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April 9, 2007

This Week’s Finds in Mathematical Physics (Week 249)

Posted by John Baez

In week249 of This Week’s Finds you can finally see Felix Klein’s famous "Erlangen program" for reducing geometry to group theory - translated into English!

Then, continue reading the Tale of Groupoidification - in which we see how group actions are just groupoids equipped with extra stuff.

Posted at April 9, 2007 4:15 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1233

9 Comments & 4 Trackbacks

Re: This Week’s Finds in Mathematical Physics (Week 249)

Thanks John! Fascinating as always:

To do this, we take two sets of figures, say X=G/HX = G/H and Y=G/KY = G/K and find all the invariant relations between them: that is, subsets of X×YX \times Y preserved by all the symmetries. I’ll say more about this next time: we can use something called “double cosets”.

These are exactly the things which crop up when you consider morphisms between 2-representations of GG. Can’t wait for next time.

Posted by: Bruce Bartlett on April 9, 2007 12:06 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

Double cosets also play a big role in ordinary group representation theory, and that’s something I want to talk about pretty soon. They lead to concepts like ‘Hecke operator’, ‘Hecke algebra’, and ‘Radon transform’. In fact, all three of these concepts are just variations on the same theme: using spans of groupoids (and in particular, double cosets) to construct intertwining operators between group representations. But nobody seems to come out and admit how closely related these concepts are! So, that’s something we’ve got to talk about…

Posted by: John Baez on April 9, 2007 3:11 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

Thanks go to Robin Houston and Lukas-Fabian Moser for creating PDF and DjVu versions of the English translation of Klein’s Erlangen program! These make it much easier to read.

Also thanks go to James Dolan for catching some (but undoubtedly far from all) errors in the first version of week249.

Namely: the morally correct group of symmetries in projective geometry is not PSLPSL but PGLPGL.

And: it’s not true in Klein geometry that each subgroup HH of our symmetry group GG is ‘a figure’. I think it’s better to say that each set XX on which GG acts transitively is ‘a type of figure’. Each figure xXx \in X has a stabilizer HH, but often different figures will have the same stabilizer. Indeed, the stabilizer of gxXg x \in X is gHg 1g H g^{-1}, and this can be the same as HH even when gxg x is different from xx. The point is that the normalizer of HH can be bigger than HH!

Note also that different sets on which GG acts transitively can be isomorphic while still not equal — so we have a vital concept of “isomorphic” types of figures. In particular, the types of figures corresponding to X=G/HX = G/H and Y=G/KY = G/K are isomorphic if the subgroups HH and KK are conjugate in GG. This issue was lurking around the edges near the end of week249.

Posted by: John Baez on April 9, 2007 7:51 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

There’s an English version now on the ArXiv.

Posted by: David Corfield on July 22, 2008 9:55 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

I am not contributing much at the moment, but at least here I can help with spotting a typo in the amazingly typo-free TWFs:

somewhere towards the last third, in the discussion of transitive actions it currently says:

“So, GG acts transitively on XX on precisely when X//GX//G is equivalent to a group!”

One superfluous “on”.

Posted by: urs on April 10, 2007 8:47 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

Thanks for catching that typo, Urs.

And guess what? Nitin C. Rughoonauth has taken the marvelous step of putting Klein’s Erlangen program paper into LaTeX! You can see a PDF file of it here. If anyone catches typos — I see one already — please email me or notify me here, and I’ll fix them.

(I don’t have the LaTeX source code yet, but I’ll get it and make it available here. Eventually I’ll try to make sure this paper gets onto the arXiv.)

Posted by: John Baez on April 10, 2007 9:31 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

p. 23 “afford us then immendiately the means for contructing a more exact geometry”

Posted by: David Corfield on April 10, 2007 10:15 PM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

A proofread version is now available, with this and other typos fixed.

Posted by: John Baez on April 14, 2007 1:27 AM | Permalink | Reply to this

Re: This Week’s Finds in Mathematical Physics (Week 249)

A very elementary introduction to this topic may be found at:

Is Beauty Truth and Truth Beauty?
How Keats’s famous line applies to math and science
By Martin Gardner
Book Review
Scientific American
April 2007

Image: ANDY ROUSE Corbis
SYMMETRY was once a synonym for beauty–recall William Blake’s “Tyger! Tyger! burning bright/In the forests of the night,/ What immortal hand or eye/Could frame thy fearful symmetry?”

WHY BEAUTY IS TRUTH: A HISTORY OF SYMMETRY
by Ian Stewart
Basic Books, 2007

[whoops, form input won’t accept the long URL, but it can be easily found at sciam.com]

Of course, both Ian Stewart and Martin Gardner have been leading popularizers of mathematics for a very long time.


Posted by: Jonathan Vos Post on April 22, 2007 5:11 PM | Permalink | Reply to this
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