garywelz/programming_framework

1# Mathematics Database Expansion Plan2 3## Overview4 5Expand the mathematics-database-table and processes to include:6- **Topic sections**: Complex analysis, complex analytic dynamics, landmark theorems (FLT, Poincaré, Riemann)7- **Named mathematicians**: Historical and modern figures with associated charts8- **Formal verification**: Lean proofs and proof assistants9- **AI mathematics**: Recent AI-assisted results10- **Overlapping collections**: Processes appear in multiple named sets (topic + mathematician + historical)11 12---13 14## 1. Metadata Schema Extension15 16### Add `namedCollections` Array to Each Process17 18```json19{20  "id": "number_theory-fermat-last-theorem",21  "name": "Fermat's Last Theorem",22  "subcategory": "number_theory",23  "namedCollections": ["fermat", "landmark_theorems", "wiles", "number_theory_milestones"]24}25```26 27**Rationale**: A process can belong to many collections. Examples:28- *Euclid's Elements* → `["euclid", "geometry_topology", "classical_geometry", "axiomatic_systems"]`29- *Galois Theory* → `["galois", "abstract_algebra", "field_theory", "landmark_theorems"]`30- *Sieve of Eratosthenes* → `["eratosthenes", "number_theory", "algorithms", "classical_algorithms"]`31 32### Optional: Add `collections` Index in metadata.json33 34```json35{36  "collections": {37    "archimedes": { "name": "Archimedes", "description": "…", "processIds": ["…"] },38    "fermat": { "name": "Pierre de Fermat", "description": "…", "processIds": ["…"] }39  }40}41```42 43Either derive from processes (scan `namedCollections`) or maintain explicitly.44 45---46 47## 2. New Subcategories48 49| Subcategory ID        | Display Name           | Notes                                      |50|-----------------------|------------------------|--------------------------------------------|51| `complex_analysis`    | Complex Analysis       | New; analytic functions, residues, etc.    |52| `landmark_theorems`   | Landmark Theorems      | FLT, Poincaré, Riemann, etc.              |53| `formal_verification` | Formal Verification    | Lean, Coq, Isabelle proofs                 |54| `ai_mathematics`      | AI Mathematics         | AlphaProof, AlphaGeometry, etc.           |55 56**Existing** (keep): `number_theory`, `geometry_topology`, `discrete_mathematics`, `linear_algebra`, `calculus_analysis`, `abstract_algebra`, `category_theory`, `foundations`, `bioinformatics`.57 58---59 60## 3. Topic Sections (New Charts)61 62### 3.1 Complex Analysis63- **Complex Analysis — Analytic Functions & Cauchy-Riemann**64- **Complex Analysis — Cauchy Integral Theorem & Residues**65- **Complex Analysis — Conformal Mappings & Riemann Surfaces**66- **Complex Analysis — Entire Functions & Picard Theorems**67 68*Collections*: `complex_analysis`, `calculus_analysis` (overlap)69 70### 3.2 Complex Analytic Dynamics (extend existing)71- Already have: Julia/Fatou, Sullivan, Hubbard-Douady, Devaney, etc.72- Add: **Complex Dynamics — Holomorphic Dynamics Overview** (hub/overview)73- Add: **Complex Dynamics — Parabolic Fixed Points & Écalle-Voronin**74 75*Collections*: `complex_dynamics`, `calculus_analysis`, `sullivan`, `hubbard_douady`, `devaney`76 77### 3.3 Landmark Theorems78| Chart                         | Subcategory        | Named Collections                    |79|------------------------------|--------------------|-------------------------------------|80| Fermat's Last Theorem        | `landmark_theorems`| `fermat`, `wiles`, `number_theory`  |81| Poincaré Conjecture          | `landmark_theorems`| `poincare`, `perelman`, `topology`   |82| Riemann Hypothesis           | `landmark_theorems`| `riemann`, `number_theory`, `analysis` |83| Four Color Theorem           | `landmark_theorems`| `appel_haken`, `graph_theory`        |84| Gödel Incompleteness         | (existing)         | `godel`, `foundations`              |85 86---87 88## 4. Named Mathematicians — Charts to Create89 90### 4.1 Classical (Ancient & Early Modern)91| Mathematician   | Charts to Create                                      | Overlaps With              |92|----------------|--------------------------------------------------------|----------------------------|93| **Archimedes** | Archimedes' Principle, Method of Exhaustion, Pi bounds | `geometry_topology`, `calculus` |94| **Eratosthenes** | Sieve (existing), Earth circumference, Prime