I’ve decided to look into topological quantization. Right now it is a pretty broad interest, but that’s kind of the point. I’d like to have a general idea of how geometry can produce quantization rules, or to know if I am even saying that correctly. The Gauss-Bonnet theorem shows that the integral of the Gaussian curvature (with some other things) gives an integer, the Euler Characteristic. Generalizing that seems to be a route into this.

I didn’t fully grasp the standard Gauss-Bonnet theorem when I first read about it. Berry phase is an example along these lines that I haven’t come to terms with. So it’s time I finally handled these things I guess. Plus, if you can understand mechanics as flows on manifolds, the next step seems to be to ask what can the topology of the space tell you about the dynamics. Lastly, there are some connections to the Hopf fibration and Dirac monopole which could be fun to explore.