning it into something else as much as they can but not more. Like the series on Hawkin's radiation that Space Time has just started. She doesn't do that, and has really disappointed me
Who is "her"?
I am missing how you can call a function that takes any number to a single one (0) an analogue to a function that takes any point and maps it to a discrete point. i.e., two points on the circle map to to different points on the disc, but two integers both map to a single integer.
Phil Maggiacomo I tried thinking about it from a set theory perspective, and ended up coming to the conclusion that it makes no sense to say that converting an infinite set to another infinite set them back to the original infinite set is the same thing as converting an infinite set to a finite set them back to an infinite set. But I’m not sure whether that proves the video, the points out a flaw.
The way that she explains things sometimes makes them more difficult to understand than I feel they should be.
...'oh boy', the cowlick theorem, the combing hair on a sphere theorem, (or in Professors' Spring Fling, the undergrad's question about collapsing a sphere while tracing Brouwer's fixed point to any interior point when there's only one such fixed point it can't return)...
To explain it more rigorous: We assume the theorem is false. That way we construct a continous function h from the disk to the circle. Let i be the function from the circle to the disc that maps each point on the boundary to itself. Then h(i(x)) is a function from the circle to the circle that maps each point to itself, h(i(x))=x.
Now homotopy theory comes into place: Let i* be the function that maps each class of loops [c(t)] in the circle to the class [i(c(t))] of loops in the disc. Since there is only one class of loops in the disc (represented by the number 0), i* maps every class of loops in the circle (represented by a whole number) to 0, so i*([c])=0 for every loop c in the circle. Let h* be the function that maps each class of loops [c(t)] in the disc to the class [h(c(t))] in the circle. Since there is only one class of loops in the disc (0), h* takes only on one value (which is also 0, because the trivial loop gets mapped to the trivial loop). So h*(i*([c(t)]))=0 for every loop c in circle. But that can't be if h(i(x))=x, because it should be [c(t)]=[h(i(c(t)))]=h*(i*([c(t)]))=0 for every loop in the circle, but the classes of loops in the circle are represented by the entire whole numbers, not just 0. Contradiction.
Excellent. Thank you, that makes it much clearer.
Wonderful explanation, thank you
I still don't get it completely, but I can finally see what the proof is built on. Thank you.
I will ponder a bit longer and look for more videos that explain it like you did. I wish they had it done in this video.
Please do more videos on Topology
+
Michael Novak videos*
Jorgete Panete Thank you. I fixed it now.
This is great! I love the way that you teach, very clear and deep at the same time :)
Thanks for this wonderful video!!
I am having trouble understanding the video after 5:10. Why are you constructing this arrow between x and g(x)?. I also don't understand the 'boring' function you reference at 5:35. It seems to me that this function leads to the loss of information, not h(x).
Russell Schwartz she draws an arrow between g(x) and x to make a mapping from the disk to the boundary. Note that this is only possible because g(x) never equals x.
The boring function is simply the function that takes as input points on the circle and places them on the boundary of the disk. I'll call it i(x).
Now we have a continuous function from the circle to the disk i(x), and we have a continuous function from the disk to the circle.
With our portal we turn continues functions into group-maps. So i gets turned into some group-map from Z to 0 let's call it i' and h gets turned into a map from 0 to Z let's call it h'.
We know that i(x) = x for all x on the circle and h(x) = x for all x on the boundary, that means h(i(x)) = x for all x on the circle. This means that h'(i'(a)) = a for all integers. This is impossible and hence the contradiction.
The real magic of this proof is in the way the portal works so it's a shame she doesn't go more into that, but you can read about it in the references in the description.
John Galmann when this comment does better job explaining the idea of the video than the video itself
you do wanna construct this function h, to get an application of the "portal" the disk of its own is just zero but the boundary, the circle, is Z so u get something to work with. Furthermore the "boring" fuction provides the same as the function h, instead of a function from the Disk to the circle, which correspondence to 0 and the integers, u want something to work with, so u first consider the inclusion of the circle to the disk and then use the function h, so u get a function from the circle to itself, which correspondence to the function from Z to Z. Then we get the observation that on topological level the map, first inclusion then h, is just x goes to x, therfore on algebraic level it is also x goes to x. But this is a contradiction because on algebraic level we are going through 0 first.
