Inter-universal Teichmüller theory is a branch of mathematics that explores the relationships between different mathematical structures, particularly in the field of geometry and topology. It extends classical Teichmüller theory, which studies the deformation of complex structures on surfaces, to a broader context involving multiple universes of mathematical objects. This theory introduces concepts such as Hochschild-Serre spectral sequences and p-adic analysis, allowing mathematicians to analyze and compare structures across different mathematical frameworks. It aims to unify various areas of mathematics, providing insights into the connections between seemingly unrelated topics and enhancing our understanding of moduli spaces and dynamics.