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Millennium Prize Problems
In physics, classical [Yang–Mills theory] is a generalization of the Maxwell theory of electromagnetism where the chromo electromagnetic field itself carries charges. 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 YangMills theory] exists without restriction to low energy scales. The problem is to establish rigorously the existence of the quantum YangMills theory and a mass gap.
The official statement of the problem was given by [Arthur Jaffe] and [Edward Witten] .
Navier–Stokes existence and smoothness
The [Navier–Stokes equations] describe the motion of [liquid] s and [gas] es. Although they were found in the 19th century, they still are not well understood. The problem is to make progress toward a mathematical theory that will give us insight into these equations.
The official statement of the problem was given by [Charles Fefferman] .
The Birch and SwinnertonDyer conjecture
The Birch and SwinnertonDyer conjecture deals with a certain type of equation, 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. [Hilbert's tenth problem] dealt with a more general type of equation, and in that case it was proven that there is no way to decide whether a given equation even has any solutions.
The official statement of the problem was given by [Andrew Wiles] .
See also
 [Hilbert's problems]
 [Smale's problems]
References

 . Available online at [http://www.claymath.org/library/] .
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