My boyfriend really admires what you do professor. Thank you for your consistent videos that i can play for him when hes irritating me! they always calm him down.
I love Alannah Kennedy's comment. It is a brief flash of illumination falling onto her life, her love, and her kind, impish personality. Thanks, Alannah!
In the proof of Brouwer's theorem, you can instead get the direction from f(x) to x and then extend the line to where it intersects the boundary circle. This way you do get the identity map when you restrict to S^1. Also, I find it funny how the contradiction in this theorem boils down to S^1 not being a retract of B^2, and the contradiction there just boils down to 0 != 1.
This is great. I'm only recently after reading about this in Mendelson + Gamelin's books. Thank you professor for continuing on with this series & this channel.
At 18:19, you say that contractibility of the unit interval is the reason we can find those finitely many closed intervals with image contained in one disc. Aren't we using compactness rather than contractibility though? (the preimages of the discs are open, so we can find finitely many open intervals that cover the unit interval and from those we can find the a_i) Also why do we need 4 "discs" to cover S^2? Wouldn't two suffice? (S^2 -- northpole and S^2 -- southpole)
I've had the same question before I saw your comment, and it seems to me that you are correct. Given that he says “cover it with - say - four disks“, he didn't quite care to optimize it, because as you've mentioned, it does not make any difference to the argument.
At 14:20, how do we prove the forward direction of the claim, that is, if the two maps are homotopic then their lift to R must have the same image of 1?
It is followed from that 1) independence of the ending point on t in the definition of a homotopy and 2) the existence of an uniquely lifted homopoty on the helix of that on S^1. It is actually proven in the proof of Theorem 1.7 on page 30 of Hatcher.
My boyfriend really admires what you do professor. Thank you for your consistent videos that i can play for him when hes irritating me! they always calm him down.
I love Alannah Kennedy's comment.
It is a brief flash of illumination falling onto her life, her love, and her kind, impish personality.
Thanks, Alannah!
I'm addicted to this channel! it's so good! thank you Prof. Borcherds!
Insight/Time is maximized in these videos, best lectures I've found on UA-cam
In the proof of Brouwer's theorem, you can instead get the direction from f(x) to x and then extend the line to where it intersects the boundary circle. This way you do get the identity map when you restrict to S^1.
Also, I find it funny how the contradiction in this theorem boils down to S^1 not being a retract of B^2, and the contradiction there just boils down to 0 != 1.
Really nice how accessible these lectures are!
This is great. I'm only recently after reading about this in Mendelson + Gamelin's books. Thank you professor for continuing on with this series & this channel.
At 18:19, you say that contractibility of the unit interval is the reason we can find those finitely many closed intervals with image contained in one disc. Aren't we using compactness rather than contractibility though? (the preimages of the discs are open, so we can find finitely many open intervals that cover the unit interval and from those we can find the a_i)
Also why do we need 4 "discs" to cover S^2? Wouldn't two suffice? (S^2 -- northpole and S^2 -- southpole)
I've had the same question before I saw your comment, and it seems to me that you are correct. Given that he says “cover it with - say - four disks“, he didn't quite care to optimize it, because as you've mentioned, it does not make any difference to the argument.
5:09
22:02 😂
loooooool
incredible content
At 14:20, how do we prove the forward direction of the claim, that is, if the two maps are homotopic then their lift to R must have the same image of 1?
It is followed from that 1) independence of the ending point on t in the definition of a homotopy and 2) the existence of an uniquely lifted homopoty on the helix of that on S^1. It is actually proven in the proof of Theorem 1.7 on page 30 of Hatcher.
@@pd-zr7fo Thank you
Wonderful!
the loop group!
Do you need algebraic topology for physics
0:23
Did you learn algebraic or differental topology first
Wasn’t some of this used by pearlman
yeeep！
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yeeep
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