Ricerc@Sapienza

Special attention will be given to investigating the geometry of band-limited spherical random fields in the high-energy (high-frequency) limit; this covers both the case of random Laplace eigenfunctions and wavelet/needlet components of random fields, of great interest for cosmological data analysis. Even stronger attention will be devoted to random sections of spin fiber bundles; these fields emerge naturally in the areas of Cosmic Microwave Background Polarization and Weak Gravitational Lensing, which are both frontiers areas in Cosmology and the object of major international experiments (such as ESA's satellite Euclid for lensing and LiteBird for Polarization). Spin random fields are random sections of complex line bundle and they carry on a very rich mathematical structure, which can also be addressed in the form of random fields on the group of rotations SO(3).

Again some cosmological applications, and in particular foreground estimation, motivate the analysis of functional data on the sphere, where we aim to develop infinite-dimensional techniques to estimate elements of function spaces indexed by spherical locations.
Spherical random fields arise naturally in several other circumstances outside astrophysics, such as medical imaging, solar physics, Earth and climate sciences just to mention a few. In these cases, we shall investigate also more complex structures where a time-dependent random component is included. Our aim will be to study random processes emerging as the evolution over time of Lipschitz-Killing curvatures for excursion sets. We aim also to consider topological processes, such as the evolution of Betti numbers for excursion sets evolving over time. These forms of topological processes will be considered also in discretized contexts, for instance studying the percolation properties of time-dependent random graphs.