{"id": "card_question_poincare", "kind": "reference", "title": "The Poincare conjecture", "body": "The Poincare conjecture. Every simply connected closed three-manifold is homeomorphic to the three-sphere. Its chart is the tick stick stick_poincare_conjecture: what is sealed, what is cited and what stays open are read live at /stick?id=stick_poincare_conjecture. PROVEN by Grigori Perelman (2002-2003) by Hamilton's Ricci flow with surgery; the prize awarded in 2010 and declined.", "source": {"label": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "url": "https://www.claymath.org/millennium-problems/", "ref": "stick_poincare_conjecture", "domain": "mathematics", "authority_tier": "reference"}, "shelf": "millennium", "box": "question", "bands": ["millennium", "open question", "clay", "poincare", "topology", "three-manifolds", "ricci flow"], "subject": "The Poincare conjecture", "connections": [{"to_card_id": "card_spine_millennium_sources", "relationship": "member_of", "evidence": "one of the seven questions the Millennium shelf is about"}, {"to_card_id": "card_floor_millennium", "relationship": "open_end_of", "evidence": "an open question hanging off the floor of what is proven"}, {"to_card_id": "card_joint_bochner_vanishing", "relationship": "connects_at", "evidence": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)"}, {"to_card_id": "card_joint_arnold_geodesics", "relationship": "connects_at", "evidence": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute"}, {"to_card_id": "card_floor_logarithm", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_src_mill_perelman_2002", "relationship": "connects_at", "evidence": "where the logarithm enters: the entropy formula: Perelman's W functional is a log-Sobolev inequality, the entropy minus the integral of u log u (sealed: the entropy of one fair coin, log 2)"}, {"to_card_id": "card_src_mill_kleiner_lott_2008", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_src_mill_cao_zhu_2006", "relationship": "builds_on", "evidence": "a later work standing on an earlier one"}, {"to_card_id": "card_floor_poincare", "relationship": "open_end_of", "evidence": "the open question this chain reaches"}, {"to_card_id": "card_chart_euler_char", "relationship": "charted_by", "evidence": "the stick seals V - E + F = 2 for the Platonic solids - the topological invariant; chi is the invariant the Poincaré question's sealed instances use (chi(S^3) = 0)"}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "call": "millennium.question", "facets": {"subject": ["clay", "millennium", "open question", "poincare", "ricci flow", "three-manifolds", "topology"]}, "presentation": {"glyph": "?", "kind_label": "A question", "by": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "authority": "reference", "posted": "", "standing": "", "link": {"url": "https://www.claymath.org/millennium-problems/", "host": "claymath.org", "provider": "", "titled": "The Poincare conjecture", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "relationship": "on the shelf of", "why": "one of the seven questions the Millennium shelf is about", "href": "/card/card_spine_millennium_sources", "resolved": true}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "relationship": "open end of", "why": "an open question hanging off the floor of what is proven", "href": "/card/card_floor_millennium", "resolved": true}, {"id": "card_joint_bochner_vanishing", "title": "Positive Ricci curvature kills harmonic one-forms", "relationship": "connects at", "why": "Positive Ricci curvature kills harmonic one-forms. bochner 1946; myers 1941; hamilton 1982. Sealed: myers_diameter_bound_on_the_unit_s3 (Poincare stick)", "href": "/card/card_joint_bochner_vanishing", "resolved": true}, {"id": "card_joint_arnold_geodesics", "title": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms", "relationship": "connects at", "why": "Euler flow is geodesic flow on the volume-preserving diffeomorphisms. arnold 1966. Cited, not sealed: no arithmetic to recompute", "href": "/card/card_joint_arnold_geodesics", "resolved": true}, {"id": "card_floor_logarithm", "title": "The logarithm - the instrument the joints share", "relationship": "open end of", "why": "the open question this chain reaches", "href": "/card/card_floor_logarithm", "resolved": true}, {"id": "card_src_mill_perelman_2002", "title": "G. Perelman 2002 — The entropy formula for the Ricci flow and its geometric applications", "relationship": "connects at", "why": "where the logarithm enters: the entropy formula: Perelman's W functional is a log-Sobolev inequality, the entropy minus the integral of u log u (sealed: the entropy of one fair coin, log 2)", "href": "/card/card_src_mill_perelman_2002", "resolved": true}, {"id": "card_src_mill_kleiner_lott_2008", "title": "B. Kleiner 2008 — Notes on Perelman's papers", "relationship": "builds on", "why": "a later work standing on an earlier one", "href": "/card/card_src_mill_kleiner_lott_2008", "resolved": true}, {"id": "card_src_mill_cao_zhu_2006", "title": "H.-D. Cao 2006 — A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow", "relationship": "builds on", "why": "a later work standing on an earlier one", "href": "/card/card_src_mill_cao_zhu_2006", "resolved": true}], "house": {"door": "FIND", "kind": "card", "trail": "connections", "seal": "/card/card_question_poincare", "next_step": {"do": "read the source itself", "door": "FIND", "web": "/reader.html?card=card_question_poincare"}, "ends": "a verdict or a card · the trail · a seal · one next step"}}