Mathematics

Reference notes on the mathematical structures used throughout the physics sections. These pages are companion material — physics docs link in for definitions and structure theorems, and the math pages stay self-contained and general.

Contents

1. Group Theory — a comprehensive folder from the group axioms through the structure theory of finite groups (Lagrange, Sylow, composition series, simple groups), representation and character theory, and Lie groups and Lie algebras (the exponential map, the classical matrix groups, root systems, highest-weight representations); the foundation behind the Lorentz, Poincaré, and internal symmetry groups ($U(1)$, $SU(2)$, $SU(3)$) used in QFT/preliminaries.md.
2. Clifford Algebras — the abstract definition and structure theory of $\mathrm{Cl}(V,Q)$, the gamma-matrix and Dirac-spinor algebra underlying relativistic-fermion fields; supplies the algebra used by QED/historical.md.
3. Topology, Manifolds, and Smooth Structure — point-set topology, smooth manifolds, tangent spaces, and smooth maps; the prerequisites that physics texts typically take for granted.
4. Calculus and Analysis — a self-contained real-analysis spine from the completeness of $\mathbb{R}$ to the generalized Stokes theorem ${\int }_{M}d\omega ={\int }_{\partial M}\omega$, recovering Green's, Stokes', and the divergence theorem as special cases.
5. Complex Analysis — a self-contained complex-analysis reference: the complex plane and its topology, holomorphic functions and the Cauchy–Riemann equations, Cauchy's theorem and the residue calculus, conformal mapping and potential theory, the Schwarz lemma and normal families (with the proof of the Riemann mapping theorem), analytic continuation, Riemann surfaces and uniformization, the factorization theorems (Weierstrass, Hadamard, Mittag-Leffler), the Gamma and Riemann zeta functions and the prime number theorem, the Picard theorems, elliptic and modular functions, and the saddle-point method; one strand supplies the technique behind QFT loop integrals and the $i\epsilon$ prescription, Wick rotation, and the analytic S-matrix.
6. Functional Analysis, Operators & Distributions — infinite-dimensional linear algebra: Banach and Hilbert spaces, bounded and unbounded self-adjoint operators, the spectral theorem and Stone's theorem, distributions and the Schwartz space, rigged Hilbert space, and C*-algebras; the rigorous foundation of the quantum postulates — states as Hilbert-space rays, observables as self-adjoint operators, and quantum fields as operator-valued tempered distributions.
7. Non-Standard Analysis — Robinson's rigorous infinitesimals: the hyperreal field ${}^{\ast }\mathbb{R}$, the transfer principle, and the reconstruction of calculus (derivative as a standard part, integral as a hyperfinite sum) that makes the physicist's infinitesimal $dx$ literally true.
8. Euclidean and Non-Euclidean Geometry — from Euclid's postulates and the two-thousand-year struggle with the parallel postulate, through the discovery of consistent hyperbolic and elliptic geometries, to the Riemannian synthesis that unifies all three as spaces of constant curvature $K=0,<0,>0$ classified by their isometry groups; the hyperbolic group $O(n,1)$ is the Lorentz group of special relativity.
9. Differential Geometry & Fiber Bundles — the global geometry of bundles over a manifold: fiber and principal bundles, connections and curvature (a gauge field is a connection, its field strength is curvature), holonomy and Wilson loops, characteristic classes and the Atiyah–Singer index theorem, and the homotopy classification of solitons, monopoles, and instantons; the mathematical backbone of QFT gauge theory, anomalies, and general relativity.
10. Computational Complexity Theory — models of computation, time/space complexity classes ($\mathsf{P}$, $\mathsf{NP}$, $\mathsf{PSPACE}$, …), reductions, NP-completeness, the polynomial hierarchy, randomized and circuit classes, and the hierarchy theorems.
11. Remarks — cross-cutting foundational fine print, including why the physicist's calculus (infinitesimal $dx$, differentials as fractions) is legitimate despite never using the $\sup /\inf$ definition of the integral.

Reading order

The physics-facing dependency arrows run

$$\text{topology-manifolds}⟶\text{group-theory}⟶\text{clifford-algebra},$$

so newcomers should read in that order. The Calculus and Analysis spine is largely independent: its point-set prerequisites come from topology-manifolds.md §1, and in turn it supplies the multivariable-calculus and integration theory that the manifold material (tangent spaces, smooth maps) presupposes — read it alongside topology-manifolds. Complex Analysis runs parallel to the real analysis spine (it assumes the latter's limits, series, and integration) and is largely self-contained; read it whenever the residue calculus, analytic continuation, or the saddle-point method is needed — it is the most broadly used of the math folders across the physics tree. Functional Analysis extends the analysis spine into infinite-dimensional operator theory and distribution theory; it is the rigorous foundation of the QM and QFT postulates (Hilbert spaces, self-adjoint operators, the spectral theorem, and operator-valued distributions), and the QM spectral-theorem material links into it rather than duplicating it. Euclidean and Non-Euclidean Geometry is self-contained through its synthetic and model-theoretic first half; its Riemannian second half (metric tensor, curvature, constant-curvature classification) reuses topology-manifolds.md §2, analysis/multivariable.md, and group-theory, and feeds directly into special relativity (the hyperbolic isometry group is the Lorentz group). Differential Geometry & Fiber Bundles sits downstream of topology-manifolds.md, analysis/differential-forms.md, and group-theory (its structure groups); it is the bundle-theoretic complement to the metric geometry folder and the direct prerequisite for the gauge-theory and topological parts of QFT. Non-Standard Analysis sits downstream of both the analysis spine and Model Theory: read it after the analysis basics (it re-derives them with infinitesimals) and after meeting ultraproducts. Computational Complexity Theory is independent of the rest and can be read on its own. Readers who only need a specific definition can jump in anywhere — each page is cross-linked to its prerequisites.