Geometric Foundations of AI (II): Data Manifolds, Symmetry, and Learned Representations

What geometry does a learning system discover in data—and what does it impose through architecture, objectives, and representation? How should we compare representations across models or modalities, and when do claims about manifolds, symmetry, topology, or emergent structure have mathematical content rather than visual appeal?

Geometric Foundations of AI (II) develops the object-side geometry of modern machine learning. It begins with group actions, representation theory, and equivariant architectures, then builds toward information bottlenecks, contrastive and self-supervised representation geometry, non-Euclidean and hyperbolic embeddings, manifold hypotheses and intrinsic dimension, representation diagnostics, alignment and transfer, topological data analysis, and the geometry of foundation-model representations. A recurring theme is that a geometric claim becomes meaningful only after the object, metric, symmetry, scale, and estimator have been specified.

Written for advanced undergraduates, graduate students, and researchers in machine learning, statistics, geometry, and applied mathematics, the book combines exact mathematics with carefully delimited model-dependent and empirical claims. It is both a technical foundation for geometric and representation learning and a guide to deciding which geometric interpretations are justified by the evidence.