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{{Short description|Seven mathematical problems with a US$1 million prize for each solution}}
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{{Millennium Problems}}

The '''Millennium Prize Problems''' are seven well-known complex [[mathematics|mathematical]] problems selected by the [[Clay 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 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 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.

==Overview==
The Clay Institute was inspired by a set of [[Hilbert's problems|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.<ref name="Jaffe_2000">{{cite journal|first=Arthur M.|last=Jaffe|url=https://www.ams.org/notices/200606/fea-jaffe.pdf|title=The Millennium Grand Challenge in Mathematics|journal=Notices of the American Mathematical Society|date=June–July 2006|volume=53|issue=6|pages=652–660}}</ref> The seven selected problems span a number of mathematical fields, namely [[algebraic geometry]], [[arithmetic geometry]], [[geometric topology]], [[mathematical physics]], [[number theory]], [[partial differential equation]]s, and [[theoretical 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.<ref>{{harvtxt|Carlson|Jaffe|Wiles|2006}}</ref>

The seven problems were officially announced by [[John Tate (mathematician)|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]].<ref>{{cite web | url=https://www.claymath.org/millennium-problems/ | title=The Millennium Prize Problems }}</ref>

[[Grigori Perelman]], who had begun work on 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 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".<ref>{{cite journal|last1=Jackson|first1=Allyn|title=Million-dollar mathematics prizes announced|journal=Notices of the American Mathematical Society|volume=47|issue=8|date=September 2000|pages=877–879}}</ref> Another board member, [[Fields Medal|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".<ref>{{cite journal|first1=David|last1=Dickson|journal=Nature|title=Mathematicians chase the seven million-dollar proofs|volume=405|issue=383|year=2000|page=383|doi=10.1038/35013216 |pmid=10839504 |s2cid=31169641 |doi-access=free}}</ref>

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.<ref>{{cite journal|last1=Vershik|first1=Anatoly|author-link=Anatoly Vershik|title=What is good for mathematics? Thoughts on the Clay Millennium prizes|journal=Notices of the American Mathematical Society|volume=54|issue=1|date=January 2007|pages=45–47}}</ref> 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 was additionally critical of the idea of a foundation taking actions to "appropriate" fundamental mathematical questions and "attach its name to them".<ref>{{cite book|author-link1=Shing-Tung Yau|last1=Yau|first1=Shing-Tung|last2=Nadis|first2=Steve|title=The shape of a life. One mathematician's search for the universe's hidden geometry|publisher=Yale University Press|location=New Haven, CT|year=2019|bibcode=2019shli.book.....Y }}</ref>

==Solved problem==

===Poincaré conjecture===
{{Main article|Poincaré conjecture}}
[[File:P1S2all.jpg|thumb|upright=1.5|A [[compact space|compact]] 2-dimensional surface without [[boundary (topology)|boundary]] is topologically [[Homeomorphism|homeomorphic]] to a 2-sphere if every loop can be continuously tightened to a point. The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.]]
In the field of [[geometric topology]], a two-dimensional [[sphere]] is characterized by the fact that it is the only [[closed manifold|closed]] and [[simply connected space|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 the Poincaré conjecture, the precise formulation of which states: {{quote|sign=|source=|Any [[3-manifold|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 [[diffeomorphism]]s.

A proof of this conjecture, together with the more powerful [[geometrization conjecture]], was given by [[Grigori Perelman]] in 2002 and 2003. Perelman's solution completed [[Richard S. Hamilton|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 equation]]s defined in 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.<ref>{{cite news |url=http://news.bbc.co.uk/2/hi/science/nature/5274040.stm |title=Maths genius declines top prize |date=22 August 2006 |publisher=[[BBC News]] |access-date=16 June 2011}}</ref> For his proof of the Poincaré conjecture, Perelman was awarded the Millennium Prize on March 18, 2010.<ref>{{cite press release|publisher=[[Clay Mathematics Institute]] |date=March 18, 2010 |title=Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman |url=http://www.claymath.org/poincare/millenniumPrizeFull.pdf |access-date=March 18, 2010 |quote=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. |url-status=dead |archive-url=https://web.archive.org/web/20100331134628/http://www.claymath.org/poincare/millenniumPrizeFull.pdf |archive-date=March 31, 2010 }}</ref> However, he declined the award and the associated prize money, stating that Hamilton's contribution was no less than his own.<ref>{{cite news|newspaper=[[Interfax]]|date=July 1, 2010|title=Последнее "нет" доктора Перельмана|url=https://www.interfax.ru/russia/143603|access-date=25 January 2024}}</ref>

