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

The Playlist for this series is at https://www.youtube.com/playlist?list=PLIljB45xT85Bmcc9ksBOAKgIZAl0BwPg7
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This is closely related to the Exceptional Structures in Mathematics and Physics lectures which are part of the Explore Research Level Mathematics video lectures.

To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at https://www.youtube.com/channel/UCriFv3G22iOUidUhkIGXuhw, or become a Patron at https://www.patreon.com/njwildberger .

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Here are all the Insights into Mathematics Playlists:

Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
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And here are the Wild Egg Maths Playlists:

Solving Polynomial Equations: https://www.youtube.com/playlist?list=PLzdiPTrEWyz7hk_Kzj4zDF_kUXBCtiGn6
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list=PLzdiPTrEWyz5HBT_Yo1G4DfeqUfI9zkKM
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-

Видео A (somewhat) new paradigm for mathematics and physics | Diffusion Symmetry 1 | N J Wildberger канала Insights into Mathematics