Hope this cleared things up for you
John Galmann
But if we are assume the theorem is true we can still construct functions h(x) and i(x), it's just that h(x) will be undefined for a single value of x meaning h(i(x))=x for almost every x while the h'(i'(a))=a still would hold only for a single value of a meaning the "portal" still doesn't work. What did I get wrong?
Женя Пермин
note that the theorem doesnt say that there is exactly one fixpoint, for example the map x goes to x from the disk to the disk has only fixpoints. For this map u also cant build the function h.
Way too hand-wavy of a proof. Didn't get a deeper understanding. 3blue1brown has a good video on this topic.
I really like the presentation and pacing now, but the only thing that I have a problem is that this channel is still trying to cram too much content like Brouwer's Fixed Point Theorem proof into a less than 10 minute video resulting in skimming over most of the details in the proof.
Future episodes about category theory would be an excellent idea!
We think so too :)
Terrific. What little I've learned about Category Theory has been fascinating, and would love to learn more.
What I feel is missing is the proof that h(x) is continuous (and thus a morphism in the category of topological spaces, so we can indeed map it using the functor described to a morphism in the category of groups).
This proof relies on the fact that there can't be a continuous function like h(x) (since that leads the to the contradiction described), however she failed to prove that the constructed h(x) is indeed continuous.
I can easily think of hand-wavy ways she could have at least explained the continuity of h(x), and how it relies on the continuity of g(x) (with simple animations).
DarkChaos110 thank you for filling in the blanks
This was awesome! I like how you got across the idea that it is important that a functor has to do with the groups/shapes and with functions. Wanting to understand this proof more fully is great motivation to learn some category theory!
Who knew someone who made such great icecream was also good at math
Really great episode!! Perhaps the best as of yet.
Really dissapointing explanation of a beautiful concept... you're too afraid of giving details where they're really needed and leave other technicalities unclear by trying too hard with empty intuition arguments.
I think it wasn't meant to be a rigorous proof, only the intuition. Anyone with the required mathematical background can put it into precise statements, however this video gave away the gist for the non-mathematically inclined.
i like the content of these recent videos , very good
Whatever happens in the land of topology stays in the land of topology
-no mathematician ever
Beautiful explanation!
category theory, topology and the tutor are all exciting.
Nice pace, keep it up!
i could listen to her all day
a great intro towards algebraic topology
You guys are doing great things!
Does this mean I can't completely mix my coffee no matter how hard I try?
you can do it by alternating between two mixing methods
Pick any ball game with two half-times: There always exist two opposing points on the ball's surface that will have exactly the same position in space at the beginning of both halves -- no matter how much you turn the ball during the first half, as long as you put it on the same spot for the second half. This is true because any two successive rotations of a sphere above any two axes can be combined into a single rotation above a third axis. Thus, any number of successive rotations (i.e. playing with the ball) can be reduced to a single rotation, and any single rotation of a sphere by definition will have two fixed points, namely the two points where the axis of rotation pierces the sphere's surface. In German this theorem is called "Satz vom Fußball" (Football Theorem).
the fact that, topologically, a disc is like the number 0 and a circle is like the intergers is very unclear to me. Therefore I didn't get the proof... And I'm a math teacher (I admit : I forgot a looooot from college).
Piffrock it has to do with what's called the fundamental group. The fundamental group of a circle is (isomorphic to) the integers. I'm sure there is a satisfactory video on UA-cam of the Fundamental Group.
I’m currently binge-watching Bartosz Milewski’s lectures on Category Theory (find it out on your YT) and it’s the best piece of maths I’ve ingested in a while.
