MATHEMATICS_DATABASE_EXPANSION_PLAN.md

Mathematics Database Expansion Plan

Overview

Expand the mathematics-database-table and processes to include:

• Topic sections: Complex analysis, complex analytic dynamics, landmark theorems (FLT, Poincaré, Riemann)
• Named mathematicians: Historical and modern figures with associated charts
• Formal verification: Lean proofs and proof assistants
• AI mathematics: Recent AI-assisted results
• Overlapping collections: Processes appear in multiple named sets (topic + mathematician + historical)

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1. Metadata Schema Extension

Add namedCollections Array to Each Process

{
  "id": "number_theory-fermat-last-theorem",
  "name": "Fermat's Last Theorem",
  "subcategory": "number_theory",
  "namedCollections": ["fermat", "landmark_theorems", "wiles", "number_theory_milestones"]
}

Rationale: A process can belong to many collections. Examples:

• Euclid's Elements → ["euclid", "geometry_topology", "classical_geometry", "axiomatic_systems"]
• Galois Theory → ["galois", "abstract_algebra", "field_theory", "landmark_theorems"]
• Sieve of Eratosthenes → ["eratosthenes", "number_theory", "algorithms", "classical_algorithms"]

Optional: Add collections Index in metadata.json

{
  "collections": {
    "archimedes": { "name": "Archimedes", "description": "…", "processIds": ["…"] },
    "fermat": { "name": "Pierre de Fermat", "description": "…", "processIds": ["…"] }
  }
}

Either derive from processes (scan namedCollections) or maintain explicitly.

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2. New Subcategories

Subcategory ID	Display Name	Notes
complex_analysis	Complex Analysis	New; analytic functions, residues, etc.
landmark_theorems	Landmark Theorems	FLT, Poincaré, Riemann, etc.
formal_verification	Formal Verification	Lean, Coq, Isabelle proofs
ai_mathematics	AI Mathematics	AlphaProof, AlphaGeometry, etc.

Existing (keep): number_theory, geometry_topology, discrete_mathematics, linear_algebra, calculus_analysis, abstract_algebra, category_theory, foundations, bioinformatics.

---

3. Topic Sections (New Charts)

3.1 Complex Analysis

• Complex Analysis — Analytic Functions & Cauchy-Riemann
• Complex Analysis — Cauchy Integral Theorem & Residues
• Complex Analysis — Conformal Mappings & Riemann Surfaces
• Complex Analysis — Entire Functions & Picard Theorems

Collections: complex_analysis, calculus_analysis (overlap)

3.2 Complex Analytic Dynamics (extend existing)

• Already have: Julia/Fatou, Sullivan, Hubbard-Douady, Devaney, etc.
• Add: Complex Dynamics — Holomorphic Dynamics Overview (hub/overview)
• Add: Complex Dynamics — Parabolic Fixed Points & Écalle-Voronin

Collections: complex_dynamics, calculus_analysis, sullivan, hubbard_douady, devaney

3.3 Landmark Theorems

Chart	Subcategory	Named Collections
Fermat's Last Theorem	landmark_theorems	fermat, wiles, number_theory
Poincaré Conjecture	landmark_theorems	poincare, perelman, topology
Riemann Hypothesis	landmark_theorems	riemann, number_theory, analysis
Four Color Theorem	landmark_theorems	appel_haken, graph_theory
Gödel Incompleteness	(existing)	godel, foundations

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4. Named Mathematicians — Charts to Create

4.1 Classical (Ancient & Early Modern)

Mathematician	Charts to Create	Overlaps With
Archimedes	Archimedes' Principle, Method of Exhaustion, Pi bounds	geometry_topology, calculus
Eratosthenes	Sieve (existing), Earth circumference, Prime counting	number_theory, algorithms
Pythagoras	Pythagorean Theorem, Pythagorean triples, Irrationals	geometry_topology, number_theory
Euclid	Elements (existing), Euclidean algorithm	geometry_topology

4.2 Early Modern

Mathematician	Charts to Create	Overlaps With
Fermat	Fermat's Last Theorem, Fermat's Little Theorem, Fermat primes	number_theory, landmark_theorems
Euler	Euler's formula (e^(iπ)+1=0), Euler characteristic, Seven Bridges	calculus_analysis, graph_theory, topology
Gauss	Fundamental Theorem of Algebra, Gaussian integers, Least squares	number_theory, linear_algebra, calculus

4.3 19th–20th Century

Mathematician	Charts to Create	Overlaps With
Galois	Galois Theory (existing), Solvability by radicals	abstract_algebra, field_theory
Cayley	Cayley's theorem (groups), Cayley-Hamilton theorem	abstract_algebra, linear_algebra
Hamilton	Quaternions, Hamiltonian mechanics, Cayley-Hamilton	linear_algebra, physics
Noether	Noether's theorems, Noetherian rings, Abstract algebra	abstract_algebra, physics
Hilbert	Hilbert's problems, Hilbert space, Basis theorem	foundations, linear_algebra, analysis
Riemann	Riemann Hypothesis, Riemann surfaces, Riemann integral	number_theory, calculus_analysis, complex_analysis

4.4 Modern (20th–21st Century)

Mathematician	Charts to Create	Overlaps With
Thurston	Geometrization conjecture, Hyperbolic 3-manifolds	geometry_topology, poincare
Milnor	Exotic spheres, Milnor's theorem, Morse theory	geometry_topology, differential_topology
Faltings	Mordell conjecture, Faltings' theorem (FLT for n>4)	number_theory, fermat, algebraic_geometry
Atiyah	Atiyah-Singer index theorem, K-theory	geometry_topology, analysis
Perelman	Ricci flow, Poincaré proof	landmark_theorems, poincare
Wiles	Modularity theorem, FLT proof	landmark_theorems, fermat

