| Millennium Prize Problems |
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TheMillennium Prize Problems are seven well-known complexmathematical problems selected by theClay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem.
The Clay Mathematics Institute officially designated the titleMillennium Problem for the seven unsolved mathematical problems, theBirch and Swinnerton-Dyer conjecture,Hodge conjecture,Navier–Stokes existence and smoothness,P versus NP problem,Riemann hypothesis,Yang–Mills existence and mass gap, and thePoincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official website of the Clay Mathematics Institute, these seven problems are officially called theMillennium 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 mathematicianGrigori Perelman in 2010. However, he declined the award as it was not also offered toRichard S. Hamilton, upon whose work Perelman built.
The Clay Institute was inspired by a set oftwenty-three problems organized by the mathematicianDavid Hilbert in 1900 which were highly influential in driving the progress of mathematics in the twentieth century.[1] The seven selected problems span a number of mathematical fields, namelyalgebraic geometry,arithmetic geometry,geometric topology,mathematical physics,number theory,partial differential equations, andtheoretical computer science. Unlike Hilbert's problems, the problems selected by the Clay Institute were already renowned among professional mathematicians, with many actively working towards their resolution.[2]
The seven problems were officially announced byJohn Tate andMichael Atiyah during a ceremony held on May 24, 2000 (at the amphithéâtreMarguerite de Navarre) in theCollège de France inParis.[3]
Grigori Perelman, who had begun work on thePoincaré 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 the choice ofUS$1 million prize money would popularize, among general audiences, both the selected problems as well as the "excitement of mathematical endeavor".[4] Another board member,Fields medalistAlain 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".[5]
Some mathematicians have been more critical.Anatoly Vershik characterized their monetary prize as "show business" representing the "worst manifestations of present-day mass culture", and thought that there are more meaningful ways to invest in public appreciation of mathematics.[6] 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 medalistShing-Tung Yau, who was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them".[7]
In the field ofgeometric topology, a two-dimensionalsphere is characterized by the fact that it is the onlyclosed andsimply-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 the Poincaré conjecture, the precise formulation of which states:
Anythree-dimensional topological manifold which is closed and simply-connected must behomeomorphic to the3-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 ofsmooth manifolds anddiffeomorphisms.
A proof of this conjecture, together with the more powerfulgeometrization conjecture, was given byGrigori Perelman in 2002 and 2003. Perelman's solution completedRichard 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'sRicci flow, which is a complicated system ofpartial differential equations defined in the field ofRiemannian geometry.
For his contributions to the theory of Ricci flow, Perelman was awarded theFields Medal in 2006. However, he declined to accept the prize.[8] For his proof of the Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010.[9] However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own.[10]
TheBirch andSwinnerton-Dyer conjecture deals with certain types of equations: those definingelliptic curves over therational 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 curveE hasrankr, then theL-functionL(E,s) associated with itvanishes to orderr ats = 1.
Hilbert's tenth problem dealt with 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 byAndrew Wiles.[11]
The Hodge conjecture is that forprojectivealgebraic varieties,Hodge cycles are rationallinear combinations ofalgebraic cycles.
We call this the group ofHodge classes of degree 2k onX.
The modern statement of the Hodge conjecture is:
The official statement of the problem was given byPierre Deligne.[12]
TheNavier–Stokes equations describe the motion offluids, and are one of the pillars offluid 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 someinitial conditions, mathematicians have not yet proven thatsmooth solutions always exist. This is called theNavier–Stokes existence and smoothness problem.
The problem, restricted to the case of anincompressible 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 byCharles Fefferman.[13]
The question is whether or not, for all problems for which an algorithm canverify a given solution quickly (that is, inpolynomial time), an algorithm can alsofind 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 inmathematics andtheoretical computer science as it has far-reaching consequences to other problems inmathematics, tobiology,[14]philosophy[15] and tocryptography (seeP versus NP problem proof consequences). A common example of an NP problem not known to be in P is theBoolean satisfiability problem.
Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven.[16]
The official statement of the problem was given byStephen Cook.[17]
TheRiemann zeta function ζ(s) is afunction whosearguments may be anycomplex number other than 1, and whose values are also complex. Itsanalytical continuation haszeros at 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 allnontrivial zeros of the analytical continuation of theRiemann zeta function have a real part of1/2. A proof or disproof of this would have far-reaching implications innumber theory, especially for the distribution ofprime numbers. This wasHilbert's eighth problem, and is still considered an importantopen problem a century later.
The problem has been well-known ever since it was originally posed byBernhard Riemann in 1860. The Clay Institute's exposition of the problem was given byEnrico Bombieri.[18]
Inquantum field theory, themass gap is the difference in energy between the vacuum and the next lowestenergy 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, we can say that the theory has a mass gap if thetwo-point function has the property
with being the lowest energy value in thespectrum of theHamiltonian and thus the mass gap. This quantity, easy to generalize to other fields, is what is generally measured in lattice computations.
QuantumYang–Mills theory is the current grounding for the majority of theoretical applications of thought to the reality and potential realities ofelementary particle physics.[19] The theory is a generalization of theMaxwell theory ofelectromagnetism where thechromo-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, the postulated phenomenon ofcolor confinement permits only bound states of gluons, forming massive particles. This is themass gap. Another aspect of confinement isasymptotic freedom which makes it conceivable thatquantum 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 the problem was given byArthur Jaffe andEdward Witten.[23]
The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture.
