Curriculum Roadmap

Topological Data Analysis — from simplices to persistence diagrams. The geometric heart of the site.

• Simplicial Complexes Foundational
• Persistent Homology Intermediate
• Čech Complexes & Nerve Theorem Planned
• Mapper Algorithm Planned
• Barcodes & Bottleneck Distance Planned
• Sheaf Theory Planned

Spectral theory, matrix decompositions, and the algebraic backbone of nearly every ML method.

• Spectral Theorem Planned
• Singular Value Decomposition Planned
• PCA & Low-Rank Approximation Planned
• Tensor Decompositions Planned

Measure-theoretic foundations through PAC learning and concentration inequalities.

• Measure-Theoretic Probability Planned
• Concentration Inequalities Planned
• PAC Learning Framework Planned
• Bayesian Nonparametrics Planned

Convex analysis, gradient methods, and the theoretical tools behind modern training algorithms.

• Convex Analysis Planned
• Gradient Descent & Convergence Planned
• Proximal Methods Planned
• Lagrangian Duality & KKT Planned

Smooth manifolds, Riemannian metrics, and information geometry for probabilistic models.

• Smooth Manifolds Planned
• Riemannian Geometry Planned
• Geodesics & Curvature Planned
• Information Geometry & Fisher Metric Planned

Entropy, divergences, and the theoretical limits that underpin compression, coding, and learning.

• Shannon Entropy & Mutual Information Planned
• KL Divergence & f-Divergences Planned
• Rate-Distortion Theory Planned
• Minimum Description Length Planned

Spectral graph theory and random walks — the mathematical foundation of graph neural networks.

• Graph Laplacians & Spectrum Planned
• Random Walks & Mixing Planned
• Expander Graphs Planned
• Message Passing & GNNs Planned

Functors, adjunctions, and monads — the abstract language that unifies disparate ML structures.

• Categories & Functors Planned
• Natural Transformations Planned
• Adjunctions Planned
• Monads & Comonads Planned