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Joint Math-Physics Seminar 10:30 am, Seminar Hall Geometry and Topology : Application to (cosmological) datasets Pratyush Pranav ENS de Lyon. 13-12-19 Abstract In the last couple of decades, geometry and topology have matured from a purely theoretical field to one with strong applicability in various research domains.The principal tools from Geometry has been the development of integral geometric quantifiers, like Minkowski functionals. From the side of topology, 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. Towards the end, I will present an analysis of the topological characteristics of the temperature fluctuations in the Cosmic Microwave Background (CMB) radiation. 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.
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