  Our goal is to discuss in detail the calculation of the mean number of
stationary points and minima for random isotropic Gaussian fields on a sphere
as well as for stationary Gaussian random fields in a background parabolic
confinement. After developing the general formalism based on the
high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory
(RMT) techniques to perform analysis of the random energy landscape of $p-$spin
spherical spinglasses and a related glass model, both displaying a
zero-temperature one-step replica symmetry breaking glass transition as a
function of control parameters (e.g. a magnetic field or curvature of the
confining potential). A particular emphasis of the presented analysis is on
understanding in detail the picture of "topology trivialization" (in the sense
of drastic reduction of the number of stationary points) of the landscape which
takes place in the vicinity of the zero-temperature glass transition in both
models. We will reveal the important role of the GOE "edge scaling" spectral
region and the Tracy-Widom distribution of the maximal eigenvalue of GOE
matrices for providing an accurate quantitative description of the universal
features of the topology trivialization scenario.
