2018-01-18

[public] 80.3K views, 2.60K likes, 371 dislikes audio only

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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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Previous Episode

The Mathematics of Diffie-Hellman Key Exchange

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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.

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

Made by Kornhaber Brown (www.kornhaberbrown.com)

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:

http://www.math3ma.com/mathema/2018/1/18/brouwers-fixed-point-theorem-proof

Algebraic Topology by Allen Hatcher, page 31:

https://www.math.cornell.edu/~hatcher/AT/AT.pdf

A Concise Course in Algebraic Topology by Peter May, page 10: https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf

Category Theory in Context by Emily Riehl, page 15: http://www.math.jhu.edu/~eriehl/context.pdf

To learn more about algebraic topology, check out:

Algebraic Topology: An Introduction by W. S. Massey https://www.amazon.com/Algebraic-Topology-Introduction-Graduate-Mathematics/dp/0387902716

Elements of Algebraic Topology by J. Munkres https://www.amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280

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

“What is category theory, anyway?” http://www.math3ma.com/mathema/2017/1/17/what-is-category-theory-anyway

“What is a functor?” http://www.math3ma.com/mathema/2017/1/31/what-is-a-functor-part-1

VSauce - Fixed Points

/youtube/video/csInNn6pfT4

Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. /youtube/video/tdOaMOcxY7U

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!