Proving Brouwer's Fixed Point Theorem | Infinite Series

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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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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.

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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
https://www.youtube.com/watch?v=csInNn6pfT4

Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. https://www.youtube.com/watch?v=tdOaMOcxY7U

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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!

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