Math foundations
Lecture-style introductions to the math, computation, and physics that everything else stands on.
Each page is the way a good lecturer might open a topic — what we're trying to do, why it's the right object, the first concrete example, just enough machinery to do something with it, then pointers to references if you want the proofs in detail.
Pages are short on purpose. One sitting per lecture, a hook at the end pointing you somewhere next, a problem set with hints and full solutions. The roadmap shows the full chapter tree with prerequisite links and your reading progress.
Foundations
Where the rest sits. Sets, algebra, analysis, topology, linear algebra.
Sets and logic
- Sets, functions, and the language of math
- Cardinality, countability, and Cantor's theorem
- The axioms of set theory and Russell's paradox
- The axiom of choice and its equivalents
Algebra
- What is a group?
- Subgroups, cosets, and Lagrange's theorem
- Homomorphisms, normal subgroups, and quotient groups
- Group actions
- What is a ring?
- Ideals and quotient rings
- Prime and maximal ideals
- Factorization domains: UFD, PID, Euclidean
- What is a field?
- Field extensions and minimal polynomials
- Finite fields
- Galois theory — a panorama
- Algebraic closure and transcendence
- Modules — vector spaces over a ring
- Applications of the structure theorem
- Tensor product of modules
- Hom, dual, and exact sequences
- Localization
- Lie groups and Lie algebras
- The classical groups — a menagerie
- Lie brackets, structure constants, and the algebra catalog
- Representations of Lie groups
- Group actions, orbits, and homogeneous spaces
Analysis
- What is a limit, really?
- Continuity
- Derivatives, formally
- Integrals — Riemann and Lebesgue
- Complex analysis — what makes it different
- Conformal maps and Möbius transformations
- Residue calculus — computing real integrals
- Zeros, identity, and the argument principle
- Riemann surfaces and analytic continuation
Topology
- Metric spaces
- Completeness and the Banach fixed-point theorem
- Compactness in metric spaces
- The Baire category theorem
- Topology — opens, closeds, continuity
- Compactness
- Connectedness
- Manifolds — spaces that look locally like Rⁿ
- Vector fields and flows
- Differential forms and de Rham cohomology
- Riemannian geometry — metrics, connections, curvature
- Vector bundles, fiber bundles, and sections
- Submanifolds, embeddings, and the Whitney theorem
Linear algebra
- Linear algebra over a field
- Linear systems, rank, and Gaussian elimination
- Determinants — volume, orientation, and invertibility
- Eigenvalues, diagonalization, and dynamics
- Inner products and orthogonality
- The spectral theorem and SVD
- Multilinear algebra and tensors
Higher structures
Built on the foundations. Category theory, type theory, functional analysis, homotopy.
Category theory
- Category theory — objects, arrows, and what they share
- Functors and natural transformations
- Adjoint functors
- Topoi
Topology and homotopy
- Homotopy
- Covering spaces and the Galois correspondence
- CW complexes and cellular homology
- Fibrations, cofibrations, and the long exact sequence
- Homological algebra
- Derived functors and resolutions
- Group cohomology
- Sheaf cohomology and derived categories
- Homotopy algebra
Type theory
- Type theory
- Homotopy type theory
- Inductive types and induction principles
- Dependent types in practice
- Proof assistants in practice
Functional analysis and beyond
- Functional analysis
- Banach spaces and the four pillars
- Lp spaces and the inequality scale
- The spectral theorem for self-adjoint operators
- Compact operators and the Fredholm alternative
- Sobolev spaces and the Fourier transform
- Fractional calculus
Computation, probability, physics, engineering
Where the math is put to work.
Theory of computation
- What is computation?
- Finite automata and regular languages
- Context-free languages and parsing
- Decidability, reducibility, and Rice's theorem
- Complexity — P, NP, and the hierarchy
- Gödel's incompleteness theorems
- Solomonoff–Kolmogorov–Chaitin complexity
Probability and stochastic processes
- Probability — measure theory, distributions, expectation
- Bayesian probability
- Markov chains and stochastic processes
Physics, SICM-flavor
- Classical mechanics, SICM-style
- Lagrangian mechanics in action
- Symmetries and conservation laws
- Phase space, Liouville, and Hamiltonian flow
- Small oscillations and normal modes
- Rigid bodies, gyroscopes, and Euler's equations
- Functional differential geometry
- FDG worked examples — sphere, hyperbolic, Schwarzschild
- Quantum mechanics, SICM-style
- Interference, amplitudes, and the Born rule
- Two-state systems and the qubit
- Wave mechanics in 1D
- Angular momentum and spin
- Entanglement, Bell, and composite systems
- General relativity, SICM-style
- Special relativity — Minkowski spacetime
- Curvature, parallel transport, and geodesics
- Black holes — Schwarzschild applied
- Cosmology and the early universe
- Gravitational waves and LIGO
Engineering applications
Sideways
Deep connections between main-spiral topics. Off the prereq DAG — nothing later requires them. Read after the prereqs land.