Video notes: http://www.antoinebourget.org/attachm... ERRATA: at 1:34:27 we should read k^2 = 1-b^2/a^2 and not k = 1-b^2/a^2 to be consistent with the notations used in the video. Be careful, the other convention also exists, that's why in the Mathematica program I used k = 1-b^2/a^2 to get the right value... ------------------------------------------------------------------- My name is Antoine Bourget, I am a theoretical physicist, and I try to convey in video what I find elegant in mathematics and physics. To follow the news of the channel, and contact me, you can join the Discord server or follow me on social networks. If you want to donate, I also have a Tipeee Discord account: / discord Twitter: / antoinebrgt My personal website: http://www.antoinebourget.org Tipeee: https://fr.tipeee.com/scientia-egregia/ ------------------------------------------------------------------- Summary This time, we will take the back roads to explore one of the most captivating subjects in mathematics, but which is buried and relatively unknown to the general public. We will see that all this wealth is hidden in the dynamics of the simple pendulum -- but without the approximation sin(u)=u! This exact resolution opens the door to the fascinating world of elliptic functions, a jewel of the 19th century, but above all a catalyst for a formidable revolution. It is indeed to understand them that Euler, Gauss, Jacobi, Abel and Cauchy will found complex analysis, but it is Riemann who will really understand the geometry behind it. In doing so, we will see that an entire discipline was created to study these functions: algebraic geometry! But that's not all, it is still this problem that will lead a few years later to the development of algebraic topology, the theory of modular forms, etc... In this video, we will travel together (the beginning of) this incredible journey, in a development where we will see before our astonished eyes the geometry revealed. ------------------------------------------------------------------- Plan 00:00 Start 4:37 Introduction: length of curves, periodic functions 17:50 Simple pendulum with approximation 30:20 Sine and Arcsine revisited 42:22 Simple pendulum without approximation 50:50 Study of solutions, sinus amplitudinus 1:01:10 Summary on periodic functions and periods 1:09:20 Rectification of the lemniscate 1:16:43 Perimeter of the ellipse 1:19:00 Arithmetic-geometric mean 1:36:40 The hidden geometry of the sine 1:59:10 General definitions of Riemann surfaces 2:11:10 Example: the Riemann sphere 2:14:30 Example: the square root 2:21:00 Riemann-Hurwitz formula 2:29:00 Interlude: Map projection on a torus 2:45:00 Return to elliptic sine and geometry 2:54:40 Periods and elliptic curve 3:04:25 Short chronology and abelian varieties 3:16:00 Group structure on elliptic curves 3:24:40 Conclusion: the trichotomy Elliptic / Trigonometric / Rational ------------------------------------------------------------------- Reference: Elliptic Curves (Henry McKean & Victor Moll), CUP, 1999.