Very nice video. The piece that I think I am not understanding: I thought that the integers corresponded to loops on the circle, not to points on the circle? Is there an obvious correspondence between the loops in the circle and the points on the circle that I am missing?
Hausdorff Groups are all you need!
Awesome stuff!
great work!!
I can finally help!!! After getting told numerous times during my complex analysis course the difference between a disk and a circle @1:41 you mentioned a disk but displayed a circle. A circle has genus 0 where a disk has genus-1, due to the puncture.
Infinite Series is my jam! 100%
I know you can't please everyone, but this is exactly the sort of video I am hoping to see from this channel. There is some irreducible complexity present in these ideas after all...
great explanation
I think that Tai-Danae has improved her presentation and the video doesn't feel as rushed as previous ones, but she really needs to explain all the steps in her proof. The correspondence between the circle and the integers is stated, but no clue is offered as to how this works.
First infinite series I've watched since Kelsey left, Still great stuff. Kudos!!
Wait how do we draw a projection from g(x) through x onto the boundary in a pure topological space? wouldn't you need some additional structure, such as an inner product in order to do that?
A) The statement of the theorem needs to say 'A continuous mapping of a CLOSED disc onto itself has a fixed point'. If it's an open disc, the theorem is not true (drag the center point to the right and all other points move with it proportionally.) B) the theorem applied to any finite-dimensional Euclidean space, not just a 2-dimensional plane, and the statement should reflect that.
I like your \cup clasp; it's shiny and draws the eye. I can't help but imagine your "portal" as a functor from Top, the category of topological groups, to Ab, the category of abelian groups. This provides a possible context for generalizing to other categories.
h is a function of a disk D (0, r) with radius r and center 0 to a circle C (0, r). h(C(0,r))=C(0,r). Because of the continuity of h, each circle C (0, x) with radius x
Those who are complaining about speed they should know .....this is not hold your hand and draw and learn maths ......this is an high profile content if you don't understand this then get lost .....😤😤😈
Please do more in-depth analysis of category theory, especially as it relates to programming and compiling / type systems.
It's very brave to try to explain homology in an elementary way, but I fear that on this occasion you have not been successful. Some suggestions: 1) prove BFPT in 1D by elementary means using the intermediate value theorem, this is very easy to illustrate pictorial, and will motivate the result in higher dimensions. 2) in the proof I think it's vital to explicitly show why the function h is not possible of three is a fixed point; is not clear, from what you say, how the assumption had been used when constructing h, and why the same can't be done with a map with fixed points. 3) this is the most important. Starting there is a "portal", that the circle is like the integers and the disc like 0, and that topological properties must manifest as arithmetic ones, is a claim entirely void of meaning, explanative power, or insight. It explains to exactly no-one why the proof works or the theorem is true. I think that some significant time needs to be spent justifying the claim of a corresponds between algebra and topology before an explanation of this proof to this audience will be possible. Keep on trying! You make really good videos generally
Too much technicality with too much hand-waving. Either clearly explain the technical details/terminologies/concepts (which, arguably, is next to impossible in a youtube video for a subject like algebraic topology), or find a way to simplify the explanation in "intuitive" terms. If neither is possible, the best thing to do will be to choose a different topic altogether where such a feat may be possible. Not much thought was put into choosing the topic nor in creating the presentation. Brouwer's fixed point theorem has a host of different proofs, some that even high-school students can understand. Why would you choose the proof that uses homology and group homomorphism, and that too presented using all the technical jargons without clearly explaining them?!
its not homology that is being used... its homotopy
Ya know part of me feels like man this is tough for a YT video. But part of me comes to infinite series to be challenged and learn something difficult. I'd rather they err to the latter than not.
I'm fine with the video not beeing self-sufficient. It's like a trailer for a movie, it draws attention to a subject that may lack attention simply because it's hard to completely explain in a youtube video.
Now I may look more into it on my onw.
Nice explanation MAm
8:00 whats difference in category theory & group theeory?
