Anomalies in the topology of the fluctuations in the Cosmic Microwave Background

Abstract: Cosmology is transitioning from a theoretical discipline towards one with increased focus on observations, resulting in a massive surge of data that demands increasingly more sophisticated methods to glean meaningful information. In a related development, geometry and topology have witnessed a tilt from purely theoretical fields towards strong focus on application. A foray into “big data” quickly brings to front two of the central statistical challenges of our times -- detection and classification of structure in extremely large, high-dimensional, data sets. Among the most intriguing new approaches to this challenge is “TDA,” or “topological data analysis,” the primary aim of which is providing topologically informative pre-analyses of data, which serve as input to more quantitative statistical analyses at a later stage. Algebraic and computational topology, at the level of homology and persistent homology, are the foundational pillars of TDA. In the first part of my talk, I will present a summary of the theoretical and computational aspects of topological data analysis. In the second part of the talk, I will present an analysis of the topological properties of the temperature and polarization maps of the Cosmic Microwave Background (CMB) radiation obtained by the Planck satellite. I will also discuss some of the anomalies that the temperature and polarization maps exhibit with respect to the simulations based on the standard cosmological model, which assumes the initial fluctuation field to be an instance of an isotropic and homogeneous Gaussian random field.