A (somewhat) new paradigm for mathematics and physics | Diffusion Symmetry 1 | N J Wildberger

The current understanding of symmetry in mathematics and physics is through group theory. However in the last 120 years, a new strand of thought has gradually appeared in a number of disciplines, from as varied as character theory, strongly regular graphs, von Neumann algebras, Hecke algebras, Lie group representation theory, cyclotomy and conformal field theory. In this video we try to put these varied approaches together to form a unified new alternative to understanding symmetry in both mathematics and physics, taking our intuition from the diffusion of a gas in an otherwise empty room -- hence the term "diffusion symmetry". We'll show through the careful analysis of a particularly rich example (familiar to all students of mathematics at some level) how both the notion of a hypergroup (going back to the 1970's with seminal papers by Dunkl, Jewett and Spector) and fusion rule algebra (appearing in physics and the study of II-1 factors and elsewhere) really do give us a rich alternative to the usual "transformational" approach to symmetry embodied in group theory. In fact diffusion symmetry meshes very well with group theory, and sheds considerable light on many aspects of it; in particular the role of duality and character tables. By going to the diffusion point of view, we somehow miraculously replace (at least in some situations) non-commutative harmonic analysis with commutative harmonic analysis -- a major simplification. Video Contents (thanks to Lucas Lofaro) 0:00 Intro 0:07 Overview 0:38 Two Types of Symmetries 2:30 Though Experiment: Gas Diffusion 5:10 History of the Subject 9:03 An Initial Example 16:33 The Multiset of Neighbors 23:43 Fusion Rule Algebra 31:00 The Probabilistic Variant 38:34 Character Table of an Alternating Group 43:17 Character Table of a Hypergroup ************** Research Gate page: https://www.researchgate.net/profile/... Blog: http://njwildberger.com/ Online courses at openlearning.com (currently Algebraic Calculus One): https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects! Patreon:   / njwildberger   Your support would be much appreciated. Insights into Mathematics YT channel:    / njwildberger   Insights into Mathematics Playlists:    • The Algebra of Boole, Logic and Circuit An...   (31 videos)    • Box Arithmetic: a new framework for Mathem...   (18 videos)    • Hypergroups and Diffusion Symmetry: an int...   (6 videos)    • Rational Trigonometry for maths, physics a...   (4 videos)    • Sociology and Pure Maths   (44 videos)    • Old Babylonian mathematics and Plimpton 322   (8 videos)    • Math Foundations   (226 videos)    • Math Seminars N J Wildberger   (26 videos)    • Math History (ancient to modern)   (45 videos)    • Geometric Linear Algebra   (43 videos)    • Algebraic Topology   (40 videos)    • Universal Hyperbolic Geometry   (55 videos)    • Differential Geometry   (34 videos)    • Elementary Probability and Statistics   (8 videos)    • Math Terminology for Incoming Uni Students   (9 videos)    • Famous Math Problems   ( 46 videos) Wild Egg Maths Playlists:    • Box Arithmetic Book: A finite multiset fou...   (NEW!!)    • Intro to Algebraic Calculus with Box Arith...   (4 videos)    • Classical to Quantum (for Members)   (64 videos)    • Solving Polynomial Equations and the Geode...   (45 videos)    • De Casteljau Bezier curves and associated ...   (20 videos)    • Exploring q-series (for Members)   (8 videos)    • Six: A mathematical exploration   (9 videos)    • Algebraic Calculus One: a new foundation f...   (52 videos)    • Advice to prospective research mathematici...   (9 videos)    • The Hexagrammum Mysticum: a Geometric Gem ...   (14 videos)    • Algebraic Calculus Two   (8 videos)    • Special (Polynomial) Functions and Maxel N...   (25 videos)    • Dynamics on Graphs: The ADE Phenomenon & B...   (30 videos)