2009 April 27 / E-Mail Me
Senior Integrative Exercise (Comps)
My 2008-2009 comps group is studying geometric algebra (not to be confused with algebraic geometry). This is an enjoyable little neighborhood of mathematics with some important residents:
- normed division algebras (real numbers, complex numbers, quaternions, octonions)
- Clifford algebras (real numbers, complex numbers, quaternions, and others, but not octonions)
- Bott periodicity (some geometric phenomena repeat every 2 or 8 dimensions)
- Lie groups (such as O(n), SO(n), Pin(n), Spin(n), and even E8 and G2)
- geometry and mathematical physics
The list could keep going, but I have to end it somewhere, especially since my own understanding is limited. The connections to representation theory and physics, in particular, relate geometric algebra to just about everything in math. In short, if you're excited by the alleged interconnectedness of all of mathematics, then geometric algebra is a good spot from which to survey that interconnectedness. Here are some materials:
- Rotations describes rotations in terms of matrices, angles, complex numbers, and quaternions.
- The Octonions by John Baez of UC-Riverside is our main text. I recommend all of Baez's Fun Stuff.
- cayleydickson.nb is a Mathematica notebook for computing with Cayley-Dickson numbers. It's quite unpolished.
- clifford.nb is a Mathematica notebook for computing with Clifford alebras. It's a work in progress.
Update: The final product of the project was a classification of the irreducible representations of the double cover of the Poincaré group, and thus some aspect of the elementary particles of physics, following Eugene Wigner's 1939 paper.