Channel: PBS Infinite Series

Category: Education

Tags: pbsinfinite serieseducationmathematicstopologymathsalgebrafunctorbrouwerfixed pointbrouwer’s fixed point theoremfunctiongeometrymathproof

Description: Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: to.pbs.org/donateinfi There is a proof for Brouwer's Fixed Point Theorem that uses a bridge - or portal - between geometry and algebra. Tweet at us! @pbsinfinite Facebook: facebook.com/pbsinfinite series Email us! pbsinfiniteseries [at] gmail [dot] com Previous Episode The Mathematics of Diffie-Hellman Key Exchange youtube.com/watch?v=ESPT_36pUFc&feature=youtu.be 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 (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: math3ma.com/mathema/2018/1/18/brouwers-fixed-point-theorem-proof Algebraic Topology by Allen Hatcher, page 31: math.cornell.edu/~hatcher/AT/AT.pdf A Concise Course in Algebraic Topology by Peter May, page 10: math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf Category Theory in Context by Emily Riehl, page 15: math.jhu.edu/~eriehl/context.pdf To learn more about algebraic topology, check out: Algebraic Topology: An Introduction by W. S. Massey amazon.com/Algebraic-Topology-Introduction-Graduate-Mathematics/dp/0387902716 Elements of Algebraic Topology by J. Munkres amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280 To learn more about category theory and functors, check out: “What is category theory, anyway?” math3ma.com/mathema/2017/1/17/what-is-category-theory-anyway “What is a functor?” math3ma.com/mathema/2017/1/31/what-is-a-functor-part-1 VSauce - Fixed Points youtube.com/watch?v=csInNn6pfT4 Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. youtube.com/watch?v=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!