{"id":"clay-millennium-problems","domain":"named-results","type":"Overview","title":"Clay Millennium Problems","slug":"clay-millennium-problems","url":"/mathematics/named-results/clay-millennium-problems/","posed":"2000, Clay Mathematics Institute","status":"6 of 7 open","summary":"Overview of Hodge Conjecture, Birch-Swinnerton-Dyer, and Yang-Mills Mass Gap — three of the seven Clay Millennium Prize Problems requiring specialized background to state precisely.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Hodge, W.V.D. (1950). The topological invariants of algebraic varieties. Proceedings of the ICM."},{"tier":1,"citation":"Birch, B.J. and Swinnerton-Dyer, H.P.F. (1965). Notes on elliptic curves II. Journal für die reine und angewandte Mathematik, 218, 79-108."},{"tier":1,"citation":"Jaffe, A. and Witten, E. (2000). Quantum Yang-Mills theory. Clay Mathematics Institute."},{"tier":3,"citation":"Devlin, K. (2002). The Millennium Problems. Basic Books."}],"relationships":[{"type":"solved_sibling","target":"poincare-conjecture","label":"Poincaré Conjecture"},{"type":"sibling_problem","target":"riemann-hypothesis","label":"Riemann Hypothesis"},{"type":"sibling_problem","target":"p-vs-np","label":"P vs NP"},{"type":"sibling_problem","target":"navier-stokes-existence-and-smoothness","label":"Navier-Stokes Existence and Smoothness"},{"type":"modelled_on","target":"hilberts-23-problems","label":"Hilbert's 23 Problems"},{"type":"tracked_in","target":"proof-status","label":"Proof Status Board"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.3"}