ARXIV_MATH_AREAS_TODO.md

Mathematics Database — arXiv Subject Areas To-Do

A prioritized list of arXiv mathematics subject areas to add for a more complete collection, aligned with arXiv math taxonomy.

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Current Coverage (What We Have)

Domain	Subcategories	arXiv codes covered	Gaps
Algebra	abstract_algebra, linear_algebra, category_theory	math.GR, math.RA, math.CT, math.AC, math.AG, math.QA	Commutative algebra, Algebraic geometry, Representation theory, Quantum algebra
Analysis	calculus_analysis	math.CA, math.CV, math.DS, math.FA, math.AP, math.NA, math.SP	Complex analysis, Functional analysis, PDEs, Numerical analysis, Spectral theory
Geometry & Topology	geometry_topology	math.GT, math.AT, math.DG, math.GN, math.MG, math.SG	Metric geometry, Symplectic geometry (light)
Number Theory	number_theory	math.NT	✓ Good
Discrete & Logic	discrete_mathematics, foundations	math.CO, math.LO	✓ Good
Applied & Other	bioinformatics, statistics_probability	math.GM, math.ST	Statistics/Probability empty

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To-Do List: Subject Areas to Add (Near Term)

Priority 1 — High Impact, Partially Covered or Empty

#	arXiv Code	Subject Area	Notes	Suggested Subcategory
1	math.ST	Statistics & Probability Theory	0 charts currently; foundational for applied math	statistics_probability (exists, populate)
2	math.PR	Probability	CLT, stochastic processes, SDEs; distinct from statistics	merge into statistics_probability or add probability
3	math.CV	Complex Variables	Holomorphic functions, residues, conformal maps; partially in calculus_analysis	add complex_analysis or extend calculus_analysis
4	math.FA	Functional Analysis	Banach spaces, Hilbert spaces, distributions	add to calculus_analysis or new functional_analysis
5	math.NA	Numerical Analysis	Newton-Raphson, bisection exist; add quadrature, linear solvers, ODE solvers	extend calculus_analysis or add numerical_analysis
6	math.AG	Algebraic Geometry	Varieties, schemes, moduli; major area	add algebraic_geometry or extend abstract_algebra
7	math.RT	Representation Theory	Representations of groups, Lie algebras	add representation_theory or extend abstract_algebra

Priority 2 — Core Pure Math Gaps

#	arXiv Code	Subject Area	Notes	Suggested Subcategory
8	math.AC	Commutative Algebra	Rings, ideals, Noetherian; differs from Ring Theory (noncommutative focus)	add commutative_algebra
9	math.AP	Analysis of PDEs	Existence, uniqueness, qualitative dynamics	add partial_differential_equations or extend analysis
10	math.DG	Differential Geometry	Curves, surfaces, Riemannian; some in geometry_topology	ensure distinct charts for differential geometry
11	math.SP	Spectral Theory	Schrödinger operators, spectral analysis	add to analysis or spectral_theory
12	math.SG	Symplectic Geometry	Hamiltonian systems, symplectic manifolds	extend geometry_topology
13	math.MG	Metric Geometry	Euclidean, hyperbolic, discrete geometry	extend geometry_topology

Priority 3 — Advanced / Specialized

#	arXiv Code	Subject Area	Notes	Suggested Subcategory
14	math.OA	Operator Algebras	C*-algebras, von Neumann algebras	add operator_algebras
15	math.KT	K-Theory and Homology	Algebraic/topological K-theory	add k_theory or extend algebraic topology
16	math.QA	Quantum Algebra	Quantum groups, operads	extend abstract_algebra
17	math.OC	Optimization and Control	Linear programming, optimal control	add optimization
18	math.IT	Information Theory	Coding, entropy, channel capacity	add information_theory
19	math.MP	Mathematical Physics	Rigorous formulations of physical theories	add mathematical_physics
20	math.HO	History and Overview	Biographies, education, philosophy	optional history_overview

Priority 4 — Already in Expansion Plan

These are in MATHEMATICS_DATABASE_EXPANSION_PLAN.md:

• Complex Analysis (math.CV) — 4 charts planned
• Landmark Theorems — FLT, Poincaré, Riemann
• Formal Verification — Lean, Coq
• AI Mathematics — AlphaProof, AlphaGeometry

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Suggested Implementation Order

Phase A (1–2 weeks): Fill Empty & High-Impact

1. Statistics & Probability — Kolmogorov axioms, Bayes, CLT (3–5 charts)
2. Complex Analysis — Cauchy, residues, conformal maps (4 charts per expansion plan)
3. Functional Analysis — Banach/Hilbert spaces basics (2–3 charts)

Phase B (2–4 weeks): Algebra & Geometry Gaps

4. Algebraic Geometry — Varieties, schemes intro (2–3 charts)
5. Representation Theory — Group representations, characters (2–3 charts)
6. Numerical Analysis — Quadrature, solvers, ODE methods (3–4 charts)

Phase C (4–6 weeks): PDEs, Operator Theory, Applied

7. PDEs — Heat, wave, Laplace; existence/uniqueness (2–3 charts)
8. Operator Algebras — C*-algebras intro (1–2 charts)
9. Optimization — Linear programming, simplex (2 charts)
10. Mathematical Physics — Lagrangian/Hamiltonian mechanics (2 charts)

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Metadata Updates Required

When adding new subcategories:

1. Add to metadata.json → subcategoryCounts
2. Add to metadata.json → subcategoryToArxiv
3. Add to metadata.json → domainHierarchy (assign to algebra, analysis, geometry_topology, or applied)
4. Run build-graph-data.js to update Whole of Mathematics
5. Update upload script if new process directories are created

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Summary: arXiv Math Codes Not Yet Represented

Code	Area	Priority
math.ST	Statistics Theory	1
math.PR	Probability	1
math.CV	Complex Variables	1
math.FA	Functional Analysis	1
math.NA	Numerical Analysis	1
math.AG	Algebraic Geometry	1
math.RT	Representation Theory	1
math.AC	Commutative Algebra	2
math.AP	Analysis of PDEs	2
math.SP	Spectral Theory	2
math.OA	Operator Algebras	3
math.KT	K-Theory	3
math.QA	Quantum Algebra	3
math.OC	Optimization & Control	3
math.IT	Information Theory	3
math.MP	Mathematical Physics	3
math.HO	History & Overview	4

Well covered: math.NT, math.CO, math.LO, math.GR, math.RA, math.CT, math.CA, math.GT, math.AT, math.DS (via complex dynamics)