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38-438: While the institute is best known for its Millennium Prize Problems , it carries out a wide range of activities, including conferences, workshops, summer schools, and a postdoctoral program supporting Clay Research Fellows. The institute is run according to a standard structure comprising a scientific advisory committee that decides on grant-awarding and research proposals, and a board of directors that oversees and approves

76-674: A given solution quickly (that is, in polynomial time ), an algorithm can also find that solution quickly. Since the former describes the class of problems termed NP, while the latter describes P, the question is equivalent to asking whether all problems in NP are also in P. This is generally considered one of the most important open questions in mathematics and theoretical computer science as it has far-reaching consequences to other problems in mathematics , to biology , philosophy and to cryptography (see P versus NP problem proof consequences ). A common example of an NP problem not known to be in P

114-824: A highly selective high school, which is alma-mater for many well-known mathematicians. During his high-school years, he won two perfect scores in International Mathematical Olympiad 1986 and 1987. He received his undergraduate and master degree in mathematics from the Saint Petersburg State University , one of the top universities in Russia, in 1992. In 1996, he defended his Ph.D. thesis, Spectral Analysis of Julia Sets , at Caltech , with Nikolai Makarov as his primary thesis advisor. After spending time at Yale University and Princeton's Institute for Advanced Study as

152-462: A more general type of equation, and in that case it was proven that there is no algorithmic way to decide whether a given equation even has any solutions. The official statement of the problem was given by Andrew Wiles . The Hodge conjecture is that for projective algebraic varieties , Hodge cycles are rational linear combinations of algebraic cycles . We call this the group of Hodge classes of degree 2 k on X . The modern statement of

190-770: A postdoc, Smirnov joined the faculty of the Royal Institute of Technology in Stockholm in 1998. In 2003, he become a professor in the Analysis, Mathematical Physics and Probability group at the University of Geneva . In 2010, Smirnov became the founding director of the Chebyshev Laboratory in Saint Petersburg State University in Russia. Smirnov has worked on percolation theory , where he proved Cardy 's formula for critical site percolation on

228-504: Is called the Navier–Stokes existence and smoothness problem. The problem, restricted to the case of an incompressible flow , is to prove either that smooth, globally defined solutions exist that meet certain conditions, or that they do not always exist and the equations break down. The official statement of the problem was given by Charles Fefferman . The question is whether or not, for all problems for which an algorithm can verify

266-462: Is the Boolean satisfiability problem . Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven. The official statement of the problem was given by Stephen Cook . The Riemann zeta function ζ(s) is a function whose arguments may be any complex number other than 1, and whose values are also complex. Its analytical continuation has zeros at

304-513: Is the current grounding for the majority of theoretical applications of thought to the reality and potential realities of elementary particle physics . The theory is a generalization of the Maxwell theory of electromagnetism where the chromo -electromagnetic field itself carries charge. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles ( gluons ). However,

342-596: The Creative Commons Attribution/Share-Alike License . 41°49′34.4″N 71°24′54.7″W  /  41.826222°N 71.415194°W  / 41.826222; -71.415194 Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $ 1 million prize for

380-528: The Poincaré conjecture in the 1990s, released his proof in 2002 and 2003. His refusal of the Clay Institute's monetary prize in 2010 was widely covered in the media. The other six Millennium Prize Problems remain unsolved, despite a large number of unsatisfactory proofs by both amateur and professional mathematicians. Andrew Wiles , as part of the Clay Institute's scientific advisory board, hoped that

418-476: The rational numbers . The conjecture is that there is a simple way to tell whether such equations have a finite or infinite number of rational solutions. More specifically, the Millennium Prize version of the conjecture is that, if the elliptic curve E has rank r , then the L -function L ( E , s ) associated with it vanishes to order r at s = 1 . Hilbert's tenth problem dealt with

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456-491: The triangular lattice , and deduced conformal invariance . The conjecture was proved in the special case of site percolation on the triangular lattice. Smirnov's theorem has led to a fairly complete theory for percolation on the triangular lattice, and to its relationship to the Schramm–Loewner evolution introduced by Oded Schramm . He also established conformality for the two-dimensional critical Ising model . Smirnov

494-501: The "worst manifestations of present-day mass culture", and thought that there are more meaningful ways to invest in public appreciation of mathematics. He viewed the superficial media treatments of Perelman and his work, with disproportionate attention being placed on the prize value itself, as unsurprising. By contrast, Vershik praised the Clay Institute's direct funding of research conferences and young researchers. Vershik's comments were later echoed by Fields medalist Shing-Tung Yau , who

