Ricerc@Sapienza

This project focuses on the analysis of geometric feature of Gaussian random fields defined as random Laplace eigenfunctions on d-dimensional manifolds. On the sphere, in particular, the geometrical analysis of random Gaussian eigenfunctions emerge naturally form the analysis of the Fourier components of Gaussian fields and has recently drawn a big amount of interest in view of strong motivating applications in cosmology, image analysis and complexity of random energy landscapes.
The goal of this research project is to obtain developments in the study of the following geometrical features:

- the effect of nontrivial boundary conditions on the nodal structure both in the vicinity of the boundary and globally,

- the introduction of ad hoc comparison methods to measure the 'degree' of clustering of critical points and the short-range repulsion of critical points of different indexes,

- the development of a general variance formula for the Lipschitz-Killing curvatures for the excursion sets of random spherical eigenfunctions (i.e. Euler characteristic of the excursion sets, length of level curves, and excursion area) that could extend the well known Gaussian Kinematic Formula for the expected value of the Lipschitz-Killing curvatures,

- the statistical analysis of random fields and in particular the application of the theoretical results on the probabilistic behaviour of the geometric functional to introduce suitable statistics for testing non-Gaussianity and isotropy.