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Sunday, June 04, 2006

Subtler Symmetry

I'm surprised that when new tools are made available to capture more subtle aspects of symmetry, mathematicians don't move a little faster to exploit their potential. The diagram on p. 12 of Ronnie Brown's notes on Nonabelian Algebraic Topology shows some of these tools and how they relate to the workhorses of symmetry - groups. The great breakthrough that was the quantum group concept doesn't explicitly appear, although they have been found to relate to double groupoids. The 2-groups John Baez and I have been discussing, vis-a-vis Kleinian geometry, are again not mentioned explicitly in Brown's diagram, but are a special case of 2-groupoids.

In view of the extraordinary pervasiveness of groups throughout mathematics, see e.g., George Mackey's wonderful The scope and history of commutative and noncommutative harmonic analysis, there must be many opportunities to find applications for their relatives. It's interesting to read then, via Ars Mathematica, an article by M. Golubitsky and I. Stewart entitled Nonlinear Dynamics of Networks: The Groupoid Formalism, which looks to exploit the more subtle symmetry that groupoids can detect in networks. Section 3 on Animal Locomotion is especially interesting. After calculating the set of possible quadruped gaits, one was not recognised as occurring, until a video from the Houston Livestock Show and Rodeo was analysed and a bucking bronco found to be performing it.


Blake Stacey said...

This is tangential to the specific topic at hand but not completely orthogonal to the general subject: Mark Chu-Carroll has started a series on category theory over at Good Math, Bad Math. He generally proves himself a lucid writer, in my eyes anyway, though (of course) not always free from error.

June 07, 2006 6:59 PM  

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