Abstract: The Anti-de-Sitter/Condensed Matter Theory (AdS/CMT) correspondence has attracted significant attention within high-energy physics, particularly in the study of holographic superconductors, which are modeled as asymptotically Anti-de-Sitter (AdS) hairy black holes. In this talk, I will present recent mathematical advancements in the construction of asymptotically AdS hairy black holes and the analysis of their interior structure. If time permits, I will explain how these mathematical results can be adapted to gain insights into the interiors of asymptotically flat black holes as well.