In recent decades, algebraic topology has become an important tool in condensed matter physics, especially around the study and classification of topological materials, which are a form of so-called “quantum matter”. In this talk, I will give a window into the very recent emergence of algebraic geometry in condensed matter physics through my work on hyperbolic band theory, which anticipates a new form of synthetic quantum matter based on hyperbolic geometries. These ideas involve stable bundles and Higgs bundles on algebraic curves. This talk will not assume any physics background.