A (somewhat) new paradigm for mathematics and physics
Insights into Mathematics
98.9K subscribers
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 other wise 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.
This is closely related to the Exceptional Structures in Mathematics and Physics lectures which are part of the Explore Research Level Mathematics video lectures. These are designed for motivated amateur, or professional, mathematicians wanting to delve into pure maths research without the abstraction and complexity that currently besets much of the subject. We will aim to always connect our abstract musings to computational reality by focusing on examples and explicit, concrete calculations - naturally without any pretense of being "able to do an infinite number of things"!
To access more Explore Research Level Mathematics, please join Norman's Wild Egg Maths channel at https://www.youtube.com/channel/UCriF..., or become a Patron at https://www.patreon.com/njwildberger . There are already dozens of videos there, and many more will appear over the next months and years. One additional advantage: you get to access pdf summaries of some of the videos along with GeoGebra worksheets.
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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...
Ancient Mathematics: https://www.youtube.com/playlist?list...
Wild West Banking: https://www.youtube.com/playlist?list...
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list...
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list...
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list...
MathFoundations: https://www.youtube.com/playlist?list...
Wild Linear Algebra: https://www.youtube.com/playlist?list...
Famous Math Problems: https://www.youtube.com/playlist?list...
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list...
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list...
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list...
Differential Geometry: https://www.youtube.com/playlist?list...
Algebraic Topology: https://www.youtube.com/playlist?list...
Math Seminars: https://www.youtube.com/playlist?list...
And here are the Wild Egg Maths Playlists:
Solving Polynomial Equations: https://www.youtube.com/playlist?list...
Exceptional Structures in Maths and Physics via Dynamics on Graphs" https://www.youtube.com/playlist?list...
Triangle Centres: https://www.youtube.com/watch?v=iLBGX...
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list...
Algebraic Calculus One: https://www.youtube.com/playlist?list...
Algebraic Calculus Two: https://www.youtube.com/playlist?list...