{"query": "Positive Ricci curvature kills harmonic one-forms", "count": 19, "results": [{"id": "card_joint_bochner_vanishing", "title": "Positive Ricci curvature kills harmonic one-forms", "shelf": "codex", "surface": "secular", "snippet": "Bochner (1946): a closed manifold with positive Ricci curvature carries no non-zero harmonic one-form, so its first Betti number is zero - a Hodge-theoretic conclusion from the very hypothesis Hamilto", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_floor_millennium", "title": "The Millennium floor - seven open questions, and where they connect", "shelf": "codex", "surface": "secular", "snippet": "The seven Millennium Prize Problems on the one map of reality. The FLOOR is what is proven or observed: the joints where two of the questions meet at one established thing - the GUE statistics shared ", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_hamilton_1982", "title": "R. S. Hamilton 1982 — Three-manifolds with positive Ricci curvature", "shelf": "millennium", "surface": "secular", "snippet": "R. S. Hamilton (1982). Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982) 255–306. DOI 10.4310/jdg/1214436922. Canonical: https://doi.org/10.4310/jdg/1214436922. Free copy:", "authority_tier": "reference", "source": "R. S. Hamilton (1982), J. Differential Geom. 17 (1982) 255–306", "readable": false, "generated": false}, {"id": "card_src_mill_bochner_1946", "title": "S. Bochner 1946 — Vector fields and Ricci curvature", "shelf": "millennium", "surface": "secular", "snippet": "S. Bochner (1946). Vector fields and Ricci curvature. Bull. Amer. Math. Soc. 52 (1946) 776–797. DOI 10.1090/S0002-9904-1946-08647-4. Canonical: https://doi.org/10.1090/S0002-9904-1946-08647-4. Free co", "authority_tier": "reference", "source": "S. Bochner (1946), Bull. Amer. Math. Soc. 52 (1946) 776–797", "readable": false, "generated": false}, {"id": "card_src_mill_myers_1941", "title": "S. B. Myers 1941 — Riemannian manifolds with positive mean curvature", "shelf": "millennium", "surface": "secular", "snippet": "S. B. Myers (1941). Riemannian manifolds with positive mean curvature. Duke Math. J. 8 (1941) 401–404. DOI 10.1215/S0012-7094-41-00832-3. Canonical: https://doi.org/10.1215/S0012-7094-41-00832-3. No f", "authority_tier": "reference", "source": "S. B. Myers (1941), Duke Math. J. 8 (1941) 401–404", "readable": false, "generated": false}, {"id": "card_chain_eells_sampson_1964", "title": "Eells 1964 — Harmonic mappings of Riemannian manifolds", "shelf": "codex", "surface": "secular", "snippet": "J. Eells, J. H. Sampson (1964). Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964) 109–160. DOI 10.2307/2373037. Canonical: https://doi.org/10.2307/2373037. Cited by its record. Licen", "authority_tier": "reference", "source": "J. Eells, J. H. Sampson (1964), Amer. J. Math. 86 (1964) 109–160", "readable": false, "generated": false}, {"id": "card_src_pron_ricci", "title": "ricci", "shelf": "pronunciation", "surface": "secular", "snippet": "ricci: pronounced (ARPABET) R IY1 CH IY0. From the CMU Pronouncing Dictionary — the standard machine-readable pronunciations of North American English.", "authority_tier": "reference", "source": "CMU Pronouncing Dictionary (cmudict) — BSD-2-Clause, Carnegie Mellon", "readable": false, "generated": false}, {"id": "card_floor_poincare", "title": "The Poincare chain - from the homology sphere and the heat flow to the closed question", "shelf": "codex", "surface": "secular", "snippet": "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 (", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_chain_hamilton_1995", "title": "Hamilton 1995 — The formation of singularities in the Ricci flow", "shelf": "codex", "surface": "secular", "snippet": "R. S. Hamilton (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry 2 (1995) 7–136. DOI 10.4310/SDG.1993.v2.n1.a2. Canonical: https://doi.org/10.4310/SDG.1993.v2.", "authority_tier": "reference", "source": "R. S. Hamilton (1995), Surveys in Differential Geometry 2 (1995) 7–136", "readable": false, "generated": false}, {"id": "card_src_mill_perelman_2003a", "title": "G. Perelman 2003 — Ricci flow with