This would have worked better if you elaborated how the circle and disk act like groups, and by extension the isomorphism from from D2 to 0 and from S1 to Z.
OH SO THATS WHY ALGEBRAIC TOPOLOGY WAS CREATED
How is it possible to have a mapping from the integers to the perimeter of a circle when the perimeter is continuous? Wouldn't you need the reals? (Not that the underlying logic is any different)
Why can use 0 vs integers to model circle vs disc? I don't really get it. Why is it not possible to map from x to x to h(x)=x? Sound trivially possible to me
I didn't understand much, but the sound effects were nice :)
You know in my Catholic high school we covered Alegbra I, Algebra II, Geometry and Calculus. I so wish they'd have taught this back then. You see in all the testing they did (This is prior to the use of tests as penalty, when they were more diagnostic.) - I turn out to have abstract reasoning skills out the wazoo. One of the biggest issues I had with math was the why it works the way it works.
I have a question!
In the argument, we create a map J:S¹ --> D where x |--> x in D. Then assuming h is well defined on S¹, we create a contradiction by showing that no map composed with a constant map from Z to 0 can send each element to itself.
But in this proof, we do not use any fact pertaining to the action of h in the interior of D. So does this not show that the fixed point not only exists, but that it should exist on the boundary of D, i.e., on S¹?
I NEED a PBS Infinite Series T-Shirt!
I think I got it :D. Step 2 was a bit of a doozie though. That may have to do more with the fact that I have very little exposure to topology. Very interesting.
I feel like you guys should be making longer videos for topics like this... I ended up with more questions than understanding. I really like that you're tackling tough concepts that go way above other UA-cam pop-math channels, but many of your recent videos have felt like they have quite a bit of hand-waving that hurts more than helps. I mean, PBS Spacetime has been getting away with 15-ish-minute-long videos for quite a while now and they've done an excellent job of explaining some really high-level physics. I don't think you'll have a problem with viewer retention or anything if you move to a slightly longer, more in-depth format.
That's just my two cents, and don't get me wrong--this is one of my favorite channels on UA-cam.
These analogies such as gadgets and portals make the video feel a tad condescending. The great thing about PBS Space Time and PBS Infinite Series is that they're not afraid to get a little bit technical. There are already dozens of great channels that are accessible to everyone (SciShow, Vsauce, Veritasium). I like PBS videos because they usually assume you have at least a vague understanding of what what a group or proof by contradiction is etc. or are willing to look it up.
Still I know that the channel in going through a transitionary period with the new hosts so it'll need some time to finds its feet. I'm confident it will return to its former glory soon.
First of all. This isn't a proof for the Brouwers Fixed Point Teorem, since the theorem is formulated for every ball of the R^n. This video just shows the case n=2.
Second, the correlation between topology an the integers is for a not math student very hard and not well explained.
Third, we don't identify the (points of the) circle with the integers, we identify the homotopy-classes of a circle with the integers. There is a big difference and I don't see why we can do things like in the video.
And last, the idea of the proof (identifying circels with integers) cannot be applied on higher dimensions, since the fundemantal group of higher spheres is π=0.
I keep seeing homotopy come up in various places. Perhaps it's time I studied it.
Yes, the leap from a circle to integers was introduced way too fast and without adequate justification. Physicist talking here so I'm not mathematically novice either. Still I felt that I could not understand the proof :/
Please resurrect this awesome channel 😭
I think she speaks and most importantly, delivers the "idea" very well. People absorb knowledge and information at different rates so there will never be one set rate that pleases everyone and in understanding that leads to a natural and kind conclusion - have patience. Just as we all ask and expect others to have with each of us. Also time (as with all things) is relative and all that's required to move between different reference frames is to change your state of mind which isn't always easy. I myself absord new knowledge that I'm grasping for the first time slowly. Not because I'm slow, my IQ is well over 140, but because I take the time to take it in and form the new concepts visually within my mind. To understand them, but most importantly to expand upon them as knowledge is nothing without imagination.