==Unsolved problems==
===Birch and Swinnerton-Dyer conjecture===
{{Main article|Birch and Swinnerton-Dyer conjecture}}
The [[Bryan John Birch|Birch]] and [[Peter Swinnerton-Dyer|Swinnerton-Dyer]] conjecture deals with certain types of equations: those defining [[elliptic curve]]s over the [[rational number]]s. 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 {{Mvar|E}} has [[Rank of an elliptic curve|rank]] {{Mvar|r}}, then the [[Hasse–Weil zeta function|''L''-function]] {{Math|''L''(''E'', ''s'')}} associated with it [[Order of vanishing|vanishes to order]] {{Mvar|r}} at {{Math|1=''s'' = 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 by [[Andrew Wiles]].<ref>{{Cite web |title=Birch and Swinnerton-Dyer Conjecture |url=https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/ |access-date=2024-11-20 |website=Clay Mathematics Institute |language=en-US}}</ref>

===Hodge conjecture===
{{Main article|Hodge conjecture}}
The Hodge conjecture is that for [[projective space|projective]] [[algebraic variety|algebraic varieties]], [[Hodge cycle]]s are rational [[linear combination]]s of [[algebraic cycle]]s.

:<math>\operatorname{Hdg}^k(X) = H^{2k}(X, \Q) \cap H^{k,k}(X).</math>

We call this the group of ''Hodge classes'' of degree 2''k'' on ''X''.

The modern statement of the Hodge conjecture is:

::Let ''X'' be a non-singular complex projective variety.  Then every Hodge class on ''X'' is a linear combination with rational coefficients of the [[cohomology|cohomology classes]] of complex subvarieties of ''X''.

The official statement of the problem was given by [[Pierre Deligne]].<ref>{{cite encyclopedia|last1=Deligne|first1=Pierre|author-link1=Pierre Deligne|title=The Hodge conjecture|pages=45–53|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|encyclopedia= The millennium prize problems|year = 2006  |location = Providence, RI|publisher = American Mathematical Society and Clay Mathematics Institute|isbn = 978-0-8218-3679-8|url=https://www.claymath.org/sites/default/files/hodge.pdf}}</ref>

===Navier–Stokes existence and smoothness===
{{Main article|Navier–Stokes existence and smoothness}}

:'''<math>
 \overbrace{
 \underbrace{\frac{\partial \mathbf{u}}{\partial t}}_{
 \begin{smallmatrix}
 \text{Variation}
 \end{smallmatrix}} +
 \underbrace{(\mathbf{u} \cdot \nabla) \mathbf{u}}_{
 \begin{smallmatrix}
 \text{Convection}
 \end{smallmatrix}}}^{\text{Inertia (per volume)}}
 \overbrace{{}-\underbrace{\nu \, \nabla^2 \mathbf{u}}_{\text{Diffusion}}=
 \underbrace{-\nabla w}_{
 \begin{smallmatrix}
 \text{Internal} \\
 \text{source}
 \end{smallmatrix}}}^{\text{Divergence of stress}} +
 \underbrace{\mathbf{g}}_{
 \begin{smallmatrix}
 \text{External} \\
 \text{source}
 \end{smallmatrix}}
 .
</math>'''

The [[Navier–Stokes equations]] describe the motion of [[fluid]]s, 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 condition]]s, mathematicians have not yet proven that [[Smooth function|smooth solutions]] always exist. This 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]].<ref>{{cite encyclopedia|last1=Fefferman|first1=Charles L.|author-link1=Charles Fefferman|title=Existence and smoothness of the Navier–Stokes equation|url=https://www.claymath.org/sites/default/files/navierstokes.pdf|pages=57–67|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|encyclopedia= The millennium prize problems|year = 2006  |location = Providence, RI|publisher = American Mathematical Society and Clay Mathematics Institute|isbn = 978-0-8218-3679-8}}</ref>

