In the last couple of decades, topology and geometry have matured from purely theoretical fields towards strong focus on applicability in various research domains. The principal tool from Geometry has been the development of integral geometric quantifiers, viz. the Minkowski functionals. On the side of topology, a combination of Morse theory, homology and persistent homology, has enabled a new branch in data analysis called topological data analysis (TDA). The central tenet is based on the identification and assessment of geometrical and topological changes that occur in a manifold as a function of the excursion sets of the field. The topological changes are accounted for by tracking the creation and destruction of p-dimensional topological holes in a d-dimensional manifold. Intuitively, in 3 spatial dimensions, these changes correspond to creation and destruction of connected components, loops/tunnels and voids. The geometric quantifiers are associated with the notions of d-dimensional volume, area etc. In the first part of my talk, I will present a non-technical summary of the methods.

In the second part, I will present two examples highlighting the application component. The first example concerns theoretical Gaussian random field models with power-law power spectra. We find that a topological characterization through Betti numbers and persistence diagrams provide information that is missed by traditional topological measures like Euler characteristic and the more familiar geometric Minkowski functionals. The second concerns the analysis of the topological characteristics of the temperature fluctuations in the Cosmic Microwave Background (CMB) from the temperature anisotropy maps measured by the Planck satellite. We find that the observed maps differ significantly from the simulations modeled as isotropic, homogeneous Gaussian random fields.