Back in the year 2000, while everyone was getting over the fact that we survived the technology rapture, the Clay Mathematics Institute announced the Millennium Prize problems.

The problems are so important that the institute offered a reward of $1 million (R15 million) to anyone who could solve any one of the seven problems.

In 2006, Grigori Perelman famously solved the Poincare Conjecture, and then turned down both the million dollar prize and the coveted Fields Medal. The remaining six problems are still unsolved.

If you have no idea what the Poincare Conjecture is, you’re not alone. There’s a reason I did a humanities degree.

If, however, you think you might have what it takes to solve six of the worlds most difficult maths problems – and score some big bucks for doing it – read on, because here they are:

P vs NP

In the world of maths and computer science, there are a lot of problems that we can program a computer to solve ‘quickly’.

These problems can be solved in ‘polynomial time’, abbreviated as ‘P’. Polynomial time refers to the number of steps it takes to add two numbers or to sort a list, which grows with the size of the numbers or the length of the list.

But there’s another group of problems for which it’s easy to check whether or not a possible solution to the problem is correct, but we don’t know how to efficiently find a solution. Finding the prime factors of a large number is such a problem — if I have a list of possible factors, I can multiply them together and see if I get back my original number. But there is no known way to quickly find the factors of an arbitrary large number. Indeed, the security of the internet relies on this fact.

Problems, where the solution can be checked quickly, are said to be solvable in ‘ non-deterministic polynomial time’, or ‘NP’.

Any problem in P is automatically in NP — if I can solve a problem quickly, I can just as quickly check a possible solution simply by actually solving the problem and seeing if the answer matches my possible solution. The essence of the P vs NP question is whether or not the reverse is true: If I have an efficient way to check solutions to a problem, is there an efficient way to actually find those solutions?

Most mathematicians and computer scientists believe the answer is no. An algorithm that could solve NP problems in polynomial time would have mind-blowing implications throughout most of maths, science, and technology, and those implications are so out-of-this-world that they suggest reason to doubt that this is possible.

Even proving that the algorithm doesn’t exist would be difficult. We currently don’t have a deep enough understanding of the nature of information and computation to make a definitive statement about the existence of the algorithm.

The Navier-Stokes equations

The Navier-Stokes equations are the fluid dynamics version of Newton’s three laws of motion. They describe how the flow of a liquid or a gas would change under various conditions.

Just as Newton’s second law gives a description of how an object’s velocity will change under the influence of an outside force, the Navier-Stokes equations describe how the speed of a fluid’s flow will change under internal forces like pressure and viscosity, as well as outside forces like gravity.

The Navier-Stokes equations are a system of differential equations. Differential equations describe how a particular quantity changes over time, given some initial starting conditions, and they are useful in describing all sorts of physical systems. In the case of the Navier-Stokes equations, we start with some initial fluid flow, and the differential equations describe how that flow evolves.

Differential equations have been used to describe a number of things like the vibration of a guitar string, or the movement of heat between a hot object and a cold object. The Navier-Stokes equations are harder.

Mathematically, the tools used to solve other differential equations have not proven as useful here. Physically, fluids can exhibit chaotic and turbulent behavior: Smoke coming off a candle or cigarette tends to initially flow smoothly and predictably, but quickly devolves into unpredictable vortices and whorls.

It’s possible that this kind of turbulent and chaotic behaviour means that the Navier-Stokes equations can’t actually be solved exactly in all cases. It might be possible to construct some idealized mathematical fluid that, following the equations, eventually becomes infinitely turbulent.

Like we said, you are earning that million dollar prize if you crack one of these.

Yang-Mills theory and the quantum mass gap

Yang-Mills theory, which describes the quantum behaviour of electromagnetism and the weak and strong nuclear forces in terms of mathematical structures that arise in studying geometric symmetries, is one of the major underpinnings of modern quantum mechanics.

The predictions of Yang-Mills theory have been verified by countless experiments, and the theory is an important part of our understanding of how atoms are put together.

Despite that physical success, the theoretical mathematical underpinnings of the theory remain unclear. One particular problem of interest is the “mass gap,” which requires that certain subatomic particles that are in some ways analogous to massless photons instead actually have a positive mass. The mass gap is an important part of why nuclear forces are extremely strong relative to electromagnetism and gravity, but have extremely short ranges.

The Millenium Prize problem asks you to show a general mathematical theory behind the physical Yang-Mills theory, and provide a mathematical explanation for the mass gap.

The Riemann Hypothesis

By the 19th century, mathematicians had discovered various formulas that give us an approximate idea of the average distance between prime numbers.

What remains unknown, however, is how close to that average the true distribution of primes stays — that is, whether there are parts of the number line where there are “too many” or “too few” primes according to those average formulas.

The Riemann Hypothesis limits that possibility by establishing bounds on how far from average the distribution of prime numbers can stray. The hypothesis is equivalent to, and usually stated in terms of, whether or not the solutions to an equation based on a mathematical construct called the “Riemann zeta function” all lie along a particular line in the complex number plane. Indeed, the study of functions like the zeta function has become its own area of mathematical interest, making the Riemann Hypothesis and related problems all the more important.

The Millenial Prize Problem asks you to definitively prove or disprove the Riemann Hypothesis.

The Birch and Swinnerton-Dyer conjecture

Polynomial equations are some of the oldest and broadest fields of mathematical study. Remember Pythagoras?

In recent years, algebraists have particularly studied elliptic curves, which are defined by a particular type of diophantine equation. These curves have important applications in number theory and cryptography, and finding whole-number or rational solutions to them is a major area of study.

One of the most stunning mathematical developments of the last few decades was Andrew Wiles’ proof of the classic Fermat’s Last Theorem, stating that higher-power versions of Pythagorean triples don’t exist. Wiles’ proof of that theorem was a consequence of a broader development of the theory of elliptic curves.

The Birch and Swinnerton-Dyer conjecture provides an extra set of analytical tools in understanding the solutions to equations defined by elliptic curves.

I’m sure they are speaking English, but I am no longer sure.

The Hodge conjecture

Algebraic geometry is the study of the higher-dimensional shapes that can be defined algebraically as the solution sets to algebraic equations.

As an extremely simple example, you may recall from high school algebra that the equation y = x2 results in a parabolic curve when the solutions to that equation are drawn out on a piece of graph paper. Algebraic geometry deals with the higher-dimensional analogues of that kind of curve when one considers systems of multiple equations, equations with more variables, and equations over the complex number plane, rather than the real numbers.

The 20th century saw a flourishing of sophisticated techniques to understand the curves, surfaces, and hyper-surfaces that are the subjects of algebraic geometry. The difficult-to-imagine shapes can be made more tractable through complicated computational tools.

The Hodge conjecture claims that certain types of geometric structures have a useful algebraic counterpart that can be used to better classify and study these shapes.

I don’t know about you, but I’m going to go and lie down in a dark room with a cloth over my eyes.

[source:businessinsider]