My Favorite Theorem: The Borsuk-Ulam Theorem

Many thanks for 10k subscribers! Fun video for you from Topology: The Borsuk-Ulam Theorem. One interpretation of this is that on the surface of the earth, there must be some point where it and its antipode (the spot exactly opposite it) have the exact same temperature and pressure. More generally, if you have a continuous map f:S^n to R^n then there exists a point x where f(x) = f(-x).

We will see some animations to get a feel for these kinds of functions, prove it from elementary methods in dimension 1 using Intermediate Value Theorem, look at a very cool set-theoretic corolloray called the Lusternik–Schnirelmann and finally see how it connects to Brower's Fixed Point theorem and various lemma's in combinatorics.

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This video was created by Dr. Trefor Bazett, an Assistant Professor, Educator at the University of Cincinnati.

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[andrewharrison]

Well, 4 years later the 10, 000subscribers seems unambitious since you now have 275, 000.
Congratulations on this deserved growth.

[particleonazock]

Your videos are excellent, explaining math concepts intuitively, but not with too much rigour so that viewers become baffled. You have earned a subscriber, sir.

[pygmalionsrobot]

This was really excellent, thanks so much for making this.

[NothingMaster]

As you know we can prove the Borsuk–Ulam theorem from the Tucker's lemma. But the beauty of it is that we could also prove the Tucker's lemma from the Borsuk–Ulam theorem; meaning that the two theorems are equivalent. What’s even more interesting is the fact that there are several fixed-point theorems that come in three equivalent variants - namely an algebraic topology variant, a combinatorial variant, and a set-covering variant. These variants could further be reduced to the other variants. For example the following:

Brouwer fixed-point theorem —-> Sperner's lemma —-> lemma
| | |
Borsuk–Ulam theorem —> Tucker's lemma. —> Lusternik–Schnirelmann theorem

[callumvlex]

The hairy ball theorem, I use this all the time as an ice-breaker

[abd-elrahmanmohamed]

Hi Dr., I'm self studying discrete mathematics now from the book "Discrete Mathematics and its Applications" by Kenneth H. Rosen . I can understand the material with no issues, but the issue is that the number of exercises (problems) after every section is too big . How can I handle that huge problems without ignoring any new ideas .

[craigjoyner]

I understand how the surface example works, with the opposite poles. I don’t understand the logical jump to how we can carry that along for temperature and pressure as well.

[yongmrchen]

This one relates me to the intermediate value theorem. I think they have some resemblance in ideas, but the IVT is much simpler and more intuitive.

[idontwantahandlethough]

Hey, I have that shirt! You look better in it though so I'll let you be the one to wear it
Oh, and cool video, this was really interesting! Keep it up dude :)

[Achill]

Thank you for this nice video showing some things the Borsum-Ulam theorem can do and a proof for S1.
Can you recommend a video with a proof of the Borsum-Ulam theorem with or without Tucker's Lemma?

[georgegammer]

Has any work been done to test these theoretical conclusions? For example. has anyone ever found antpidol points that have the same temperature and pressure?

[Fine_Mouche]

what i want to see is a heatmap of all point having same temperature, another heatmap showing same temp + pressure for see that the total area is inferior, and continuing adding parameters until the area of the heatmap is reduce to a single point.

[abdulsamadkhan]

i want more videos like these...mannnn

[zray]

First time seeing the Lusternik-Schnirelmann theorem, and it seems weird to my non-mathematician eyes. The image shown seems to imply that the sets are disjoint, but I don't think that's possible.

[angelaruiz]

Hi! Can you recommend me some bibliography? c:

[broccoloodle]

I am doing a category theory course, the diagram at 15:21 scares me 😢

[Michael-hspr]

Thanks for the video! I'm holding a seminar on the Borsuk-Ulam theorem soon myself. Do you have any recommendations on how to code some animations e.g. in LaTeX or Python?

[charlesbrowne]

FYI: Antipodes is pronounced an-TIP-o-deez.