{"id": "card_floor_poincare", "kind": "note", "title": "The Poincare chain - from the homology sphere and the heat flow to the closed question", "body": "Two trees. Topology: Poincaré's homology sphere and the question (1904), Thurston's geometrization (1982). Geometric analysis: the heat-flow method of Eells and Sampson (1964), Hamilton's Ricci flow (1982). They become one in Hamilton's program for the singularities (1995), which Perelman completed (2002-2003); the write-ups of Kleiner-Lott, Morgan-Tian and Cao-Zhu are the receipts. The end is CLOSED.", "source": {"label": "Narrow Highway - a chain on the one map (operator seed)", "url": "", "ref": "docs/MILLENNIUM_PREPAREDNESS.md", "authority_tier": "engine_derived"}, "shelf": "codex", "box": "floor", "bands": ["floor", "chain", "two trees", "poincare", "ricci flow", "geometrization", "three-manifolds"], "subject": "The Poincare chain - from the homology sphere and the heat flow to the closed question", "connections": [{"to_card_id": "card_k_floor_of_discovery", "relationship": "part_of", "evidence": "the poincare chain rests on the one Floor of Discovery"}, {"to_card_id": "card_chain_poincare_1904", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_chain_eells_sampson_1964", "relationship": "has_part", "evidence": "a root of this chain - one of the two trees it began from"}, {"to_card_id": "card_question_poincare", "relationship": "has_open_end", "evidence": "the open question this chain reaches"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "codex.floor", "facets": {"subject": ["chain", "geometrization", "poincare", "ricci flow", "three-manifolds", "two trees"]}, "presentation": {"glyph": "•", "kind_label": "floor", "by": "Narrow Highway - a chain on the one map (operator seed)", "authority": "engine_derived", "posted": "", "standing": ""}, "neighbors": [{"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "part of", "why": "the poincare chain rests on the one Floor of Discovery", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_chain_poincare_1904", "title": "Poincaré 1904 — Cinquième complément à l'analysis situs", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_poincare_1904", "resolved": true}, {"id": "card_chain_eells_sampson_1964", "title": "Eells 1964 — Harmonic mappings of Riemannian manifolds", "relationship": "has part", "why": "a root of this chain - one of the two trees it began from", "href": "/card/card_chain_eells_sampson_1964", "resolved": true}, {"id": "card_question_poincare", "title": "The Poincare conjecture", "relationship": "has open end", "why": "the open question this chain reaches", "href": "/card/card_question_poincare", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_floor_poincare", "next_step": {"do": "follow a connection: what this card rests on, and what rests on it", "door": "FIND", "tool": "card_connections", "params": {"id": "card_floor_poincare"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}