Alladi Ramakrishnan Hall

Geometry and Topology: Application to (cosmological) datasets

Pratyush Pranav

Ecole Normale Supérieure de Lyon, France

In the last couple of decades, algebraic topology has matured from a purely theoretical field to one with strong applicability in various research domains. Combining Morse theory, homology and persistent homology, this has enabled a new branch in data analysis called topological data analysis (TDA). The central tenet is based on the identification of 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.

As an illustration, I will present an example of application of the formalism to the Cosmic Microwave Background (CMB) radiation, towards the end. The CMB is the earliest visible light in the universe, and studying its properties has the potential to reveal information about the physical processes occuring in the nascent stages of the Universe. Stemming from inflationary theories, the CMB radiation is hypothesised to be an isotropic and homogeneous Gaussian random field. We find that the observations disagree significantly with the model.