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Clay Mathematics Institute Search Clay Mathematics Institute About About About History Principal Activities Who’s Who CMI Logo Policies Programs & Awards Programs & Awards Programs & Awards Funded programs Fellowship Nominations Clay Research Award Dissemination Award People The Millennium Prize Problems The Millennium Prize Problems The Millennium Prize Problems Birch and Swinnerton-Dyer Conjecture Hodge Conjecture Navier-Stokes Equation P vs NP Poincaré Conjecture Riemann Hypothesis Yang-Mills & The Mass Gap Rules for the Millennium Prize Problems Online resources Online resources Online resources Books Video Library Lecture notes Collections Collections Collections Euclid’s Elements Ada Lovelace’s Mathematical Papers Collected Works of James G. Arthur Klein Protokolle Notes of the talks at the I.M.Gelfand Seminar Quillen Notebooks Riemann’s 1859 Manuscript Events News2024 Clay Research Conference and Workshops The 2024 Clay Research Conference will be held on Wednesday, 2 October. Associated workshops will be held during the week of the conference, 30 September-4 October. Read more Call for Proposals CMI invites proposals under the Enhancement and Partnership Program for fiscal year 2025 (1 October 2024-30 September 2025) and later. The principal aim of the program is to enhance activities that are already planned and financially viable. Read more Call for Nominations The Clay Mathematics Institute (CMI) calls for nominations for its competition for the 2025 Clay Research Fellowships. Read more The Clay Mathematics Institute is a global organisation dedicated to furthering the beauty, power and universality of mathematical thought. Read more The Millennium problems See all Hodge Conjecture The answer to this conjecture determines how much of the topology of the solution set of a system of algebraic equations can be defined in terms of further algebraic equations. The Hodge conjecture is known in certain special cases, e.g., when the solution set has dimension less than four. But in dimension four it is unknown. Birch and Swinnerton-Dyer Conjecture Supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points. Elliptic curves, defined by cubic equations in two variables, are fundamental mathematical objects that arise in many areas: Wiles’ proof of the Fermat Conjecture, factorization of numbers into primes, and cryptography, to name three. P vs NP If it is easy to check that a solution to a problem is correct, is it also easy to solve the problem? This is the essence of the P vs NP question. Typical of the NP problems is that of the Hamiltonian Path Problem: given N cities to visit, how can one do this without visiting a city twice? If you give me a solution, I can easily check that it is correct. But I cannot so easily find a solution. See all Upcoming events See all events 8 - 13 September 2024 Number Theory in the Americas 2 Casa Matemática Oaxaca Read more 10 - 13 September 2024 Geometry from the Model Theorist’s Point of View University of Oxford Read more 21 January - 16 May 2025 Extremal Combinatorics Simons Laufer Mathematical Research Institute Read more 22 - 26 July 2024 AGGITaTE 2024 University of Essex Read more See all events Latest news See all news 01 May 2024 2024 Clay Research Award Read more See all news Privacy Policy Contact CMI © 2024 Clay Mathematics Institute Site by...