【前編】幾何学の極致　ポアンカレ予想【ペレルマン】

This time we'll be discussing the Poincaré Conjecture, one of the most pinnacle problems in geometry! We'll explain the magnificent history of geometry behind the Poincaré Conjecture, its mathematical meaning, and even its proof. (This video is the first part.) Here is the video structure and characters. [Video Structure] (Part 1) Introduction 00:00 Chapter 1: The Shape of the World: The Curvature of Spacetime 02:02 The Pythagorean Order Philosophers and Geometry The Shape of the Earth The World of Manifolds Chapter 2: The Cosmological Principle: The Geometry of the Divine Comedy 16:58 Dimensions: Space, Bounded, Boundary, Closed Set Atlas, Manifold, Simply Connected Capturing the Shape of the Earth The Cosmological Principle The Geometry of the Divine Comedy Chapter 3: Euclid's Legacy: The Three Geometricians 30:35 The Founder of Geometry The Fifth Postulate The Three Doctors of Geometry The Secret of the Fifth Postulate Euclid's Legacy Chapter 4: A Mathematician from the Future: Gauss's Successor 47:54 ・Gauss's Successors ・On the Fundamental Hypothesis of Geometry ・The Nobleman and the Monster (Part 2) Chapter 5: Two Successors: Poincaré and Klein ・Felix Klein ・Rivale I ・Henri Poincaré ・Rivale II ・Aftermath ・The Golden Age of German Mathematics: What Happened to Klein Afterward ・Hilbert: Klein's Successor ・The Last Universalist ・Poincaré Conjecture Chapter 6: The Poincaré Conjecture: Mathematics Beyond Human Reach ・The Groundwork for Proving the Poincaré Conjecture ・Thurston: The Ultra-Intuitive Mathematician ・Hamilton: The Man Who Came Closest to Proving the Poincaré Conjecture ・A Mathematician's Challenge Lemma: Millennium Mathematics: Hilbert's Testament Chapter 7: Complete Proof: The Greatest Mathematicians of Our Time ・The Greatest Mathematicians of Our Time ・Preface ・Nonlocal Collapse Theorem collapsing theorem -Finite time extinction -Concerning the proof -Is that man crazy? [Part 1: Characters] Pythagoras (ancient mathematician, proponent of the spherical Earth theory) Euclid (ancient mathematician, founder of geometry, author of the Elements) Plato and Aristotle (ancient philosophers, researchers of geometry) Dante (the greatest poet of the Middle Ages, creator of the Divine Comedy, with its unique worldview) Gauss (the greatest mathematician in human history, first to unravel the secret of the fifth postulate) Farkás (mathematician, rival of Gauss, father of Bolyai) Bolyai (mathematician, unraveled the secret of the fifth postulate) Lobachevsky (mathematician, unravelled the secret of the fifth postulate) Riemann (mathematician) Gauss's last disciple) [Part 2: Characters] Poincaré (One of the founders of topology, proponent of the Poincaré conjecture) Klein (Poincaré's rival) Hilbert (Father of modern mathematics) Dehn (Developer of the Dehn surgery) Milner (Exotic spheres, Fields Medal winner) Smale (Solving the Poincaré conjecture in higher dimensions, Fields Medal winner) Friedman (Solving the Poincaré conjecture in higher dimensions, Fields Medal winner) Papakyriakopoulos (Proving Dehn's lemma, devoted his life to the Poincaré conjecture) Thurston (Proposing Thurston's geometrization conjecture, Fields Medal winner) Hamilton (Proving the Ricci flow) (Fields Medal Winner) Perelman (The Greatest Mathematician of Our Time, Proving the Poincaré Conjecture) Recommended Related Videos "Part 2: The Consummation of Geometry: The Poincaré Conjecture"    • 【後編】幾何学の極致　ポアンカレ予想【天才数学者】   "Perelman, the Greatest Mathematician of Our Time"    • 【ポアンカレ予想】現代最高峰の天才数学者ペレルマン【ミレニアム懸賞問題】   "Complete Edition: The Consummation of Geometry: The Poincaré Conjecture" Part 1 + Part 2    • 【完全版】幾何学の極致　ポアンカレ予想   "Explaining Fermat's Last Theorem: The Greatest Problem in the History of Mathematics"    • 【余白が狭すぎる】『数学史上最高の難問　フェルマーの最終定理を解説』【ワイルズ】   "The Greatest Mathematician of Our Time, Proving the Poincaré Conjecture" "Explaining Hilbert, the Father of Mathematics"    • 【ノイマン】『現代数学の父ヒルベルトを解説』【ゲーデル】   "Erdős, the Genius Mathematician Who Devoted His Life to Mathematics"    • 【天才】『数学に全てを捧げた天才数学者エルデシュ』【数学の狂気】   "Explaining Plato's Philosophy and Thought"    • 【洞窟の比喩】『プラトンの哲学と思想を解説』【国家論】   "Explaining Aristotle's Philosophy and Thought"    • 【形而上学】『アリストテレスの哲学と思想を解説』【四原因説】   Recommended Related Books "Low-Dimensional Geometry and the Poincaré Conjecture" https://amzn.to/4djm15w This book provides a rough overview of the mathematics surrounding the Poincaré Conjecture. Anyone with undergraduate-level knowledge of mathematics should be able to read it. "Foundations of Manifolds" This book is the go-to for manifolds. https://amzn.to/4gus6il "Seven Unsolved Problems in Mathematics" The Poincaré conjecture is explained in more of a topological sense than in differential geometry. It's a fairly serious book. https://amzn.to/3TzuRoG #TheBiggestDifficultProbleminMathematics #UnsolvedProblem #MillenniumPrizeQuestion