Mathematical Structures in AI

Scientific data exhibits complex, multiscale structure that standard statistical and machine learning models often struggle to capture. In modern machine learning, the choice of mathematical structure acts as the strongest inductive bias, determining not just how a model learns, but what it can learn. In this talk, we show how tools from differential, discrete, and spectral geometry, and topology guide both the foundations and the applications of modern AI. Our examples range from learning graph embeddings in symmetric spaces and tropical geometries to modeling complex dynamics on networks using discrete and spectral tools, with case studies in chemistry, physics, and neuroscience. Lastly, we illustrate the payoff on both theory and practice, e.g., establishing theoretical links between model behavior and mathematical invariants in data and model structure (e.g., curvature), achieving stable long-range forecasting, and providing diagnostics that explain why some of these gains occur. We conclude with an outlook on open theoretical problems and how these methods pave the way for AI to become a partner in mathematical discovery itself.