===P versus NP===
[[File:P np np-complete np-hard.svg|thumb|300px|right|[[Euler diagram]] for '''[[P (complexity)|P]]''', '''[[NP (complexity)|NP]]''', '''NP'''-complete, and '''NP'''-hard set of problems (excluding the empty language and its complement, which belong to '''P''' but are not '''NP'''-complete)]]
{{Main article|P versus NP problem}}
The question is whether or not, for all problems for which an algorithm can ''verify'' a given solution quickly (that is, in [[Polynomial time#Polynomial time|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 [[computation|theoretical computer science]] as it has far-reaching consequences to other problems in [[mathematics]], to [[biology]],<ref>{{Cite journal |last=Rajput |first=Uday Singh |date=2016 |title=P Versus NP: More than just a prize problem |url=http://bharataganitaparisad.com/wp-content/uploads/2016/12/ch8.pdf |url-status=live |journal=Ganita |publication-place=Lucknow, India |volume=66 |pages=90 |issn=0046-5402 |archive-url=https://web.archive.org/web/20220617115240/http://bharataganitaparisad.com/wp-content/uploads/2016/12/ch8.pdf |archive-date=17 June 2022 |access-date=17 June 2022}}</ref> [[philosophy]]<ref>{{cite web |url=http://eccc.hpi-web.de/report/2011/108/ |title=Why Philosophers Should Care About Computational Complexity |date=14 August 2011 |author=Scott Aaronson |author-link=Scott Aaronson |publisher=Technical report}}</ref> and to [[cryptography]] (see [[P versus NP problem#Consequences of solution|P versus NP problem proof consequences]]). A common example of an NP problem not known to be in P is the [[Boolean satisfiability problem]].

Most mathematicians and computer scientists expect that P&nbsp;≠&nbsp;NP; however, it remains unproven.<ref>{{cite journal|author=William Gasarch|author-link=William Gasarch|title=The P=?NP poll.|journal=SIGACT News|volume=33|issue=2|pages=34–47|date=June 2002| url=http://www.cs.umd.edu/~gasarch/papers/poll.pdf|doi=10.1145/1052796.1052804|s2cid=18759797}}</ref>

The official statement of the problem was given by [[Stephen Cook]].<ref>{{cite encyclopedia|last1=Cook|first1=Stephen|author-link1=Stephen Cook|title=The P versus NP problem|pages=87–104|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|encyclopedia= The millennium prize problems|year = 2006  |location = Providence, RI|publisher = American Mathematical Society and Clay Mathematics Institute|isbn = 978-0-8218-3679-8|url=https://www.claymath.org/wp-content/uploads/2022/06/pvsnp.pdf}}</ref>

===Riemann hypothesis===
[[File:RiemannCriticalLine.svg|thumb|The real part (red) and imaginary part (blue) of the Riemann zeta function along the critical line Re(''s'') = 1/2. The first nontrivial [[Zero of a function|zeros]] can be seen at Im(''s'') = ±14.135, ±21.022 and ±25.011.]]

{{Main article|Riemann hypothesis}}
:<math>\zeta(s) = \sum_{n=1}^\infty n^{-s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math>
The [[Riemann zeta function]] ζ(s) is a [[Function (mathematics)|function]] whose [[Argument of a function|arguments]] may be any [[complex number]] other than 1, and whose values are also complex. Its [[analytical continuation]] has [[Zero of a function|zeros]] 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 real part of every nontrivial zero of the Riemann zeta function is 1/2.

The Riemann hypothesis is that all [[Triviality (mathematics)|nontrivial]] zeros of the analytical continuation of the [[Riemann zeta function]] have a real part of {{sfrac|1|2}}. A proof or disproof of this would have far-reaching implications in [[number theory]], especially for the distribution of [[prime number]]s. 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 the problem was given by [[Enrico Bombieri]].<ref>{{cite encyclopedia|last1=Bombieri|first1=Enrico|author-link1=Enrico Bombieri|title=The Riemann hypothesis|pages=107–124|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|encyclopedia= The millennium prize problems|year = 2006  |location = Providence, RI|publisher = American Mathematical Society and Clay Mathematics Institute|isbn = 978-0-8218-3679-8|url=https://www.claymath.org/sites/default/files/official_problem_description.pdf}}</ref>

===Yang–Mills existence and mass gap===
{{Main article|Yang–Mills existence and mass gap}}
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 <math>\phi(x)</math>, we can say that the theory has a mass gap if the [[Correlation function (quantum field theory)|two-point function]] has the property

