  The formalism to compute the geometrical and topological one-point statistics
of mildly non-Gaussian 2D and 3D cosmological fields is developed. Leveraging
the isotropy of the target statistics, the Gram-Charlier expansion is
reformulated with rotation invariant variables. This formulation allows us to
track the geometrical statistics of the cosmic field to all orders. It then
allows us to connect the one point statistics of the critical sets to the
growth factor through perturbation theory, which predicts the redshift
evolution of higher order cumulants. In particular, the cosmic non-linear
evolution of the skeleton's length, together with the statistics of extrema and
Euler characteristic are investigated in turn. In 2D, the corresponding
differential densities are analytic as a function of the excursion set
threshold and the shape parameter. In 3D, the Euler characteristics and the
field isosurface area are also analytic to all orders in the expansion.
Numerical integrations are performed and simple fits are provided whenever
closed form expressions are not available. These statistics are compared to
estimates from N-body simulations and are shown to match well the cosmic
evolution up to root mean square of the density field of ~0.2. In 3D,
gravitational perturbation theory is implemented to predict the cosmic
evolution of all the relevant Gram-Charlier coefficients for universes with
scale invariant matter distribution. The one point statistics of critical sets
could be used to constrain primordial non-Gaussianities and the dark energy
equation of state on upcoming cosmic surveys; this is illustrated on idealized
experiments.
