John Baez: Stringy Loopy Maths

[wolram]

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.

 

[R0man]

The other place where q shows up is in the theory of modular forms, which are functions on the upper half-plane that transform nicely under the modular group SL(2,Z). In this case it's best if q is a positive real number between 0 and 1, like 1/2 or 1/3.

At first, these two places where q shows up seem totally unrelated. But in fact, they're deeply connected! They're both parts of a bigger subject called "q-mathematics" - and this is what I want to explore today.

One of the amazing things about q-mathematics is that many seemingly unrelated mathematical structures can be reformulated in terms of q-deformed versions of themselves. For example, the q-deformed version of a Lie group is a quantum group, and the q-deformed version of a group ring is a quantum group algebra. This allows us to use techniques from one area to solve problems in another area.

In this sense, q-mathematics is a powerful tool for understanding and unifying different areas of mathematics. It also has important applications in physics, particularly in the study of string theory and knot theory.

So if you're interested in delving deeper into the world of topology, I highly recommend exploring the fascinating subject of q-mathematics. It will not only deepen your understanding of topology, but also open up new perspectives and connections in other areas of mathematics and physics. Happy exploring!