:<math>\langle\phi(0,t)\phi(0,0)\rangle\sim \sum_nA_n\exp\left(-\Delta_nt\right)</math>

with <math>\Delta_0>0</math> being the lowest energy value in the [[Spectrum (functional analysis)|spectrum]] of the [[Hamiltonian (quantum mechanics)|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]] is the current grounding for the majority of theoretical applications of thought to the reality and potential realities of [[elementary particle physics]].<ref>{{cite web
| url = https://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap
| title = Yang–Mills and Mass Gap
| work = www.claymath.org ([[Clay Mathematics Institute|Claymath]])
| archive-url = https://archive.today/20151122222730/http://www.claymath.org/millennium-problems/yang%E2%80%93mills-and-mass-gap
| archive-date = 22 November 2015
| access-date = 29 June 2021
| url-status = live
}}</ref> The theory is a generalization of the [[James Clerk Maxwell|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 ([[gluon]]s). However, 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.

::Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on <math>\mathbb{R}^4</math> and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964),<ref>{{cite book |last1=Streater |first1=R. |last2=Wightman |first2=A. |title=PCT, Spin and Statistics and all That|url=https://archive.org/details/pctspinstatistic0000stre |url-access=registration |publisher=W. A. Benjamin|year= 1964}}</ref> Osterwalder & Schrader (1973),<ref>{{cite journal |last1=Osterwalder |first1=K. |last2=Schrader |first2=R. |title=Axioms for Euclidean Green's functions |journal=[[Communications in Mathematical Physics]] |volume=31 |issue=2 |pages=83–112 |year=1973|doi=10.1007/BF01645738 |bibcode = 1973CMaPh..31...83O |s2cid=189829853 |url=http://projecteuclid.org/euclid.cmp/1103858969 }}</ref> and Osterwalder & Schrader (1975).<ref>{{cite journal |last1=Osterwalder |first1=K. |last2=Schrader |first2=R. |title=Axioms for Euclidean Green's functions II |journal=[[Communications in Mathematical Physics]] |volume=42 |issue=3 |pages=281–305 |year=1975|doi=10.1007/BF01608978 |bibcode = 1975CMaPh..42..281O |s2cid=119389461 |url=http://projecteuclid.org/euclid.cmp/1103899050 }}</ref>

The official statement of the problem was given by [[Arthur Jaffe]] and [[Edward Witten]].<ref>{{cite encyclopedia|last1=Jaffe|first1=Arthur|last2=Witten|first2=Edward|author-link1=Arthur Jaffe|author-link2=Edward Witten|title=Quantum Yang–Mills theory|pages=129–152|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|encyclopedia= The millennium prize problems|year = 2006  |location = Providence, RI|publisher = American Mathematical Society and Clay Mathematics Institute|isbn = 978-0-8218-3679-8|url=http://www.claymath.org/sites/default/files/yangmills.pdf}}</ref>

==See also==
{{Portal|Mathematics}}
* [[Beal conjecture]]
* [[Hilbert's problems]]
* [[List of mathematics awards]]
* [[List of unsolved problems in mathematics]]
* [[Smale's problems]]
* [[Paul Wolfskehl]] (offered a cash prize for the solution to [[Fermat's Last Theorem]])
* [[abc conjecture]]

==References==
{{Reflist}}
* {{PlanetMath attribution|id=8716|title=Millennium Problems}}

==Further reading==
* {{Cite book|editor1-last = Carlson|editor1-first = James|editor2-last = Jaffe|editor2-first = Arthur|editor2-link = Arthur Jaffe|editor3-last = Wiles|editor3-first = Andrew|editor3-link = Andrew Wiles|title = The Millennium Prize Problems|year = 2006  |location = Providence, RI|publisher = [[American Mathematical Society]] and [[Clay Mathematics Institute]]|isbn = 978-0-8218-3679-8 |url=http://www.claymath.org/library/}}
* {{Cite book|last = Devlin|first = Keith J.|author-link = Keith Devlin|title = The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles of Our Time|publisher = Basic Books|orig-year = 2002|location = New York|isbn = 0-465-01729-0|year = 2003|url-access = registration|url = https://archive.org/details/millenniumproble00keit}}

==External links==
{{Sister project links| wikt=no | commons=no | b=no | n=no | q=Millennium Prize Problems | s=no | v=no | voy=no | species=no | d=no}}

* [http://www.claymath.org/millennium-problems The Millennium Prize Problems]

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