I think the video should have a bigger duration in order to understand it without stopping the video several times. I think that explains the bad feedback on this great video.
Can someone explain to me why you can just replace a disc with 0 and the corresponding circle with the integers, and why does it make sense that mapping from the integers to 0 is analogous to mapping a boundary to the disc itself??
I didn't understand why I could asign a zero to a disc's area and the integers to the border. I wish you only used this episode to show how to make this particular functor and a following video to using it for the theorem.
Brilliant👌👌👌
This proposition has the principal characteristic of the Universal Singularity, wave-particle, centre of mass, picture plane and vanishing point, ..dependent on the same mathematical methodology of a logical connection (Quantum) 1-0D and Euler's Formula, in nothing, that the hierarchical strata phase-state probabilities of a temporally (including temporary) everted/reciprocal-field vanishing point layering becomes the nodal rubber band etc suggested by String Theory. Ie Universe-Singularity is Eternity-now superposition.
Loved the video, very well presented. Can't understand why so many people aren't happy in the comments. I'm glad there's a channel that doesn't spoon feed us like primary school students. Keep up the good work :D
Two travelers--polar opposite each other in time, set out to change the others history.
There will be some event at some fixed point in time can never be changed by either of them.
Holy shit.. wondered in here on accident. Everyone in the comments is so unbelievably smart to me. I've always believed myself sufficiently intelligent (which is to say I can start fires and feed myself) but I can honestly say that I didn't understand one damn word this lady said. My favorite math trick is to turn 9 toothpicks into 10 by spelling TEN with the 9 toothpicks. That's the level of intelligence I bring to the table. Thanks for reminding me I'm a caveman. Goodluck proving whatever you were talking about. ✌
Can anyone explain how to do this proof properly?
3:20 Is that some kinda algebra-topology isomorphism?
Very interesting. I'm not well versed in this stuff but I'm seeing some parallels with database design theory. The (illegal) scenario of many-many relationships is much like the function h you described (0 can be assigned to exactly 1 integer making it impossible to do a set-set map)
I wish we would have spent more time on the function that links the areas to 0 or Z.
Because my question here is:
If I have an area it is "equal" to 0, right? But this area consists out of the inner part (still 0) and a boundary (Z).
Of cause not ever function is linear f(full area) = f(inner + boundary) = f(inner space) + f(boundary).
However my intuition tells my to assume f(a+b)=f(a)+f(b) in this case.
VSauce is probably the reason I'm on this channel today
If I understood why the portal is valid and the disk is like 0, then yeah, it works out as being a contradiction. :)
I don't have a strong math background but I'm not convinced the portal is valid (it apparently is but in the video, it's just stated that we can use it). Why is the disk is like zero? I'm not sure how the analogy applies. I understand the function, given the transformation function g(x), that maps points on the disk to the circle and why the boring function is needed to handle the edge cases. It would have helped to say why the boring function was needed but now it makes sense.
aren't there two possible contradictions? one would be that there are no fixed points, and the other one would be that there is more than one fixed point...
Why not a video about Theorema Egregium ?
Not sure we got the full story there.. the x=>x=>h(x) route would be the equivalent of 0=>0=>0 which is perfectly fine as far as I can tell, since all of those x's were on the boundary (which you assigned to be equivalent to algebraic zero, for reasons that also weren't fully made clear..)
Need the tie in from g(x)=>x=>h(x) in order to properly demonstrate.. I think.. assuming I'm following along to any degree of accuracy at all..
The argument was too subtle for me to follow. All I understood was:
1. Assume g such that there is no x for which g(x)=x.
2. Assume h such that h((g(x)) lies on the boundary.
3. Assume f=Id => h(f(x))=x for all x on the boundary.
4. Show that multiplication by zero is not bijective.
?: why does point 3 imply that multiplication by 0 is bijective?
The biggest mystery or fail of this video is that it never explain clearly why a disc is zero and circle is integer. It make the following part of the video meaningless
Exactly. It feels weird that the full circle is part of the disc, but the analogy to the circle (the integers) is not completely included in the analogy of the disc (zero), which is basically what makes the proof work. Really just raises the question of how/why that analogy works even more.
