Who Wants to be a Millionaire?

27 September 2011
Who Wants to be a Millionaire?
There are many careers that mathematicians can go into, some of which can be very lucrative.
However if you really want to hit the mathematical big time, and get rich in the process, solving one of the Millennium Problems would be a great place to start…

The Millennium Problems are a collection of mathematical problems to which a $1 million prize is offered for finding a solution. The prizes were offered in 2000 by the Clay Mathematics Institute, and as at the time of this blog, six of the seven problems remain unsolved.

1.    P versus NP problem
This is a problem based in theoretical computer science and asks whether, for all problems that a computer can verify a given solution quickly (NP), can it also find that solution quickly (P)? Most mathematicians and computer scientists expect that P does not equal NP, but a proof has been elusive since Stephen Cook first posed the problem in 1971.

2.    Hodge conjecture
This is a problem based in algebraic geometry and was formulated by the Scottish mathematician William Vallance Douglas Hodge as a result of a work in between 1930 and 1940.

3.    Poincaré conjecture
This question asks whether a three-dimensional sphere (the set of points in 4-D space at a distance of 1 unit from the origin) is simply connected (can you shrink a rubber band from the surface down to a point without it tearing or leaving the surface of the sphere?).
The problem was formulated in 1904 by Henri Poincare, but if you can find a solution to this one unfortunately you are too late! It was solved by Grigoriy Perelman in 2002-3, although he declined the prize.

4.    Riemann hypothesis
The Riemann Hypothesis is a conjecture about a function called the Riemann zeta function and where it crosses the real axis.
The hypothesis implies results about the distribution of prime numbers and was posed by Bernhard Riemann in 1859.

5.    Yang–Mills existence and mass gap
This is a problem based in mathematical physics and uses geometry to describe the behaviour of elementary particles. The problem begins with proving the Yang-Mills theory, first introduced almost 50 years ago.

6.    Navier–Stokes existence and smoothness
The Navier-Stokes equations were first written down in the 19th century and attempt to describe the behaviour of waves (for example in lakes and in air currents)
The problem is to add to understanding of the equations and the insights they can give.

7.    Birch and Swinnerton-Dyer conjecture
This problem is based in number theory and is related to the solvability of alegebraic equations of a particular form. It was first posed in the 1960s.
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