The overall goal of the project under consideration for the second period is to push forward the analysis of random Riemannian manifolds — or more precisely, random metric measure spaces — obtained through random perturbations of Riemannian manifolds, and to deepen our understanding of the associated discrete approximations. Of particular interest will be the enhanced study of the conformally invariant random objects, the construction and analysis of which are among the groundbreaking results in the first funding period. In detail, the focus will be on: i) construction of random fields, Liouville measure and Polyakov measure in cases beyond the previous approach including manifolds of odd dimension, non-compact manifolds and non-admissible manifolds; ii) detailed study of discrete approximations for higher dimensional random Riemannian manifolds (more precisely, random metric measure spaces) including convergence — or at least sub-convergence — of the re- normalized distance functions; iii) modification of the Polyakov-Liouville measure via vertex insertion and derivation of corre- sponding Seiberg bounds in higher dimensions; iv) analysis of the semiclassical limit of the Polyakov-Liouville measure and characterization of the limit points as manifolds with constant Q-curvature.

We will study random perturbations of Riemannian manifolds (M,g) by means of the so-called “Fractional Gaussian Fields” defined intrinsically by the given manifold. The fields will act on the manifolds via conformal transformation. Our focus will be on the regular case with Hurst parameter H>0, the celebrated Liouville geometry in even dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, spectral bound, spectral gap, hitting probabilities of the Brownian motions, or heat kernel estimates will change under the influence of the noise. Another goal is to define and analyze in detail the Fractional Gaussian Field on a general Riemannian manifold, a fascinating object of independent interest.