Stringy loopy maths

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wolram
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http://math.ucr.edu/home/baez/week184.html

John Baez

To really know a subject you've got to learn a bit of its history. If that subject is topology, you've got to read this:

1) I. M. James, editor, History of Topology, Elsevier, New York, 1999.

From a blow-by-blow account of the heroic papers of Poincare to a detailed account by Peter May of the prehistory of stable homotopy theory... it's all very fascinating. You'll probably want to study some more of the subject by the time you're done!

In order to satisfy that craving, I want to tell you how to compute some homology groups. But we'll do it a strange way: using "q-mathematics". I began talking about q-mathematics last week, but now I want to dig deeper.

At first, it looks like there are two really different places where this q-stuff shows up. One is when you do mathematics with q-deformed quantum groups replacing the Lie groups you know and love - this is important in string theory, knot theory, and loop quantum gravity. In this case it's best if q is a unit complex number, especially an nth root of unity:

q = exp(2 pi i / n)

You'll notice that in string theory, knot theory and loop quantum gravity, loops play a big role. This is no coincidence; in a way, quantum groups are just a technical device for studying "loop groups", which are groups consisting of functions from a circle to some specified Lie group.
 

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