532-578: The 2024 Clay Research Fellowships. Both are completing their PhDs at the Massachusetts Institute of Technology and will start their five-year fellowships on July 1, 2024. The 2024 Clay Research Conference will be held on October 2, 2024, at the Mathematical Institute, University of Oxford. The conference will be accompanied by workshops from September 30 to October 4, 2024. Notable workshops include: Daniel Graham from

570-597: The CMI. In announcing the prize, CMI drew a parallel to Hilbert's problems , which were proposed in 1900, and had a substantial impact on 20th century mathematics. Of the initial 23 Hilbert problems, most of which have been solved, only the Riemann hypothesis (formulated in 1859) is included in the seven Millennium Prize Problems. For each problem, the Institute had a professional mathematician write up an official statement of

608-544: The Hodge conjecture is: The official statement of the problem was given by Pierre Deligne . The Navier–Stokes equations describe the motion of fluids , and are one of the pillars of fluid mechanics . However, theoretical understanding of their solutions is incomplete, despite its importance in science and engineering. For the three-dimensional system of equations, and given some initial conditions , mathematicians have not yet proven that smooth solutions always exist. This

646-572: The Millennium Prize Problems, the Clay Mathematics Institute supports mathematics via the awarding of research fellowships (which range from two to five years and are aimed at younger mathematicians), as well as shorter-term scholarships for programs, individual research, and book writing. The institute also has a yearly Clay Research Award , recognizing major breakthroughs in mathematical research. Finally,

684-469: The Poincaré conjecture, the precise formulation of which states: Any three-dimensional topological manifold which is closed and simply-connected must be homeomorphic to the 3-sphere . Although the conjecture is usually stated in this form, it is equivalent (as was discovered in the 1950s) to pose it in the context of smooth manifolds and diffeomorphisms . A proof of this conjecture, together with

722-572: The University of Surrey won the Gold Medal for Mathematical Sciences at the 2024 STEM for Britain competition for his work on quantum authentication methods. The institute is best known for establishing the Millennium Prize Problems on May 24, 2000. These seven problems are considered by CMI to be "important classic questions that have resisted solution over the years." For each problem, the first person to solve it will be awarded US$ 1,000,000 by

760-485: The analytical continuation of the Riemann zeta function have a real part of ⁠ 1 / 2 ⁠ . A proof or disproof of this would have far-reaching implications in number theory , especially for the distribution of prime numbers . This was Hilbert's eighth problem , and is still considered an important open problem a century later. The problem has been well-known ever since it was originally posed by Bernhard Riemann in 1860. The Clay Institute's exposition of

798-568: The choice of US$ 1 million prize money would popularize, among general audiences, both the selected problems as well as the "excitement of mathematical endeavor". Another board member, Fields medalist Alain Connes , hoped that the publicity around the unsolved problems would help to combat the "wrong idea" among the public that mathematics would be "overtaken by computers". Some mathematicians have been more critical. Anatoly Vershik characterized their monetary prize as "show business" representing

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836-471: The committee's decisions. As of September 2024, the board is made up of members of the Clay family, whereas the scientific advisory committee is composed of Simon Donaldson , Michael Hopkins , Andrei Okounkov , Gigliola Staffilani , Andrew Wiles , and Martin R. Bridson . Bridson is the current president of CMI. The Clay Mathematics Institute has announced that Ishan Levy and Mehtaab Sawhney have been awarded

874-595: The field of Riemannian geometry . For his contributions to the theory of Ricci flow, Perelman was awarded the Fields Medal in 2006. However, he declined to accept the prize. For his proof of the Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010. However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own. The Birch and Swinnerton-Dyer conjecture deals with certain types of equations: those defining elliptic curves over

912-624: The first correct solution to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch and Swinnerton-Dyer conjecture , Hodge conjecture , Navier–Stokes existence and smoothness , P versus NP problem , Riemann hypothesis , Yang–Mills existence and mass gap , and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on

