We study the limiting distribution, in the high energy limit, of critical points and extrema of random spherical harmonics.

In particular, we first derive the density functions of extrema and saddles and then we provide analytic expressions for the variances. Our arguments require a careful investigation of the validity of the Kac-Rice formula in nonstandard circumstances, entailing degeneracies of covariance matrices for first and second derivatives of the processes being analyzed. It is well known that after proper rescaling random spherical harmonics converge to Berry's random plane waves;

in the second part of the talk we focus on the spatial distribution of critical points of random plane waves.

Based on joint works with Dmitry Beliaev, Domenico Marinucci and Igor Wigman.