Introduction to the spectral scheme for spacetime physics

The spectral scheme is a framework for describing geometry of spaces in terms of the spectra of some elliptic operator. Its motto is "Let us hear the shape of the Universe!" Three central components are: (I) The spectral distance; (II) The space of spaces; (III) The spectral evolution equations. The basics of the spectral scheme along with some of its applications have been presented here. Some recent developments of the scheme, including the spectral representation of the Einstein-Hilbert action and its application to quantum spacetimes, have also been presented.