My Favorite Theorem: The Borsuk-Ulam Theorem

[@andrewharrison8436]

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

[@pygmalionsrobot1896]

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

[@particleonazock2246]

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.

[@DrTrefor]

Thanks so much!

[@aashsyed1277]

HI CONDENCED PHYSICS AGAIN!!!!!!!

[@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 --> Knaster-Kuratowski-Mazurkiewicz lemma | | | Borsuk-Ulam theorem -> Tucker's lemma. -> Lusternik-Schnirelmann theorem

[@DrTrefor]

That's right, isn't it cool! btw if you haven't seen it before I have a video providing Brouwer from Sperner that I think you might enjoy

[@callumvlex7059]

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

[@craigjoyner9857]

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.

[@abdurrafeh8899]

Every point on the surface of the earth has a temperature and pressure associated with it; therefore, if you represent the temperature and pressure on a 2D xy-plane, every point on the surface of the sphere (earth) is essentially mapped to a specific point on the xy-plane representing the temprerature and pressure (one axis representing temperature and the other axis representing the pressure), since the surface of the sphere also has a certain temperature and pressure associated with every point on its surface. You're basically mapping the surface of the sphere to the plane, which is no different than any random mapping.

[@craigjoyner9857]

@@abdurrafeh8899 , I guess what I struggled with originally was how could you guarantee that the temperature and pressure combination would hold in the same way. Once I took into consideration that temperature and pressures on the earth will all have continuous gradients, it made more sense. For instance the only way to get to an area with 80 def F temp, to an area with 77 degree temp, would be to go through a series of discrete points along the continuous temperature profile, to get from one point to the other. Same thing with pressure.

[@justinswag3403]

Yay cool video!

[@marwaassem1087]

Amazing Video

[@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 :)

[@DrTrefor]

haha nice!

[@broccoloodle]

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

[@abd-elrahmanmohamed9839]

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 .

[@aashsyed1277]

SUPER nifty theorem

[@georgegammer2515]

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?

[@petros_adamopoulos]

A mathematical proof doesn't require any physical confirmation. Besides, both temperature and pressure don't have actual precise values. You couldn't even say *any* two things/locations have the same temperature or pressure, or even that one thing has the same than it had before. Fundamentally, both measure basically "count" how many molecules did hit the detector in some small amount of time. You could argue forever whether the same temperature was actually the same, and the other way around, you could always argue that two temperatures are close enough.

[@abdulsamadkhan9743]

i want more videos like these...mannnn

[@Achill101]

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?

[@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.

[@Fine_Mouche]

reduce to a single couple of points*

[@ashlyeli8442]

Do you talk about ring theory?

[@ycombinator765]

for the algorithm 🔥🔥🤞👍

[@zray2937]

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.

[@angelaruiz5599]

Hi! Can you recommend me some bibliography? c:

[@Michael-hs7pr]

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?

[@Michael-hs7pr]

@@DrTrefor Thanks! I'll try and do it in MatLab then :)

[@dickybannister5192]

are there any examples of 'nice looking' cont. diff. dynamic functions which have a small number of such pairs which look cool when they are animated (i.e. time lapse). I cant think of anything?

[@charlesbrowne9590]

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

[@DrTrefor]

Haha thanks, I “know” that but also forget that every time:D