Spectral Triples

Spectral triples are a mathematical framework used to capture the geometry of space, especially in contexts like quantum physics where traditional notions of shape and size may not apply. They consist of three components: an algebra of functions describing the space, a Hilbert space that provides a setting for analyzing these functions, and a special operator called the Dirac operator, which encodes geometric information such as distances and curvature. Together, spectral triples offer a way to study and understand geometric structures through the analysis of spectral, or frequency-based, data, bridging algebraic and geometric perspectives.