  We study random perturbations of Riemannian manifolds
$(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields,
which are defined intrinsically by the given manifold. The fields $h^\bullet:
\omega\mapsto h^\omega$ will act on the manifolds via conformal transformation
$\mathsf{g}\mapsto \mathsf{g}^\omega\colon\!\!= e^{2h^\omega}\,\mathsf{g}$. Our
focus will be on the regular case with Hurst parameter $H>0$, the celebrated
Liouville geometry in two dimensions being borderline. We want to understand
how basic geometric and functional analytic quantities like diameter, volume,
heat kernel, Brownian motion, spectral bound, or spectral gap will change under
the influence of the noise. And if so, is it possible to quantify these
dependencies in terms of key parameters of the noise. Another goal is to define
and analyze in detail the Fractional Gaussian Fields on a general Riemannian
manifold, a fascinating object of independent interest.