4.5 Additional Candidates (for later)

• Gödel (existing via Peano)
• Turing (computability, halting problem)
• Kolmogorov (probability, complexity)
• Grothendieck (schemes, topos theory)
• Serre (algebraic geometry, number theory)
• Deligne (Weil conjectures)
• Tao (existing: Green-Tao)
• Szemerédi (existing)
• Sullivan (existing)
• Hubbard, Douady, Devaney (existing)

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5. Formal Verification (Lean Proofs)

5.1 New Subcategory: formal_verification

Chart	Description
Lean 4 — Proof Assistant Overview	What Lean is, tactic language, type theory
Mathlib — Library Structure	Mathlib dependency graph, key namespaces
Fermat's Last Theorem in Lean	FLT statement and proof status in Lean
Kepler Conjecture (Flyspeck)	Hales' proof, formalization in HOL Light
Four Color Theorem in Coq	Gonthier's formalization
Odd Order Theorem (Feit-Thompson)	Gonthier et al. formalization

Collections: lean, formal_verification, landmark_theorems (where applicable)

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6. AI Mathematics

6.1 New Subcategory: ai_mathematics

Chart	Description
AlphaProof (DeepMind 2024)	IMO results, statement proving
AlphaGeometry (DeepMind 2024)	Synthetic geometry, IMO-style problems
AI-Assisted Proof Discovery	Overview: GPT, Lean, collaboration
Ramanujan Machine / Conjecture Generation	Automated conjecture generation
Formalization Gaps (AI + Human)	What remains to be formalized

Collections: ai_mathematics, formal_verification (overlap)

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7. Table Structure — Section Headers & Breaks

7.1 Proposed Table Sections (with breaks)

1. Algorithms — Flowcharts (existing)
2. Axiomatic Theories — Dependency Graphs (existing)
3. Landmark Theorems (new section)
4. Complex Analysis & Dynamics (new or merged into Calculus & Analysis)
5. Formal Verification (Lean, Coq, etc.) (new)
6. AI Mathematics (new)

7.2 Named Collections Panel (expand)

Current: Euclid, Tao, Peano, Gödel, Sullivan, Hubbard & Douady, Devaney, Smale, Bioinformatics

Add:

• Archimedes, Eratosthenes, Pythagoras
• Fermat, Euler, Gauss
• Galois, Cayley, Hamilton, Noether, Hilbert
• Riemann, Thurston, Milnor, Faltings, Atiyah
• Wiles, Perelman
• Lean / Formal Verification
• AI Mathematics

Implementation: Either (a) one link per collection → landing page listing all processes in that collection, or (b) first/representative process. Prefer (a) for multi-process collections.

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8. Overlap Handling

8.1 Process in Multiple Collections

Example: Fermat's Last Theorem

• subcategory: landmark_theorems
• namedCollections: ["fermat", "wiles", "number_theory", "landmark_theorems"]

Appears in:

• Landmark Theorems table section
• Fermat collection page
• Wiles collection page
• Number Theory subcategory filter

8.2 Collection Landing Pages

Create processes/collections/ (or similar):

• collections/fermat.html — lists all processes with namedCollections containing fermat
• collections/euler.html
• collections/landmark_theorems.html
• etc.

These can be generated from metadata or static HTML with links derived from metadata.

8.3 Table Filtering (Optional)

Add filter dropdown: "Show by collection: All | Fermat | Euler | Landmark Theorems | …"

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9. Implementation Phases

Phase 1: Schema & Infrastructure

• Add namedCollections to metadata schema
• Add new subcategories to metadata
• Create collection landing page template
• Update table to support new sections and breaks

Phase 2: Landmark Theorems

• Fermat's Last Theorem
• Poincaré Conjecture
• Riemann Hypothesis
• (Optional) Four Color, Gödel as landmark)

Phase 3: Complex Analysis

• 3–4 complex analysis charts
• Ensure overlap with existing complex dynamics

Phase 4: Named Mathematicians (Batch 1)

• Archimedes, Eratosthenes, Pythagoras
• Fermat, Euler, Gauss
• Tag existing processes (Euclid, Sieve, etc.) with namedCollections

Phase 5: Named Mathematicians (Batch 2)

• Galois, Cayley, Hamilton, Noether, Hilbert
• Riemann, Thurston, Milnor, Faltings, Atiyah
• Wiles, Perelman

Phase 6: Formal Verification

• Lean overview
• 2–3 key formalized results (FLT, Four Color, etc.)

Phase 7: AI Mathematics

• AlphaProof, AlphaGeometry
• AI-assisted proof overview

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10. File Naming Conventions

• number_theory-fermat-last-theorem.html
• landmark_theorems-poincare-conjecture.html
• landmark_theorems-riemann-hypothesis.html
• complex_analysis-cauchy-integral-theorem.html
• formal_verification-lean-flt.html
• ai_mathematics-alphaproof.html
• collections/fermat.html (collection index)

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11. Summary: New Content Counts (Estimate)

Category	New Charts (approx)
Complex Analysis	4
Landmark Theorems	3–5
Named Mathematicians	15–25 (many overlap)
Formal Verification	4–6
AI Mathematics	3–5
Total new	~30–45

Many of these overlap (e.g., Fermat chart counts for Fermat, Wiles, Landmark Theorems, Number Theory). The namedCollections array is the key to supporting this overlap cleanly.