The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. They have resisted solving for decades. The mind-numbing challenges are important and famous enough that a $1 million reward has been offered for each correct answer. These are seven of the greatest unsolved mathematical puzzles of our time. Each has been around a long time, is well known within mathematics, and has resisted attempts at solution by the best mathematical minds in the world.
The most recent of the seven millennium problems is the P vs. NP problem which was developed in 1970. The Riemann hypothesis, which is concerned with the pattern behind prime numbers, remains unsolved despite being first raised in 1859. The Riemann Hypothesis is the one problem mathematicians would like to solve most of all.
As of June 2013, six of the problems remain unsolved. The Poincaré conjecture, the only Millennium Prize Problem to be solved so far, was solved by Grigori Perelman, but he declined the $1 million award in 2010.
The seven problems are:
- P versus NP
The question is whether, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. The former describes the class of problems termed NP, while the latter describes P. The question is whether or not all problems in NP are also in P.
- The Hodge conjecture
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
- The Poincaré conjecture (proved)
In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every two-dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.
- 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.
- 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.
- Navier–Stokes existence and smoothness
The Navier–Stokes equations describe the motion of fluids. 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 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.
So, good luck in solving the Millennium problems. Remember, if you solve one of the remaining six problems, send your solution to Mr. P, and he will take care of forwarding it on to the Clay Mathematics institute for you!! (-:
Here are a couple of jokes about the Riemann Hypothesis:
A mathematician asked a fortune teller, "Tell me, are the proofs to unsolved theorems found in heaven?"
"I have good news and I have some bad news," she replies.
"What's the good news?"
"Not only are all of the proofs revealed in heaven, but they are the most elegant proofs possible!"
"That's awesome! What's the bad news?"
"By this time tomorrow, You'll have an elegant proof of the Riemann Hypothesis."
A mathematician is invited to speak at a math conference. His talk is entitled, "Proof of the Riemann Hypothesis."
When the conference takes place, he speaks about an entirely different topic.
A colleague comes up to him afterwards and says, "Did you find an error in you proof?"
He replies, "No, I never had one."
"So, why did you make this announcement?"
"That's my standard precaution -- in case I die on the way to the conference."
The question is whether, for all problems for which an algorithm can verify a given solution quickly (that is, in polynomial time), an algorithm can also find that solution quickly. The former describes the class of problems termed NP, while the latter describes P. The question is whether or not all problems in NP are also in P.
The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.
In topology, a sphere with a two-dimensional surface is essentially characterized by the fact that it is simply connected. It is also true that every two-dimensional surface which is both compact and simply connected is topologically a sphere. The Poincaré conjecture is that this is also true for spheres with three-dimensional surfaces. The question had long been solved for all dimensions above three. Solving it for three is central to the problem of classifying 3-manifolds.
The Riemann hypothesis is that all nontrivial zeros of the analytical continuation of the Riemann zeta function have a real part of 1/2.
In physics, classical Yang–Mills theory is a generalization of the Maxwell theory of electromagnetism where the chromo-electromagnetic field itself carries charges.
The Navier–Stokes equations describe the motion of fluids. 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 insight into these equations.
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.