A beautiful combinatorical proof of the Brouwer Fixed Point Theorem - Via Sperner's Lemma
Using a simple combinatorical argument, we can prove an important theorem in topology without any sophisticated machinery.
Brouwer's Fixed Point Theorem: Every continuous mapping f(p) from between closed balls of the same dimension have a fixed point where f(p)=p.
Sperner's Lemma: Every Sperner covering of a triangulation - that is, unique colors on exterior vertices, edges inherent the colors between the vertices, and insides inherent any of the colors - contains at least one triangle with vertices of all three colors.
We require exactly no technical machinery to prove Sperner's Lemma. To prove Brouwer from Sperner we do need one major theorem, Bolzano-Weierstrauss, which shows the existence of convergent subsequences on closed and bounded regions.
Leave the proof of the general case for n greater than 2 in the comments!
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett
Видео A beautiful combinatorical proof of the Brouwer Fixed Point Theorem - Via Sperner's Lemma канала Dr. Trefor Bazett
Brouwer's Fixed Point Theorem: Every continuous mapping f(p) from between closed balls of the same dimension have a fixed point where f(p)=p.
Sperner's Lemma: Every Sperner covering of a triangulation - that is, unique colors on exterior vertices, edges inherent the colors between the vertices, and insides inherent any of the colors - contains at least one triangle with vertices of all three colors.
We require exactly no technical machinery to prove Sperner's Lemma. To prove Brouwer from Sperner we do need one major theorem, Bolzano-Weierstrauss, which shows the existence of convergent subsequences on closed and bounded regions.
Leave the proof of the general case for n greater than 2 in the comments!
BECOME A MEMBER:
►Join: https://www.youtube.com/channel/UC9rTsvTxJnx1DNrDA3Rqa6A/join
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: https://www.amazon.com/shop/treforbazett
Видео A beautiful combinatorical proof of the Brouwer Fixed Point Theorem - Via Sperner's Lemma канала Dr. Trefor Bazett
Показать
Комментарии отсутствуют
Информация о видео
Другие видео канала
NYT: Sperner's lemma defeats the rental harmony problemWhat is Sperner's Lemma and how does it relate to Graph Theory? [Discrete Mathematics]My Favorite Theorem: The Borsuk-Ulam TheoremPi is IRRATIONAL: animation of a gorgeous proofFixed PointsBrouwer's fixed point theoremFixed Point Theorem and its Application | Under what condition function has a fixed point | #REALSolipsism and philosophy of mathematics: a comparison among intuitionists, Miriam FranchellaProving Brouwer's Fixed Point Theorem | Infinite SeriesSperner's LemmaSneaky Topology | The Borsuk-Ulam theorem and stolen necklacesThe Kreisel Lawvere Debate on Categories and the Foundations of MathematicsEdge Colorings and Chromatic Index of Graphs | Graph TheoryMath317 OpR Class27 Nov12 2014F. William Lawvere - What are Foundations of Geometry and Algebra? (2013)Banach Fixed Point TheoremSplitting Rent with Triangles | Infinite SeriesTopology For Beginners: Brouwer Fixed Point TheoremM 04 08 Brouwer's Fixed Point Theorem