Proving Brouwer's Fixed Point Theorem | Infinite Series

There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra.

Tweet at us! @pbsinfinite
Email us! pbsinfiniteseries [at] gmail [dot] com

Previous Episode
The Mathematics of Diffie-Hellman Key Exchange

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.

Written and Hosted by Tai-Danae Bradley
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow and Meah Denee Barrington

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:

Algebraic Topology by Allen Hatcher, page 31:

To learn more about algebraic topology, check out:

To learn more about category theory and functors, check out:

VSauce - Fixed Points

Special thanks to Roman Pinchuk for supporting us on our Converse level on Patreon.

Along with thanks to Matthew O'Connor, Yana Chernobilsky, and John Hoffman who are supporting us on Patreon at the Identity level!

And thanks to Nicholas Rose, Jason Hise, Thomas Scheer, Marting Sergio H. Faester, CSS, and Mauricio Pacheco who are supporting us at the Lemma level!

[pbsinfiniteseries]

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!

[jagoandlitefoot]

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

[ChazyK]

Disc is zero and circle is integers? Why? The rest of the proof is obvious, but the most important thing is not explained.

[telotawa]

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

[tuchapoltr]

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."

[timh.]

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.

[martinkuffer]

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

[Theraot]

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.

[jeffreybernath]

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.

[michelfug]

As an intuitionist mathematician, Brouwer would turn in his grave if he saw this proof using contradiction.
Nice introduction to functors though :)

[conoroneill]

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.

[arthur]

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.

[philmaggiacomo]

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.

[Theraot]

See also "Who (else) cares about topology? Stolen necklaces and Borsuk-Ulam" by 3Blue1Brown

[SultanLaxeby]

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 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.

[rkpetry]

...'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)...

[deepdata]

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.

[viktort]

People are all complaining because they like to complain, but you are doing a very amazing work (although very amibitious).

[subh]

Too much technicality with too much hand-waving. Either clearly explain the technical (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?!

[blackflan]

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.