Proving Brouwer's Fixed Point Theorem | Infinite Series
96.3 هزار بار بازدید -
7 سال پیش
-
Viewers like you help make
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://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
The Mathematics of Diffie-Hellman Key...
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...
Algebraic Topology by Allen Hatcher, page 31:
https://www.math.cornell.edu/~hatcher...
A Concise Course in Algebraic Topology by Peter May, page 10: https://www.math.uchicago.edu/~may/CO...
Category Theory in Context by Emily Riehl, page 15: http://www.math.jhu.edu/~eriehl/conte...
To learn more about algebraic topology, check out:
Algebraic Topology: An Introduction by W. S. Massey https://www.amazon.com/Algebraic-Topo...
Elements of Algebraic Topology by J. Munkres https://www.amazon.com/Elements-Algeb...
To learn more about category theory and functors, check out:
“What is category theory, anyway?” http://www.math3ma.com/mathema/2017/1...
“What is a functor?” http://www.math3ma.com/mathema/2017/1...
VSauce - Fixed Points
Fixed Points
Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. Topology vs "a" Topology | Infinite S...
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!
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
The Mathematics of Diffie-Hellman Key...
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...
Algebraic Topology by Allen Hatcher, page 31:
https://www.math.cornell.edu/~hatcher...
A Concise Course in Algebraic Topology by Peter May, page 10: https://www.math.uchicago.edu/~may/CO...
Category Theory in Context by Emily Riehl, page 15: http://www.math.jhu.edu/~eriehl/conte...
To learn more about algebraic topology, check out:
Algebraic Topology: An Introduction by W. S. Massey https://www.amazon.com/Algebraic-Topo...
Elements of Algebraic Topology by J. Munkres https://www.amazon.com/Elements-Algeb...
To learn more about category theory and functors, check out:
“What is category theory, anyway?” http://www.math3ma.com/mathema/2017/1...
“What is a functor?” http://www.math3ma.com/mathema/2017/1...
VSauce - Fixed Points
Fixed Points
Congratulations to Robby Weintraub for being the winner of the Topology vs "a" Topology Challenge Question. Topology vs "a" Topology | Infinite S...
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!
7 سال پیش
در تاریخ 1396/10/28 منتشر شده
است.
96,350
بـار بازدید شده