Quantum Gravity: Unification of Principles and Interactions, and Promises of Spectral Geometry

Quantum gravity was born as that branch of modern theoretical physics that tries to unify its guiding principles, i.e. quantum mechanics and general relativity. Nowadays it is providing new insight into the unification of all fundamental interactions, while giving rise to new developments in modern mathematics. It is however unclear whether it will ever become a falsifiable physical theory, since it deals with Planck-scale physics. Reviewing a wide range of spectral geometry from index theory to spectral triples, we hope to dismiss the general opinion that the mere mathematical complexity of the unification programme will obstruct that programme.