THE GEOMETRY OF RANDOM EIGENFUNCTIONS AND NEEDLET RANDOM - ANNA PAOLA TODINO – DIPARTIMENTO DI SCIENZE MATEMATICHE
Recently, considerable interest has been drawn by the analysis of geometric functionals (Lipschitz-Killing curvatures, hereafter LKCs) for the excursion sets of random eigenfunctions on the unit sphere (spherical harmonics). In dimension 2, LKCs correspond to the area, half of the boundary length and the Euler-Poincaré characteristic. The asymptotic behavior of their expected values and variances have been investigated and quantitative central limit theorems have been established in the high energy limits, after exploiting Wiener chaos expansions and Stein-Malliavin techniques. These results have been then extended to local behavior; more precisely Nodal Lengths in shrinking domains and excursion area in a spherical cap were considered. These studies are strongly motivated by cosmological applications, in particular in connection to the Cosmic Microwave Background.