Arcadian Functor

occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

Wednesday, April 22, 2009

Breakfast Ideas

Breakfast at my new house is very civilised. We sit down together at the table, chat a little and eat slowly. This morning a visiting materials physicist lamented about previous unsuccessful attempts to study category theory, the usefulness of which he was anyways greatly in doubt. I promised to come up with a recommended reading list, tailored to a materials scientist, but having thought about it further today I must confess to being totally stumped. Clearly there are still some gaps to fill in the introductory category theory genre. Now we can all find the good arxiv papers and standard textbooks. Any other recommendations?

These days one often comes across people in other (ie. usually not physics) departments who have some interest in category theory. I already mentioned Lawvere and Rosebrugh's book, Sets for Mathematics, to the lovely young philosopher who is studying Frege, amongst other things. And I suspect that Ross Street's book on Quantum Groups is also useful. Poor Carl is presently struggling with the ubiquitous text by the late Mac Lane. This book is very good, but perhaps not for the beginner. My favourite is Sheaves in Geometry and Logic, but that betrays a bias towards topos theory.

Aside: Now you can support Abtruse Goose with this groovy cap. I thought of merchandising for funds, but unfortunately it is entirely against my green anti-materialist ethos. Maybe I'll buy a cap though. AG deserves it.


Blogger CarlBrannen said...

I admit I burst out laughing as soon as I got to the part where the practical physicist wanted to learn category theory, but it does seem that this is gradually becoming ubiquitous and esoteric in a manner similar to group theory a couple generations ago.

Meanwhile, I continue to butt my head against problem of proving that all unitary 3x3 matrices are equivalent to magic ones. I am more and more convinced that it's true, but still no actual proof.

Right now I'm about to dive into the concept of looking at the problem in terms of the relationship between u(3) and a new Lie algebra, call it mu(3), the Lie group of magic matrices. Also of interest, the special magic unitary matrices, smu(3).

U = exp(iu) turns Hermitian matrices u into unitary matrices U. This is ugly in that iu is anti-Hermitian, so one cannot use the fact that Hermitian matrices are a subalgebra of the matrices. But if m is a magic matrix, then so is im, so it's a more pleasant place to work.

Computing exp(m) is very easy if m is idempotent or nilpotent. I think this should give a nice one-line proof that my parameterization works, but I haven't checked it yet.

April 22, 2009 5:53 PM  
Blogger CarlBrannen said...

Okay, here's the paper I probably need: Differential Geometry on SU(3) with Applications to Three State Systems.

April 22, 2009 9:39 PM  
Anonymous Anonymous said...

What about "Conceptual Mathematics" by Lawvere and Schanuel? I haven't read it, it just lies around, but the intro does say it has been used with bright high schoolers. Could serve as an introduction to introduction? I'm in the same position as the physicist.

April 23, 2009 1:34 PM  
Blogger Kea said...

That's an excellent recommendation, Anonymous. Sorry I didn't think of it.

April 23, 2009 9:10 PM  
Blogger L. Riofrio said...

It looks like victory is near for Category theory. Over here the discussion are mostly about Space flight, though I try to sneak physics in wherever possible.

In Xanadu did Kubla Kahn,
The stately pleaure domes decree..
--Coleridge, from memory

April 25, 2009 8:09 AM  

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