Quantum geometry and quantum gravity
The aim of quantum geometry is to find generalizations of smooth manifolds and differential geometry suitable for applications in quantum physics. This area was inspired by the quest for a theory of quantum gravity, which had been originally defined as quantization of General Relativity, but later it was realised that a generalization of the notion of a manifold is necessary in order to describe the spacetime at very short distances.
There are several approaches to the problem of quantum geometry and quantum gravity, and we have been working on two approaches: Loop Quantum Gravity (LQG) and Noncommutative Geometry (NG). In the LQG approach the quantum geometry of space is represented by spin networks, which are graphs colored by spins (irreducible representations of SU(2)). The time evolution of spin networks creates the spin foams, which are two-complexes whose faces are colored by spins. The path integral for General Relativity can be represented as a sum over colorings of an amplitude for a spin foam. The spin foam state sums can be defined by using the representations of quantum groups or by using the propagators on coset spaces, and we use both techniques in order to obtain quantum gravity models. The quantum group technique is also useful for obtaining the manifold invariants.
In the NG approach, the commutative algebra of functions an a manifold is replaced by a non-commutative algebra, and according to the approach of A. Connes, there is a precise formalism how to define a geometry on a such algebra. One way of introducing this algebra is through the star product for the algebra of functions on a simplectic manifold. This approach is known as the deformation quantization approach, and it is also useful for quantization of particles whose coordinates are non-commutative and for defining the corresponding quantum field theories.