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What are the Seven Million Dollar Millenium Prize Math Problems?

Updated on July 7, 2020
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I hold both a bachelor's and a master's degree in applied mathematics.

The Millennium Prize Problems

On the 24th of May 2000 the Clay Mathematics Institute named seven unsolved problems that became known as the seven Millennium Prize Problems. The Clay Mathematics Institute is a non-profit organisation based in Petersborough, New Hampshire. It was formed in 1998 by Landon T. Clay and has a postdoctoral program and organizes various conferences and workshops in the field of mathematics.

However, the millennium problems are what they are known for. Solving one of these seven problems correctly will earn the person who does it one million dollars.

As of 2020, only the Poincaré Conjecture was solved. This was done by Grigori Perelman from Russia in 2003. However, he did not claim his prize because he said that his contribution to proving the problem was not larger than the contribution of Richard S. Hamilton. Hamilton had sent in a proof himself earlier for the Poincaré conjecture, but it had some flaws. What Perelman basically did was adapt the proof to get rid of these flaws.

The other six problems are still unsolved. Many mathematicians have tried. Also, some thought that they had found the solution to one of the problems. But every time this happened a mistake was found.

Grigori Perelman
Grigori Perelman | Source

The Seven Problems That Could Earn You A Million Dollars

The Poincaré Conjecture

This is the problem that was solved. The problem statement was as follows:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

This is a very abstract mathematical statement that was conjectured by Henri Poincaré in 1904. In fact, the conjecture of Poincaré spoke about n-manifolds and n-spheres, but it was already proven to be true for all n except for n is 3 by the time the millennium prize problems were chosen.

Easily said, the theorem states that any simply connected 3-dimensional topological space can be transformed to a sphere and back without creating or removing any holes.

P = NP

This problem is about the complexity of problems. The class P is the class of decision problems for which the solution can be found in polynomial time. A decision problem is a problem that can be answered with yes or no. The class NP is the class of decision problems for which a yes-instance can be verified in polynomial time. So clearly everything in P is also in NP, since we can just solve the problem in polynomial time to verify if the answer is yes. However, the question is whether there are problems that are in NP, but not in P. This seems to be the case. There are numerous problems for which we don't know a polynomial time algorithm. These problems are called NP-Hard, which means: at least as difficult as any problem in NP. For none of these problems a polynomial time algorithm is known, and if one can find a polynomial time algorithm for one of them, we know that there is a polynomial time algorithm for all of them. However, most mathematicians believe that P is not equal to NP.

The Hodge Conjecture

The Hodge Conjecture is formulated as follows:

Let X be a non-singular complex projective manifold. Then every Hodge class on X is a linear combination with rational coefficients of the cohomology classes of complex subvarieties of X.

It was formulated by William Hodge in the first half of the twentieth century. The first time it really got attention was at the 1950 International Congress of Mathematics in Cambridge Massachussets. Again, this is a very abstract statement which is very difficult to explain when you are not an expert in the field of algebra.

Georg Riemann
Georg Riemann | Source

The Riemann Hypothesis

The Riemann Hypothesis states the following:

The real part of every non-trivial zero of the Riemann zeta function is 1/2.

If this is true than this has important consequences for finding prime numbers. Riemann found a formula to determine the number of primes smaller or equal than some number x. This is related to this real part of zeros of the Riemann Zeta function. Being able to calculate this would make have considerable impact on cryptology and other number theory related topics.

Yang-Mills Existence and Mass Gap

To claim the million dollar prize you will have to do the following:

Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R^4 and has a mass gap Δ > 0.

Yang-Mills theory is an important concept in particle physics. If it would exist on R^4 with mass gap larger than zero it would be the simplest non-trivial quantum field theory in four dimensions and therefore easier than the existing theories.

Birch and Swinnerton-Dyer conjecture

The statement of the Birch and Swinnerton-Dyer conjecture is as follows:

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.

Here L(E,s) is the Hasse Weil L-function. Again this is a very abstract theorem that requires very advanced knowledge about analytical number theory to even understand what is asked to prove. The theorem is proven for some special cases, but not in general.

The Navier-Stokes Equations

Prove, or give a counterexample:

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.

The Navier-Stokes equation describe the motions of fluids. With numerical tools we are a able to get good approximations of the solutions of these equations, however we don't have an analytical solution. We do not even know if an analytical solution exists. The numerical solutions show turbulence, which means that chaotic changes in pressure and velocity occur. Turbulence is not understand well in science. Therefore finding an analytical solution could be very helpful to get better understanding of it.

Navier-Stokes Equations
Navier-Stokes Equations | Source

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