Astronomical surveys with planets, stars, or galaxies are available from different types of instruments. Analysis of these data have been considered as statistical challenges in the modern field of astrostatistics. This thesis lies at the intersection of statistics and astronomy, with a focus on Topological data analysis (TDA). TDA is an approach for learning about the underlying structure of a dataset, within which persistent homology is a tool to detect different dimensional holes in a dataset. In this thesis, we present a method called Significant Cosmic Holes in Universe (SCHU) for identifying cosmic voids and loops of filaments in cosmological datasets and assigning their statistical significance using techniques from TDA. Also, we propose a method called Empty Territory (EmT) to provide representations of different dimensional holes with a specified level of complexity of the hole boundary. In addition, we introduce a procedure for using persistent homology on a large dataset, such as a large-scale cosmological simulation. This is carried out by dividing the data volume into subsets, running persistent homology on the individual subsets, and combining the results.

Along with the TDA focus, this thesis also presents a work of continuum normaliza- tion of echelle spectra. Spectroscopy is a powerful observational technique for understand- ing fundamental astrophysics. The Alpha-shape Fitting to Spectrum algorithm (AFS) is completely data-driven, the Alpha-shape and Lab Source Fitting to Spectrum algorithm (ALSFS) incorporates a continuum constraint from a lab source reference spectrum.