We've all joined a consortium of brilliant minds in order to spend our free time solving physics' six remaining most glaring unsolved mysteries. Which are:
P vs. NP
The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time), it can also find that solution quickly. This is generally considered the most important open question in theoretical computer science as it has far-reaching consequences in mathematics, philosophy and cryptography (see P=NP proof consequences).
The Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
The Riemann hypothesis
The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. 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.
Yang-Mills existence and mass gap
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 (gluons). 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.
Navier–Stokes existence and smoothness
The Navier-Stokes equations describe the motion of liquids and gases. 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 Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, 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. 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.
So far, Killagram and I have spent most of our time snorting oxycotin and eating those cereal sticks that you can drink milk through. Chuck has spent the majority of his time shaving his head and listening to the Beatles. TJ Rapist, Ghetto Sideburns and Batt Gurl came up with a stunningly accurate hypothesis for the Navier-Stokes equations by mathematically measuring their farts, which they sometimes magnify by sticking lit lighters to their assholes. However, they would not accept this as "scientific" enough to win the one million dollar prize. We are not making much headway.
P vs. NP
The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time), it can also find that solution quickly. This is generally considered the most important open question in theoretical computer science as it has far-reaching consequences in mathematics, philosophy and cryptography (see P=NP proof consequences).
The Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
The Riemann hypothesis
The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2. 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.
Yang-Mills existence and mass gap
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 (gluons). 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.
Navier–Stokes existence and smoothness
The Navier-Stokes equations describe the motion of liquids and gases. 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 Birch and Swinnerton-Dyer conjecture
The Birch and Swinnerton-Dyer conjecture deals with a certain type of equation, 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. 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.
So far, Killagram and I have spent most of our time snorting oxycotin and eating those cereal sticks that you can drink milk through. Chuck has spent the majority of his time shaving his head and listening to the Beatles. TJ Rapist, Ghetto Sideburns and Batt Gurl came up with a stunningly accurate hypothesis for the Navier-Stokes equations by mathematically measuring their farts, which they sometimes magnify by sticking lit lighters to their assholes. However, they would not accept this as "scientific" enough to win the one million dollar prize. We are not making much headway.