Millennium Prize Problems

The Millennium Prize Problems are seven well-known complex

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 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.^[1] 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, 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 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.^[3]

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.

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 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".^[7]

Solved problem

Poincaré conjecture

In the field of

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 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 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.^[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]

Unsolved problems

Birch and Swinnerton-Dyer conjecture

The

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.^[11]

Hodge conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

${\displaystyle \operatorname {Hdg} ^{k}(X)=H^{2k}(X,\mathbb {Q} )\cap H^{k,k}(X).}$

We call this the group of Hodge classes of degree 2k 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 classes of complex subvarieties of X.

The official statement of the problem was given by Pierre Deligne.^[12]

Navier–Stokes existence and smoothness

${\displaystyle \overbrace {\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\begin{smallmatrix}{\text{Variation}}\end{smallmatrix}}+\underbrace {(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\begin{smallmatrix}{\text{Convection}}\end{smallmatrix}}} ^{\text{Inertia (per volume)}}\overbrace {{}-\underbrace {\nu \,\nabla ^{2}\mathbf {u} } _{\text{Diffusion}}=\underbrace {-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}} ^{\text{Divergence of stress}}+\underbrace {\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.}$

The

The question is whether or not, for all problems for which an algorithm can verify a given solution quickly (that is, in

Most mathematicians and computer scientists expect that P ≠ NP; however, it remains unproven.[16]

The official statement of the problem was given by Stephen Cook.^[17]

Riemann hypothesis

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }n^{-s}={\frac {1}{1^{s}}}+{\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+\cdots }$

The

The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of ½. 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 the problem was given by Enrico Bombieri.^[18]

Yang–Mills existence and mass gap

In

For a given real field ${\displaystyle \phi (x)}$, we can say that the theory has a mass gap if the two-point function has the property

${\displaystyle \langle \phi (0,t)\phi (0,0)\rangle \sim \sum _{n}A_{n}\exp \left(-\Delta _{n}t\right)}$

with ${\displaystyle \Delta _{0}>0}$ 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

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on ${\displaystyle \mathbb {R} ^{4}}$ and has a mass gap Δ > 0. Existence includes establishing axiomatic properties at least as strong as those cited in Streater & Wightman (1964),^[20] Osterwalder & Schrader (1973),^[21] and Osterwalder & Schrader (1975).^[22]

The official statement of the problem was given by Arthur Jaffe and Edward Witten.^[23]

See also

• 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