Intro to the Fundamental Group // Algebraic Topology with @TomRocksMaths

In this video I teach the amazing @TomRocksMaths a little bit of algebraic topology, specifically the fundamental group. Tom also taught me some really cool fluid dynamics and you can find our collab over at his channel here: ►►► https://www.youtube.com/watch?v=bpeCfwY4qa0&ab_channel=TomRocksMaths 0:00 What is Algebraic Topology? 4:01 The alphabet to a topologist 8:20 The algebra of loops about a ring 14:50 Defining Homotopy Equivalence 18:54 The Fundamental Group 23:58 Fundamental Group of R^2 25:50 Fundamental Group of a Sphere 28:32 Fundamental Group of a Circle 31:45 Fundamental Group of a Torus 34:18 Proof of Brouwer's Fixed Point Theorem We begin by talking about what connotations the words "algebraic" and "topology" have; "algebra" has a certain concreteness to it as we can add or multiply things, have explicit formulas, etc while "topology" is all about considering spaces that are thought of the same even if we continuously deformed like they were playdoh. Under this perspective, the alphabet only has three letters to a topologist, a single point, a single circle, and a double circle (the letter B). We then define more precisely the notion of a homotopy equivalence between two maps into a space X. There is an operation we call multiplication on such paths which captures the idea of doing one path followed by the next. It turns out that the homotopy equivalence classes of loops in a space X starting and finishing from a basepoint x_0 with this notion of multiplication form the fundamental group which we often write is Pi_1(X, x_0). Tom then computes the fundamental group of many spaces, the plane, the 2-sphere, the 1-sphere or circle, and finally - triumphantly - the torus! Finally we finish with a nice proof of Brouwer's Fixed Point theorem that uses the power of the fundamental group to arrive at a contradiction. COURSE PLAYLISTS: ►DISCRETE MATH: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxersk8fUxiUMSIx0DBqsKZS ►LINEAR ALGEBRA: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfUl0tcqPNTJsb7R6BqSLo6 ►CALCULUS I: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfT9RMcReZ4WcoVILP4k6-m ► CALCULUS II: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc4ySKTIW19TLrT91Ik9M4n ►MULTIVARIABLE CALCULUS (Calc III): https://www.youtube.com/playlist?list=PLHXZ9OQGMqxc_CvEy7xBKRQr6I214QJcd ►VECTOR CALCULUS (Calc IV) https://www.youtube.com/playlist?list=PLHXZ9OQGMqxfW0GMqeUE1bLKaYor6kbHa ►DIFFERENTIAL EQUATIONS: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxde-SlgmWlCmNHroIWtujBw ►LAPLACE TRANSFORM: https://www.youtube.com/watch?v=xeeM3TT4Zgg&list=PLHXZ9OQGMqxcJXnLr08cyNaup4RDsbAl1 OTHER PLAYLISTS: ► Learning Math Series https://www.youtube.com/watch?v=LPH2lqis3D0&list=PLHXZ9OQGMqxfSkRtlL5KPq6JqMNTh_MBw ►Cool Math Series: https://www.youtube.com/playlist?list=PLHXZ9OQGMqxelE_9RzwJ-cqfUtaFBpiho 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 SOCIALS: ►Twitter (math based): http://twitter.com/treforbazett ►Instagram (photography based): http://instagram.com/treforphotography