NEXT_STEPS_PLAN.md

Mathematics Database — Next Steps Plan

Three initiatives: Search (near-term), Comprehensive Collection (mid-term), and Research Frontier (long-term).

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1. Search the Collection

Goal: Place a search bar near the top of the table page. Users can search by theorem name, mathematician name, subcategory, or keyword and get links to individual charts or collection pages.

1.1 Search UI Placement

• Add a search box immediately after the header (before or alongside "Start Here")
• Design: Single input, optional filters (All / Algorithms / Axiomatic / Collection)
• Live/filter-as-you-type or "Search" button — both viable

1.2 Search Data Source

• Client-side: Load metadata.json (already fetched for the table); search in memory
• Indexable fields (extend metadata if needed):
  • name (process title) — e.g. "Fermat's Last Theorem", "Sieve of Eratosthenes"
  • subcategory / subcategory_name — e.g. "Number Theory", "Calculus & Analysis"
  • namedCollections (when added) — e.g. "euclid", "fermat", "sullivan"
  • Optional: add keywords or searchTerms array for aliases ("FLT", "Poincaré", "ZFC")

1.3 Search Algorithm

• Simple: Case-insensitive substring match on name, subcategory_name
• Better: Tokenize query, match against name + subcategory + collections
• Fuzzy (optional): Use a small library (e.g. Fuse.js) for typo tolerance

1.4 Results Display

• Single process match → link directly to process page
• Collection match (e.g. "Euclid") → link to collection landing page (or list of processes in that collection)
• Multiple matches → show dropdown or results panel with:
  • Process name + subcategory
  • Link to process page
  • "Part of: Euclid, Geometry & Topology" (when namedCollections exists)

1.5 Metadata Enhancements for Search

• Add namedCollections to processes (per expansion plan)
• Optional: keywords: ["FLT", "Fermat", "Wiles"] for common aliases
• Optional: theorems: ["Modularity Theorem", "Fermat's Last Theorem"] for axiomatic theories

1.6 Implementation Scope

Task	Effort
Add search input + results dropdown	Small
Client-side search over metadata.json	Small
Add namedCollections to metadata (partial)	Medium
Collection landing pages for multi-result	Medium

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2. Plan to Fill Out the Collection (Comprehensive)

Build on MATHEMATICS_DATABASE_EXPANSION_PLAN.md. Aim for a representative, well-structured set across major areas.

2.1 Coverage Goals by Domain

Domain	Current	Target	Priority Additions
Algebra	Strong	Maintain + expand	Cayley-Hamilton, Noether, Representation theory
Analysis	Good	Expand	Complex analysis (4 charts), Functional analysis basics
Geometry & Topology	Good	Expand	Milnor exotic spheres, Thurston geometrization
Number Theory	Good	Expand	Landmark theorems (FLT, Riemann), Fermat's Little Theorem
Discrete & Logic	Strong	Maintain	Add combinatorics algorithms (inclusion-exclusion, generating functions)
Applied	Bioinformatics only	Expand	Statistics/probability, optimization basics

2.2 Landmark Theorems (High Impact)

• Fermat's Last Theorem (Wiles, modularity)
• Poincaré Conjecture (Perelman, Ricci flow)
• Riemann Hypothesis (statement, equivalent forms)
• Four Color Theorem (Appel–Haken, formalization)
• Gödel Incompleteness (already present via Peano)

2.3 Gaps to Fill

• Complex Analysis: Cauchy, residues, conformal maps
• Statistics & Probability: Kolmogorov axioms, Central Limit Theorem, Bayes
• Numerical Methods: More algorithms (Newton, Euler methods, quadrature)
• Representation Theory: Basics (groups, characters)
• Differential Geometry: Curves, surfaces, Riemannian basics