counting | `number_theory`, `algorithms` |95| **Pythagoras** | Pythagorean Theorem, Pythagorean triples, Irrationals | `geometry_topology`, `number_theory` |96| **Euclid**    | Elements (existing), Euclidean algorithm              | `geometry_topology`        |97 98### 4.2 Early Modern99| Mathematician | Charts to Create                                      | Overlaps With        |100|---------------|--------------------------------------------------------|----------------------|101| **Fermat**    | Fermat's Last Theorem, Fermat's Little Theorem, Fermat primes | `number_theory`, `landmark_theorems` |102| **Euler**    | Euler's formula (e^(iπ)+1=0), Euler characteristic, Seven Bridges | `calculus_analysis`, `graph_theory`, `topology` |103| **Gauss**    | Fundamental Theorem of Algebra, Gaussian integers, Least squares | `number_theory`, `linear_algebra`, `calculus` |104 105### 4.3 19th–20th Century106| Mathematician      | Charts to Create                                      | Overlaps With        |107|-------------------|--------------------------------------------------------|----------------------|108| **Galois**        | Galois Theory (existing), Solvability by radicals      | `abstract_algebra`, `field_theory` |109| **Cayley**        | Cayley's theorem (groups), Cayley-Hamilton theorem     | `abstract_algebra`, `linear_algebra` |110| **Hamilton**     | Quaternions, Hamiltonian mechanics, Cayley-Hamilton     | `linear_algebra`, `physics` |111| **Noether**      | Noether's theorems, Noetherian rings, Abstract algebra | `abstract_algebra`, `physics` |112| **Hilbert**      | Hilbert's problems, Hilbert space, Basis theorem       | `foundations`, `linear_algebra`, `analysis` |113| **Riemann**      | Riemann Hypothesis, Riemann surfaces, Riemann integral | `number_theory`, `calculus_analysis`, `complex_analysis` |114 115### 4.4 Modern (20th–21st Century)116| Mathematician | Charts to Create                                      | Overlaps With        |117|--------------|--------------------------------------------------------|----------------------|118| **Thurston** | Geometrization conjecture, Hyperbolic 3-manifolds       | `geometry_topology`, `poincare` |119| **Milnor**   | Exotic spheres, Milnor's theorem, Morse theory         | `geometry_topology`, `differential_topology` |120| **Faltings** | Mordell conjecture, Faltings' theorem (FLT for n>4)    | `number_theory`, `fermat`, `algebraic_geometry` |121| **Atiyah**   | Atiyah-Singer index theorem, K-theory                 | `geometry_topology`, `analysis` |122| **Perelman** | Ricci flow, Poincaré proof                             | `landmark_theorems`, `poincare` |123| **Wiles**    | Modularity theorem, FLT proof                        | `landmark_theorems`, `fermat` |124 125### 4.5 Additional Candidates (for later)126- **Gödel** (existing via Peano)127- **Turing** (computability, halting problem)128- **Kolmogorov** (probability, complexity)129- **Grothendieck** (schemes, topos theory)130- **Serre** (algebraic geometry, number theory)131- **Deligne** (Weil conjectures)132- **Tao** (existing: Green-Tao)133- **Szemerédi** (existing)134- **Sullivan** (existing)135- **Hubbard, Douady, Devaney** (existing)136 137---138 139## 5. Formal Verification (Lean Proofs)140 141### 5.1 New Subcategory: `formal_verification`142 143| Chart                              | Description                                      |144|-----------------------------------|--------------------------------------------------|145| Lean 4 — Proof Assistant Overview | What Lean is, tactic language, type theory       |146| Mathlib — Library Structure       | Mathlib dependency graph, key namespaces         |147| Fermat's Last Theorem in Lean    | FLT statement and proof status in Lean          |148| Kepler Conjecture (Flyspeck)      | Hales' proof, formalization in HOL Light         |149| Four Color Theorem in Coq        | Gonthier's formalization                         |150| Odd Order Theorem (Feit-Thompson)| Gonthier et al. formalization                   |151 152*Collections*: `lean`, `formal_verification`, `landmark_theorems` (where applicable)153 154---155 156## 6. AI Mathematics157 158### 6.1 New Subcategory: `ai_mathematics`159 160| Chart                                  | Description                                      |161|----------------------------------------|--------------------------------------------------|162| AlphaProof (DeepMind 2024)             | IMO results, statement proving                  |163| AlphaGeometry (DeepMind 2024)          | Synthetic geometry, IMO-style problems          |164| AI-Assisted Proof Discovery            | Overview: GPT, Lean, collaboration               |165| Ramanujan Machine / Conjecture Generation | Automated conjecture generation              |166| Formalization Gaps (AI + Human)        | What remains to be formalized                    |167 168*Collections*: `ai_mathematics`, `formal_verification` (overlap)169 170---171 172## 7. Table Structure — Section Headers & Breaks173 174### 7.1 Proposed Table Sections (with breaks)175 1761. **Algorithms — Flowcharts** (existing)1772. **Axiomatic Theories — Dependency Graphs** (existing)1783. **Landmark Theorems** (new section)1794. **Complex Analysis & Dynamics** (new or merged into Calculus & Analysis)1805. **Formal Verification (Lean, Coq, etc.)** (new)1816. **AI Mathematics** (new)182 183### 7.2 Named Collections Panel (expand)184 185Current: Euclid, Tao, Peano, Gödel, Sullivan, Hubbard & Douady, Devaney, Smale, Bioinformatics186 187**Add**:188- Archimedes, Eratosthenes, Pythagoras189- Fermat, Euler, Gauss190- Galois, Cayley, Hamilton, Noether, Hilbert191- Riemann, Thurston, Milnor, Faltings, Atiyah192- Wiles, Perelman193- Lean / Formal Verification194- AI Mathematics195 196**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.197 198---199 200## 8. Overlap Handling201 202### 8.1 Process in Multiple Collections203 204Example: **Fermat's Last Theorem**205- `subcategory`: `landmark_theorems`206- `namedCollections`: `["fermat", "wiles", "number_theory", "landmark_theorems"]`207 208Appears in:209- Landmark Theorems table section210- Fermat collection page211- Wiles collection page212- Number Theory subcategory filter213 214### 8.2 Collection Landing Pages215 216Create `processes/collections/` (or similar):217- `collections/fermat.html` — lists all processes with `namedCollections` containing `fermat`218- `collections/euler.html`219- `collections/landmark_theorems.html`220- etc.221 222These can be generated from metadata or static HTML with links derived from metadata.223 224### 8.3 Table Filtering (Optional)225 226Add filter dropdown: "Show by collection: All | Fermat | Euler | Landmark Theorems | …"227 228---229 230## 9. Implementation Phases231 232### Phase 1: Schema & Infrastructure233- Add `namedCollections` to metadata schema234- Add new subcategories to metadata235- Create collection landing page template236- Update table to support new sections and breaks237 238### Phase 2: Landmark Theorems239- Fermat's Last Theorem240- Poincaré Conjecture241- Riemann Hypothesis242- (Optional) Four Color, Gödel as landmark)243 244### Phase 3: Complex Analysis245- 3–4 complex analysis charts246- Ensure overlap with existing complex dynamics247 248### Phase 4: Named Mathematicians (Batch 1)249- Archimedes, Eratosthenes, Pythagoras250- Fermat, Euler, Gauss251- Tag existing processes (Euclid, Sieve, etc.) with `namedCollections`252 253### Phase 5: Named Mathematicians (Batch 2)254- Galois, Cayley, Hamilton, Noether, Hilbert255- Riemann, Thurston, Milnor, Faltings, Atiyah256- Wiles, Perelman257 258### Phase 6: Formal Verification259- Lean overview260- 2–3 key formalized results (FLT, Four Color, etc.)261 262### Phase 7: AI Mathematics263- AlphaProof, AlphaGeometry264- AI-assisted proof overview265 266---267 268## 10. File Naming Conventions269 270- `number_theory-fermat-last-theorem.html`271- `landmark_theorems-poincare-conjecture.html`272- `landmark_theorems-riemann-hypothesis.html`273- `complex_analysis-cauchy-integral-theorem.html`274- `formal_verification-lean-flt.html`275- `ai_mathematics-alphaproof.html`276- `collections/fermat.html` (collection index)277 278---279 280## 11. Summary: New Content Counts (Estimate)281 282| Category              | New Charts (approx) |283|-----------------------|---------------------|284| Complex Analysis      | 4                   |285| Landmark Theorems     | 3–5                 |286| Named Mathematicians  | 15–25 (many overlap)|287| Formal Verification   | 4–6                 |288| AI Mathematics        | 3–5                 |289| **Total new**         | **~30–45**          |290 291Many 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.292