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.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA