Abstract. Application of the intersection of algebraic topology and computational geometry, especially in Cosmology become a growing interest. Indeed, it provides a proper pipeline to elucidate and clarify any high-dimensional data using low-dimensional algebraic representations. Cosmological and astrophysical fields in 1, 2, and 3 dimensions such as CMB, Gravitational wave (GW), and Large-scale structures (LSS) are some appropriate cases to implement the topological and geometrical-based analysis. Some relevant purposes are classified into the morphological, the invariant properties, evaluation for model discrimination, and putting constraints on the relevant free parameters.
In this talk, I will give a brief review of the different categories of geometrical and topological measures, then I will show some of our results incorporating the mentioned criteria for the Planck CMB map, Stochastic Gravitational wave background (SGWB) and emphasizing the counterpart of real-space, namely red-shift space and corresponding distortion (RSD).

Inspired by Minkowski Functionals, we introduced a new measure which is so-called the “conditional moments of the first derivative” (cmd) to examine RSD. It successfully recognizes the impact of linear Kaiser and Finger of God effects. We can follow the brute force approach to distinguish the preferred direction. I will also show that taking into account the information on the plane perpendicular to the line of sight in redshift space can well reduce the degeneracy of non-Gaussianity due to distortion. In addition, the cmd is sensitive enough to the magnitude and direction of anisotropy, thought for breaking the degeneracy between matter density and linear bias. We also advocate the cmd measure instead of Minkowski tensors to constrain the RSD parameter, more precisely.