The fundamental group of a disk is zero because any loop can be shrunk continuously to a point without leaving the disk. The fundamental group of a circle is the integers because there are distinct loops which wrap around the circle some integral number of times which can't be continuously transformed into each other without leaving the circle. The video did try to explain this, but algebraic topology is a pretty difficult subject.
As for the circle being part of the disk but the fundamental group being larger, the fundamental group is essentially a tool for detecting holes in a space, and the disk has no holes, while the circle has a hole made from removing the interior part of the disk, so the circle contains more information about holes than the disk, making its fundamental group larger. At least that's a (hopefully) intuitive explanation.
Thanks Seán O'Neil! Your explanation gave me more intuition on what's going on. Can you explain me why those two different (topologic and algebraic) descriptons relate to one another though? I feel like this is what I'm failing to understand in order to get why this proof works
Andre Angelo, there are many relations between algebra and topology, and the whole field of algebraic topology is dedicated to them. The one used here is the fundamental group. Given some topological space, this is an operation which produces a group from that space in a natural way, as the equivalence classes of loops with a fixed base point up to homotopy (which can be thought of as a continuous deformation). The group operation here takes two loops and produces a new loop which goes through one loop followed by the other loop. Going through the loop in reverse gives the inverse element in the group, and staying at the base point gives the identity. Loops that can be continuously shrunk to a point without leaving the space are equivalent to the identity, up to homotopy. The hard part of proving that this forms a group is that it behaves well under the homotopy equivalence (if [a] gives the equivalence class of the loop a, show that [ab] = [a][b], [a]^-1 = [a^-1], etc), and that is something which involves a lot of tedious detail checking, which is probably why the video didn't want to go through it. Proving that the fundamental group of a topological space is a functor requires checking that any continuous function between any two topological spaces induces a group homomorphism between their fundamental groups. This is a pretty general idea in category theory, and the purpose of functors. A functor is a map between two different categories which preserves their structure (morphisms... Morphisms of topological spaces are continuous functions and morphisms of groups are group homomorphisms).
If you're asking why topology and algebra are connected in this way, that's a harder question which algebraic topology attempts to answer, but it goes much deeper than just the fundamental group.
I agree with the point that assuming Brouwer's Fixed Point theorem to be false, one can constuct the function h(x) from the unit disk to the the unit circle. But shouldn't we also prove that the constructed function h(x) is continuous for the functorial argument to work? I think we should because only continuous maps preserve fundamental groups.
Absolutely! Fantastic observation. The function h *is* continuous, and it would have been helpful to state this in the video: a small perturbation in x corresponds to a small perturbation in h(x). (This relies on the continuity of g.) Good catch!
-Tai-Danae
I have so many questions i don't know where to begin.
But if we are assume the theorem is true we can still construct functions h(x, g(x)) and g(x), it's just that g(x) will be undefined for a single value of x meaning h(x, g(x)) = x for almost every x while the h'(a, g'(a)) = a still would hold only for a single integer of a meaning the "portal" from topology to algebra still doesn't work.
What did I get wrong?
Step 4: Find a contradiction
Step 5: Objection!
I'm not math expert but not totally clueless. This portal seem like "cheat" and proof, well... I might as well assign other numbers based on another property. I felt like there was not enough explanation why it works that way and oversimplifications like "gadget" or "portal' were expected to be taken for granted like in elementary school.
I felt like this video was meant to get people "excited" about what the hell that "portal" is, leading to a follow up video, which I don't really like. There's a fine balance between doing topics that feel meaningful but also explaining in a way which the viewer feels it learned something and I don't think you're hitting the right spot. I do very much enjoy more complex and advanced topic, but at the same time those topics need o be covered in much more detail, just like what Gabe did with his cryptography series. I don't think the topic was poorly chosen, it just needs to be explained better and in more detail.