Douglas Lundholm: Quantum cosmology and the Big Bounce?

In the seminar two weeks ago we learned about the mathematical origin of the Big Bang in cosmology (a symmetry-reduced model of general relativity). Here I will sketch how in a recent framework called Loop Quantum Cosmology (a reduced model of quantum gravity), the Big Bang becomes replaced by a Big Bounce and the singularity is thereby resolved in a similar spirit as the classical singularity of the hydrogen atom is resolved by quantum mechanics. The procedure is surprisingly rich in mathematics and involves holonomies of connections, Poisson structures, representation theory of Weyl algebras, Bohr compactification, non-separable Hilbert spaces, and both differential and difference equations.