To answer your question. Louisiana State University is not on your list. You should add them to your list and consider them for your PhD. They have Jorge Pullin, Kristina Giesel, and Parampreet Singh working in Loops.
Good luck in your studies.
Hi, I think (and someone can correct me if I am wrong) that LQG is noncommutatitive in the sense that certain geometrical operators fail to commute. For instance, the area operators associated with 2 surfaces that intersect fail to commute.
For example, see section V in Ashtekar's review...