{"query": "The Hodge conjecture", "count": 20, "results": [{"id": "card_question_hodge", "title": "The Hodge conjecture", "shelf": "millennium", "surface": "secular", "snippet": "The Hodge conjecture. On a projective non-singular algebraic variety over the complex numbers, every Hodge class is a rational linear combination of classes of algebraic cycles. Its chart is the tick ", "authority_tier": "reference", "source": "Clay Mathematics Institute, the Millennium Prize Problems (2000)", "readable": false, "generated": false}, {"id": "card_joint_weil_conjectures", "title": "The Weil conjectures (Deligne 1974): the Riemann hypothesis over finite fields, a theorem", "shelf": "codex", "surface": "secular", "snippet": "The Weil conjectures are the analogue of the Riemann hypothesis for the zeta functions of varieties over FINITE FIELDS, and they are a theorem (Deligne 1974): every eigenvalue of Frobenius on the i-th", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_deligne_1974", "title": "P. Deligne 1974 — La conjecture de Weil. I", "shelf": "millennium", "surface": "secular", "snippet": "P. Deligne (1974). La conjecture de Weil. I. Publ. Math. IHÉS 43 (1974) 273–307. DOI 10.1007/BF02684373. Canonical: https://doi.org/10.1007/BF02684373. Free copy: http://www.numdam.org/item/PMIHES_197", "authority_tier": "reference", "source": "P. Deligne (1974), Publ. Math. IHÉS 43 (1974) 273–307", "readable": false, "generated": false}, {"id": "card_src_mill_deligne_2000", "title": "P. Deligne 2000 — The Hodge conjecture", "shelf": "millennium", "surface": "secular", "snippet": "P. Deligne (2000). The Hodge conjecture. Clay Mathematics Institute, the official problem description. Canonical: https://www.claymath.org/millennium/hodge-conjecture/. Free copy: https://www.claymath", "authority_tier": "reference", "source": "P. Deligne (2000), Clay Mathematics Institute, the official problem description", "readable": false, "generated": false}, {"id": "card_floor_hodge", "title": "The Hodge chain - from periods and topology to the classes that are not cycles", "shelf": "codex", "surface": "secular", "snippet": "Two trees. The transcendental: Riemann's periods (1857). The topological: Lefschetz's analysis situs of a variety and the (1,1) theorem (1924), de Rham's forms (1931). They become one in Hodge's harmo", "authority_tier": "engine_derived", "source": "Narrow Highway - a chain on the one map (operator seed)", "readable": false, "generated": false}, {"id": "card_joint_tate_conjecture", "title": "The Tate conjecture: Hodge's arithmetic twin, and BSD over function fields", "shelf": "codex", "surface": "secular", "snippet": "Tate's conjecture (1965) is the Hodge conjecture with Galois representations in place of Hodge structures. For an elliptic surface over a finite field, the Tate conjecture is equivalent to the Birch a", "authority_tier": "engine_derived", "source": "Narrow Highway - the Millennium floor, a joint found in the literature (operator seed)", "readable": false, "generated": false}, {"id": "card_src_mill_tate_1965", "title": "J. Tate 1965 — Algebraic cycles and poles of zeta functions", "shelf": "millennium", "surface": "secular", "snippet": "J. Tate (1965). Algebraic cycles and poles of zeta functions. Arithmetical Algebraic Geometry (Purdue 1963), Harper & Row, 1965, 93–110. Canonical: https://www.claymath.org/millennium/hodge-conjecture", "authority_tier": "reference", "source": "J. Tate (1965), Arithmetical Algebraic Geometry (Purdue 1963), Harper & Row, 1965, 93–110", "readable": false, "generated": false}, {"id": "card_chain_deligne_1971", "title": "Deligne 1971 — Théorie de Hodge, II", "shelf": "codex", "surface": "secular", "snippet": "P. Deligne (1971). Théorie de Hodge, II. Publ. Math. IHÉS 40 (1971) 5–57. DOI 10.1007/BF02684692. Canonical: https://doi.org/10.1007/BF02684692. Free copy: http://www.numdam.org/item/PMIHES_1971__40__", "authority_tier": "reference", "source": "P. Deligne (1971), Publ. Math. IHÉS 40 (1971) 5–57", "readable": false, "generated": false}, {"id": "card_spine_millennium_sources", "title": "The Millennium sources — the papers behind the seven sticks", "shelf": "spine", "surface": "secular", "snippet": "63 sources located for the seven Millennium sticks (Riemann, Birch and Swinnerton-Dyer, Navier-Stokes, Yang-Mills, P versus NP, Hodge, Poincare), one reference card each: the bibliographic record, DOI", "authority_tier": "reference", "source": "The Millennium sources — a spine of located records", "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_hodge_1950", "title": "W. V. D. Hodge 1950 — The topological invariants of algebraic