Interplay between Loop Quantum Gravity and Mathematics It is known that there are many interesting connections between String Theory and modern Mathematics, with a rich feedback going on in both directions: there have been advances in mathematics thanks to research in String Theory and vice-versa. For the interested reader, some of the interactions between String Theory and Mathematics have been reviewed in
http://rsta.royalsocietypublishing.org/content/roypta/368/1914/913.full.pdf
by Atiyah, Dijkgraaf and Hitchin. However, String Theory is not the only quantum theory of gravity on the market: it has a much less popular competitor called Loop Quantum Gravity. My question is, are there any interesting interactions between mathematics and LQG? Has there been any advances in mathematics thanks to LQG research, as it has happened with ST?
Thanks.
 A: Here is one area of interplay between loop quantum gravity and combinatorics:
Oriented Matroids – Combinatorial Structures Underlying Loop Quantum Gravity, Brunnemann and Rideout (2010).

We analyze combinatorial structures which play a central role in
  determining spectral properties of the volume operator in loop quantum
  gravity (LQG). We demonstrate that these structures are a special
  representation of the general mathematical concept of an oriented
  matroid. This opens up a new mathematical arena to study LQG, and to
  overcome certain technical and conceptual obstacles which, at present,
  make concrete analytical and numerical investigations of the full
  theory very difficult. In particular it becomes feasible to develop a
  systematic and unified treatment of global (connectedness) and local
  (embedding) properties of the graph poset underlying LQG.

A: One topic which started in the general area of LQG but now has taken a life of its own is the theory random tensors and group field theories. For its connections to mathematics you can have a look at the talks and courses from the program I co-organized last summer in Vienna:
http://www.mat.univie.ac.at/~kratt/esi3/
