The year 2000 brought about a whirlwind of changes and chaos. Not only was it the subject of the Y2K concerns (the fears that the computers would not be able to properly switch from 1999 to 2000 due to the rollover from 99 to 00), but it was designated as the International Year For The Culture Of Peace (by United Nations) and the World Mathematics Year. In order to celebrate mathematics in the New Year, the Clay Mathematics Institution of Cambridge (CMI) established seven prize problems, known as the Millennium Prize Problems. The prize problems were chosen by the founding Scientific Advisory Board (SAB) of CMI and they were announced at a meeting in Paris, held on May 24, 2000. The Board of Directors at CMI has actually set aside 7 million dollars for prize funds, and to whoever can be the first to solve (prove the statement true or false) a problem will receive a million dollars. The seven problems are: 1. P versus NP problem: Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly? 2. Hodge Conjecture: Let X be a projective complex manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X. 3. Poincaré Conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. 4. Riemann Hypothesis: The distribution of such prime numbers among all natural numbers does follow any regular pattern 5. Yang-Mills existence and mass gap Proving that the quantum field theory underlying the Standard Model of particle physics, called Yang–Mills theory, satisfies the standard of rigor that characterizes contemporary mathematical physics, i.e. constructive quantum field theory. The winner must also prove that the mass of the smallest particle predicted by the theory be strictly positive, i.e., the theory must have a mass gap. 6. Navier-Stokes existence and smoothness In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations. 7. Birch and Swinnerton-Dyer conjecture It is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K. As of January 2011, six problems remain unsolved. The one problem that has been solved is the Poincaré Conjecture, by Grigori Yakovlevich Perelman. It was announced on March 18, 2010 that he had met the criteria to receive the first Millennium Prize Problem award, but he turned it down because he believed that his contribution was no greater than that of Richard Hamilton, who first suggested a solution to the problem. So I’ll leave you with that- there are six problems left, six million dollars left and six billion people in the world that can give it a shot. So go ahead, try one. http://www.youtube.com/watch?v=9sfkw8IWkl0 A video about the Poicare Conjecture (the only problem that has been proven)
Millennium Prize Problems
The year 2000 brought about a whirlwind of changes and chaos. Not only was it the subject of the Y2K concerns (the fears that the computers would not be able to properly switch from 1999 to 2000 due to the rollover from 99 to 00), but it was designated as the International Year For The Culture Of Peace (by United Nations) and the World Mathematics Year.
In order to celebrate mathematics in the New Year, the Clay Mathematics Institution of Cambridge (CMI) established seven prize problems, known as the Millennium Prize Problems. The prize problems were chosen by the founding Scientific Advisory Board (SAB) of CMI and they were announced at a meeting in Paris, held on May 24, 2000. The Board of Directors at CMI has actually set aside 7 million dollars for prize funds, and to whoever can be the first to solve (prove the statement true or false) a problem will receive a million dollars. The seven problems are:
1. P versus NP problem:
Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also be computed quickly?
2. Hodge Conjecture:
Let X be a projective complex manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.
3. Poincaré Conjecture:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
4. Riemann Hypothesis:
The distribution of such prime numbers among all natural numbers does follow any regular pattern
5. Yang-Mills existence and mass gap
Proving that the quantum field theory underlying the Standard Model of particle physics, called Yang–Mills theory, satisfies the standard of rigor that characterizes contemporary mathematical physics, i.e. constructive quantum field theory. The winner must also prove that the mass of the smallest particle predicted by the theory be strictly positive, i.e., the theory must have a mass gap.
6. Navier-Stokes existence and smoothness
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.
7. Birch and Swinnerton-Dyer conjecture
It is conjectured that the rank of the abelian group E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.
As of January 2011, six problems remain unsolved. The one problem that has been solved is the Poincaré Conjecture, by Grigori Yakovlevich Perelman. It was announced on March 18, 2010 that he had met the criteria to receive the first Millennium Prize Problem award, but he turned it down because he believed that his contribution was no greater than that of Richard Hamilton, who first suggested a solution to the problem.
So I’ll leave you with that- there are six problems left, six million dollars left and six billion people in the world that can give it a shot. So go ahead, try one.
http://www.youtube.com/watch?v=9sfkw8IWkl0
A video about the Poicare Conjecture (the only problem that has been proven)