Millennium Prize Problems
The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010.
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.
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".
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.
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".
In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes.
[10] The Birch and Swinnerton-Dyer conjecture deals with certain types of equations: those defining elliptic curves over 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.
For the three-dimensional system of equations, and given some initial conditions, mathematicians have not yet proven that smooth solutions always exist.
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 of color confinement permits only bound states of gluons, forming massive particles.
Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales.