varieties", "shelf": "millennium", "surface": "secular", "snippet": "W. V. D. Hodge (1950). The topological invariants of algebraic varieties. Proc. International Congress of Mathematicians 1950 (Cambridge, Mass.), vol. 1, 182–192. Canonical: https://www.mathunion.org/", "authority_tier": "reference", "source": "W. V. D. Hodge (1950), Proc. International Congress of Mathematicians 1950 (Cambridge, Mass.), vol. 1, 182–192", "readable": false, "generated": false}, {"id": "card_src_mill_voisin_2002", "title": "C. Voisin 2002 — A counterexample to the Hodge conjecture extended to Kähler varieties", "shelf": "millennium", "surface": "secular", "snippet": "C. Voisin (2002). A counterexample to the Hodge conjecture extended to Kähler varieties. Int. Math. Res. Not. 2002, no. 20, 1057–1075. DOI 10.1155/S1073792802111135. arXiv: math/0112247. Canonical: ht", "authority_tier": "reference", "source": "C. Voisin (2002), Int. Math. Res. Not. 2002, no. 20, 1057–1075", "readable": false, "generated": false}, {"id": "card_src_mill_lewis_1999", "title": "J. D. Lewis (appendix by B. B. Gordon) 1999 — A Survey of the Hodge Conjecture (2nd ed.)", "shelf": "millennium", "surface": "secular", "snippet": "J. D. Lewis (appendix by B. B. Gordon) (1999). A Survey of the Hodge Conjecture (2nd ed.). CRM Monograph Series 10, AMS, 1999; ISBN 0-8218-0568-1. Canonical: https://bookstore.ams.org/CRMM/10. No free", "authority_tier": "reference", "source": "J. D. Lewis (appendix by B. B. Gordon) (1999), CRM Monograph Series 10, AMS, 1999; ISBN 0-8218-0568-1", "readable": false, "generated": false}, {"id": "card_src_mill_hirzebruch_1966", "title": "F. Hirzebruch 1966 — Topological Methods in Algebraic Geometry (3rd ed.)", "shelf": "millennium", "surface": "secular", "snippet": "F. Hirzebruch (1966). Topological Methods in Algebraic Geometry (3rd ed.). Springer, 1966 (Grundlehren 131). DOI 10.1007/978-3-642-62018-8. Canonical: https://doi.org/10.1007/978-3-642-62018-8. No fre", "authority_tier": "reference", "source": "F. Hirzebruch (1966), Springer, 1966 (Grundlehren 131)", "readable": false, "generated": false}, {"id": "card_src_mill_atiyah_hirzebruch_1962", "title": "M. F. Atiyah 1962 — Analytic cycles on complex manifolds", "shelf": "millennium", "surface": "secular", "snippet": "M. F. Atiyah, F. Hirzebruch (1962). Analytic cycles on complex manifolds. Topology 1 (1962) 25–45. DOI 10.1016/0040-9383(62)90094-0. Canonical: https://doi.org/10.1016/0040-9383(62)90094-0. Free copy:", "authority_tier": "reference", "source": "M. F. Atiyah, F. Hirzebruch (1962), Topology 1 (1962) 25–45", "readable": false, "generated": false}, {"id": "card_chain_hodge_1941", "title": "Hodge 1941 — The Theory and Applications of Harmonic Integrals", "shelf": "codex", "surface": "secular", "snippet": "W. V. D. Hodge (1941). The Theory and Applications of Harmonic Integrals. Cambridge University Press, 1941. Cited by its record. License as found: Cambridge University Press — publisher's copyright, c", "authority_tier": "reference", "source": "W. V. D. Hodge (1941), Cambridge University Press, 1941", "readable": false, "generated": false}, {"id": "card_src_pron_hodges", "title": "hodges", "shelf": "pronunciation", "surface": "secular", "snippet": "hodges: pronounced (ARPABET) HH AA1 JH IH0 Z. 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_src_book_28394", "title": "William Penn — George Hodges", "shelf": "gutenberg", "surface": "secular", "snippet": "William Penn, by George Hodges. Subjects: Penn, William, 1644-1718. Read the full text (public domain): https://www.gutenberg.org/ebooks/28394", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_src_book_62459", "title": "Usury; Or, Interest, Premium and Discount — S. H. (Salmon Hodges) Crittenden", "shelf": "gutenberg", "surface": "secular", "snippet": "Usury; Or, Interest, Premium and Discount, by S. H. (Salmon Hodges) Crittenden. Subjects: Usury. Read the full text (public domain): https://www.gutenberg.org/ebooks/62459", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "generated": false}, {"id": "card_src_book_52581", "title": "Fountains Abbey: The story of a mediæval monastery — George Hodges", "shelf": "gutenberg", "surface": "secular", "snippet": "Fountains Abbey: The story of a mediæval monastery, by George Hodges. Subjects: Fountains Abbey (North Yorkshire, England). Read the full text (public domain): https://www.gutenberg.org/ebooks/52581", "authority_tier": "reference", "source": "Project Gutenberg (public domain)", "readable": true, "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_question_hodge"}}, "ends": "a verdict or a card · the trail · a seal · one next step"}}