2.4 Phased Rollout (from expansion plan, refined)

Phase	Focus	Charts (approx)
1	Schema + search + namedCollections	0 new charts
2	Landmark theorems (FLT, Poincaré, Riemann)	3–5
3	Complex analysis	4
4	Named mathematicians (batch 1: Fermat, Euler, Gauss, Euclid tag)	5–8
5	Named mathematicians (batch 2: Galois, Noether, Hilbert, Riemann)	5–8
6	Statistics & probability	3–5
7	Formal verification (Lean, Four Color in Coq)	3–4
8	AI mathematics (AlphaProof, AlphaGeometry)	2–3

2.5 Definition of "Fairly Comprehensive"

• All 6 domains have ≥5 distinct charts
• Every subcategory has at least 1 chart
• Landmark theorems (FLT, Poincaré, Riemann) represented
• Major figures (Euclid, Euler, Gauss, Fermat, Gödel, Galois) have at least one chart
• ~150–200 total processes as a stretch goal

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3. Long-Term: Research Frontier & Conjectures

Goal: Update axiomatic theory trees to show recent theorems, open conjectures, and the frontier of research — making the dependency graphs reflect the state of the field, not just classic textbook material.

3.1 What "Frontier" Means

• Recent theorems: Results from the last 20–30 years (e.g. Perelman/geometrization, Taylor–Wiles modularity)
• Conjectures: Stated but unproven (Riemann, Birch–Swinnerton-Dyer, Hodge, P vs NP)
• Formalization status: What is in Mathlib/Lean, what remains to be formalized

3.2 Data Sources for Frontier Content

• arXiv: Recent math.NT, math.GT, math.AG, etc. — identify major theorems
• Mathlib / formalization: Lean 4, Coq, Isabelle — which theorems are proved
• Surveys & encyclopedias: Wikipedia, Encyclopaedia of Mathematics, Scholarpedia
• Clay Institute, Hilbert problems: Lists of major open problems

3.3 Schema Extensions

• Node metadata in dependency graphs:
  • status: proved | conjecture | open_problem | formalized
  • year: publication or proof year
  • prover: e.g. "Wiles", "Perelman", "Gonthier et al."
  • formalization: e.g. { "tool": "Lean", "status": "in_progress" }
• Process-level:
  • frontierLevel: classical | modern | recent | conjecture
  • openProblems: array of conjecture names

3.4 Visualization Ideas

• Color coding: Green (proved), yellow (recent), orange (conjecture), grey (formalized)
• "Expand to frontier" control: Toggle to show/hide conjectures and recent theorems
• Year annotations: Small labels on nodes (e.g. "1995", "2003")
• Separate "Conjectures" section: Page listing open problems with links to related axiom–theorem trees

3.5 Implementation Phases (Long-Term)

Phase	Focus
A	Add status, year to process metadata (manual curation)
B	Extend Mermaid/diagram format to support status annotations
C	Curate 5–10 landmark theorems with frontier metadata
D	Build "Open Problems" index page
E	Integrate formalization status (Mathlib, etc.) where available

3.6 Challenges

• Curation effort: Requires domain expertise to classify and annotate
• Currency: Frontier changes; need update process (annual review?)
• Formalization: Mathlib evolves; linking to specific commits or versions
• Scope creep: Easy to expand; need clear criteria for "frontier"

3.7 Sample Implemented: Number Theory Research Frontier

• Page: number-theory-research-frontier.html — static view of proved vs conjecture
• Metadata: frontierStatus, year, prover added to Sieve, Szemerédi, Green–Tao in metadata.json
• Linked from database table "Start Here" section
• Contents: Classical (Sieve, Extended Euclidean, Gödel), recent (Szemerédi 1975, Green–Tao 2004, Fermat 1995, Mordell 1983), conjectures (Riemann, BSD, Goldbach, Twin Primes)

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Summary: Immediate Next Steps

1. Search (1–2 days): Add search input, client-side search over metadata, results dropdown with links.
2. Expansion plan (ongoing): Execute phases from MATHEMATICS_DATABASE_EXPANSION_PLAN.md; use this doc for prioritization.
3. Frontier (quarterly/yearly): Start with schema additions and manual curation of a few landmark results; build out as capacity allows.