surgery on three-manifolds", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2003). Ricci flow with surgery on three-manifolds. arXiv (2003). arXiv: math/0303109. Canonical: https://arxiv.org/abs/math/0303109. Free copy: https://arxiv.org/abs/math/0303109. License", "authority_tier": "reference", "source": "G. Perelman (2003), arXiv (2003)", "readable": false, "generated": false}, {"id": "card_question_poincare", "title": "The Poincare conjecture", "shelf": "millennium", "surface": "secular", "snippet": "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", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_src_mill_morgan_tian_2007", "title": "J. Morgan 2007 — Ricci Flow and the Poincaré Conjecture", "shelf": "millennium", "surface": "secular", "snippet": "J. Morgan, G. Tian (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Monographs 3, AMS, 2007. arXiv: math/0607607. Canonical: https://bookstore.ams.org/CMIM/3. Free copy: https://arxiv.", "authority_tier": "reference", "source": "J. Morgan, G. Tian (2007), Clay Mathematics Monographs 3, AMS, 2007", "readable": false, "generated": false}, {"id": "card_src_mill_perelman_2002", "title": "G. Perelman 2002 — The entropy formula for the Ricci flow and its geometric applications", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv (2002). arXiv: math/0211159. Canonical: https://arxiv.org/abs/math/0211159. Free copy: https://arxiv.or", "authority_tier": "reference", "source": "G. Perelman (2002), arXiv (2002)", "readable": false, "generated": false}, {"id": "card_src_mill_perelman_2003b", "title": "G. Perelman 2003 — Finite extinction time for the solutions to the Ricci flow on certain three-manifolds", "shelf": "millennium", "surface": "secular", "snippet": "G. Perelman (2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv (2003). arXiv: math/0307245. Canonical: https://arxiv.org/abs/math/0307245. Free copy: ", "authority_tier": "reference", "source": "G. Perelman (2003), arXiv (2003)", "readable": false, "generated": false}, {"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", "shelf": "millennium", "surface": "secular", "snippet": "H.-D. Cao, X.-P. Zhu (2006). A complete proof of the Poincaré and geometrization conjectures — application of the Hamilton-Perelman theory of the Ricci flow. Asian J. Math. 10 (2006) 165–492. DOI 10.4", "authority_tier": "reference", "source": "H.-D. Cao, X.-P. Zhu (2006), Asian J. Math. 10 (2006) 165–492", "readable": false, "generated": false}, {"id": "card_seal_3ebe3d579fb944e9d872c6bc2177416604c941331423210f7370df2221e8cc84", "title": "Receipt 3ebe3d579fb9… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_b39343d13df1d99291d2ce33190e8d2b615a2b2aeac8e9fee875b91fb2c6d967", "title": "Receipt b39343d13df1… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_seal_f1ec0104100e340c16ca54d1b6cbf21b7d922078ca24c1fa60f759be3e38dfe4", "title": "Receipt f1ec0104100e… — sealed: record", "shelf": "seals", "surface": "secular", "snippet": "{\n \"anchors\": [],\n \"axis_coords\": {\n  \"axis\": \"mathematics\",\n  \"dimensions\": [\n   \"reasoning\"\n  ]\n },\n \"gate_results\": [\n  {\n   \"details\": {\n    \"broken_at\": null,\n    \"confirmed_steps\": 1,\n    \"error", "authority_tier": "engine_derived", "source": "Narrow Highway engine — sealed record", "readable": false, "generated": false}, {"id": "card_isbe_baptism_lutheran_doctrine", "title": "ISBE: Baptism (Lutheran Doctrine)", "shelf": "encyclopedia", "surface": "witness", "snippet": "I. THE TERM\n1. The Derivation\n2. The Meaning\n3. The Application\n4. Equivalent Terms\nII. THE ORDINANCE\n1. The Teaching of Scripture\n(1) An Authoritative Command\n(2) A Clear Declaration of the Object in", "authority_tier": "reference", "source": "International Standard Bible Encyclopedia (1915), ed. James Orr — Public Domain (CrossWire SWORD module ISBE v2.2)", "readable": false, "generated": false}], "house": {"door": "FIND", "kind": "cards", "trail": "results", "seal": null, "next_step": {"do": "open the top card", "door": "FIND", "tool": "card_get", "params": {"id": "card_joint_bochner_vanishing"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}