Information Geometry of smooth densities on the Gaussian space: Poincare inequalities -- On Normalization Functions and ϕ-families of Probability Distributions -- Affine Connections with Torsion in (para-)complexified Structures -- Contact Hamiltonian systems for probability distribution functions and expectation variables: A study based on a class of master equations -- Invariant Koszul Form of Homogeneous Bounded Domains and Information Geometry Structures -- Gauge freedom of entropies on q-Gaussian measures -- On geodesic triangles with right angles in a dually flat space -- Chain Rule Optimal Transport -- Towards the "Shape" of Cosmological Observables and the String Theory Landscape with Topological Data Analysis -- A review of two decades of correlations, hierarchies, networks and clustering in financial markets.