950-989: The institute has an annual prize – the Clay Research Award. Its recipients to date are Ian Agol , Manindra Agrawal , Yves Benoist , Manjul Bhargava , Tristan Buckmaster, Danny Calegari , Alain Connes , Nils Dencker , Alex Eskin , David Gabai , Ben Green , Mark Gross, Larry Guth , Christopher Hacon , Richard S. Hamilton , Michael Harris , Philip Isett, Jeremy Kahn , Nets Katz , Laurent Lafforgue , Gérard Laumon , Aleksandr Logunov, Eugenia Malinnikova, Vladimir Markovic , James McKernan , Jason Miller, Maryam Mirzakhani , Ngô Bảo Châu , Rahul Pandharipande , Jonathan Pila , Jean-François Quint , Peter Scholze , Oded Schramm , Scott Sheffield, Bernd Siebert, Stanislav Smirnov , Terence Tao , Clifford Taubes , Richard Taylor , Maryna Viazovska , Vlad Vicol, Claire Voisin , Jean-Loup Waldspurger , Andrew Wiles , Geordie Williamson , Edward Witten and Wei Zhang. Besides

988-450: The institute organizes a number of summer schools, conferences, workshops, public lectures, and outreach activities aimed primarily at junior mathematicians (from the high school to the postdoctoral level). CMI publications are available in PDF form at most six months after they appear in print. This article incorporates material from Millennium Problems on PlanetMath , which is licensed under

1026-423: The more powerful geometrization conjecture , was given by Grigori Perelman in 2002 and 2003. Perelman's solution completed Richard Hamilton 's program for the solution of the geometrization conjecture, which he had developed over the course of the preceding twenty years. Hamilton and Perelman's work revolved around Hamilton's Ricci flow , which is a complicated system of partial differential equations defined in

1064-414: The negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. However, the negative even integers are not the only values for which the zeta function is zero. The other ones are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The Riemann hypothesis is that all nontrivial zeros of

1102-524: The official website of the Clay Mathematics Institute, these seven problems are officially called the Millennium Problems . To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton , upon whose work Perelman built. The Clay Institute

1140-429: The postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the mass gap . Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum Yang–Mills theory and a mass gap. The official statement of

1178-490: The problem was given by Arthur Jaffe and Edward Witten . Stanislav Smirnov Stanislav Konstantinovich Smirnov ( Russian : Станисла́в Константи́нович Cмирно́в ; born 3 September 1970) is a Russian mathematician currently working as a professor at the University of Geneva . He was awarded the Fields Medal in 2010. His research involves complex analysis , dynamical systems and probability theory . Stanislav Smirnov graduated Saint Petersburg Lyceum 239 ,

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1216-471: The problem was given by Enrico Bombieri . In quantum field theory , the mass gap is the difference in energy between the vacuum and the next lowest energy state . The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of as particles in plane-waves, the mass gap is the mass of the lightest particle. For a given real field ϕ ( x ) {\displaystyle \phi (x)} , we can say that

1254-452: The problem, which will be the main standard against which a given solution will be measured. The seven problems are: Some of the mathematicians who were involved in the selection and presentation of the seven problems were Michael Atiyah , Enrico Bombieri , Alain Connes , Pierre Deligne , Charles Fefferman , John Milnor , David Mumford , Andrew Wiles , and Edward Witten . In recognition of major breakthroughs in mathematical research,

1292-512: The problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution. The seven problems were officially announced by John Tate and Michael Atiyah during a ceremony held on May 24, 2000 (at the amphithéâtre Marguerite de Navarre ) in the Collège de France in Paris . Grigori Perelman , who had begun work on

1330-458: The theory has a mass gap if the two-point function has the property with Δ 0 > 0 {\displaystyle \Delta _{0}>0} being the lowest energy value in the spectrum of the Hamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations. Quantum Yang–Mills theory

1368-1033: Was awarded the Saint Petersburg Mathematical Society Prize (1997), the Clay Research Award (2001), the Salem Prize (joint with Oded Schramm , 2001), the Göran Gustafsson Prize (2001), the Rollo Davidson Prize (2002), and the Prize of the European Mathematical Society (2004). In 2010 Smirnov was awarded the Fields medal for his work on the mathematical foundations of statistical physics , particularly finite lattice models. His citation reads "for

1406-471: Was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them". In the field of geometric topology , a two-dimensional sphere is characterized by the fact that it is the only closed and simply-connected two-dimensional surface. In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as

1444-477: Was inspired by a set of twenty-three problems organized by the mathematician David Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century. The seven selected problems span a number of mathematical fields, namely algebraic geometry , arithmetic geometry , geometric topology , mathematical physics , number theory , partial differential equations , and theoretical computer science . Unlike Hilbert's problems,

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