And three ways in which mathematics has made some people famous.

1. Carol Vorderman is the best known maths graduate and obtained this position through her position as a star on the game show Countdown.
2. Simon Singh has become famous by writing about cracking codes. To crack one of the well-used code systems would be even better.
3. Solve one of the 7 grand challenges, each of which has a $1,000,000 prize attached. It is this last method that will be described here.

Mathematicians have always thrived on challenging problems, whether derived from a practical application, or pushing theory to its limits. The Greeks had tried to develop solutions of geometrical problems using only a straight edge and compass, but they left three problems that they could not solve:

1. Trisection of an angle, that is to find an angle which is one third the size of a given angle. They knew how to find an angle equal to half the size, but not one third.
2. Find the side of a cube whose volume is twice that of a given cube. The story is that during the time of bad harvests, an oracle had asked for her altar to be "doubled", but when merely doubling the length of a side failed to alleviate the famine, the oracle was again consulted and it became clear that the volume had to be doubled.
3. Find the side of a square whose area is that of a given circle.

These remained unsolved for over 2000 years, and all three were eventually proved to be insoluble by Evariste Galois (1811–1832), a romantic and revolutionary who died in a duel over a lady at the age of 21. He was a brilliant example of what has often been observed that most (but not all) breakthroughs in mathematics are made by people before they are 30.

In 1900, David Hilbert, one of the foremost mathematicians of his day, gave an address at the International congress in Paris, and presented 23 problems that he though would occupy and challenge the best during the 20^th century. All of these have led to great advances as they have been tackled. Two examples from his list are:

1. Fermat's Last Theorem, about which there are a now a number of popular and semi-popular books. It claims that the equation

has no solution for

2^z 3^z

is zero lie on a line in the complex plane.

If this could be shown to be true, then a lot of deep results in the theory of prime numbers will follow.

Another famous problem not on the list was the so-called 4-colour problem. Map makers had found that they never needed more than 4 colours to colour a map. It was quickly shown that any map could be coloured in 5 colours, and that some maps needed 4, but the gap between these two took 150 years to close and when it was finally done about 25 years ago, a computer was used to exhaustively test a very large number of cases. This was the first time a computer has been essential for proving a significant theorem, and we still do not have a more elegant method of demonstrating it. It has effects in graph theory, which is becoming more important with the growth of the Internet.

In 2000, the Clay Institute in Cambridge, Massachusetts, offered a prize of $1,000,000 for a proof of any one of 7 outstanding problems. The solution had to be published in a recognised journal and had to withstand 2 years of scrutiny by the mathematical world. These 7 are:

1. P v NP
2. Hodge conjecture
3. Poincaré conjecture (a submission has been received)
4. Riemann Hypothesis
5. Yang – Mills existence
6. Navier – Stokes existence and smoothness theorem
7. Birch – Swinnerton-Dyer conjecture

All are very hard, and 1 & 6 possibly have the most immediate relevance to practical problems.

This is about the difficulty of solving a problem; are computers better than mathematicians? It may be possible to compute a solution to a particular problem in a reasonable amount of time that increases in a polynomial (P)manner with the complexity, or it may be turn out that it is impossible to do so (NP), and the time increases exponentially, or worse.

To give an example, consider the selection of guests to a party when it is known that certain pairs are incompatible. Suppose that we have 5 friends to invite, named conveniently Annabel, Brian, Colin, Daphne and Edward. We know that Annabel hates both Brian and Daphne; Brian and Daphne hate Colin; and Edward hates Daphne. If we want three guests, who do we invite? It is simple to see that Annabel, Colin and Edward would fit the bill.

But now suppose that our circle of friends expands to 400 and we wish to invite 100 and there are a large number of constraints to satisfy. How long would it take to devise a party list? Setting a computer on the task would entail us trying all possible combinations, where each trial would consist of

A: select a combination of 100 from the 400.

B: check to see whether the combination is a valid possibility.

B is very easy to test for. We just check each of the constraints against the proposed solution – an example of P.

But A is much harder. To make every selection and test it would mean we would have to make 400!/100!300! tests, which works out at about 2.24 x 10⁹⁶. The fastest computers currently available would take about 7.11 x 10⁷⁹ years to complete the task; a number which is somewhat larger than the age of the universe of 15 x 10⁹ years, and even larger than the number of particles in the universe which is presently estimated at ~10⁷⁰. This is a task thought to be NP and we must find a better method if we are to make any progress.

It turns out that there are many problems like this and that if one can be shown to have an easier computational solution, then so will a lot of others.

The factorisation of large numbers is difficult, although testing whether a set of factors is correct is easy – just multiply them up. A fact that is at the heart of modern commercial cryptographic systems used for electronic security. Fame and money awaits anyone who could crack these systems, or prove they are as safe as is assumed now.

Other examples of difficult problems are doing a jigsaw (it is easy to see it is finished, but hard to do); packing a knapsack with various shaped packets; and timetabling – a perennial practical problem at any educational establishment.

The best attack by computers at the moment is by a sort of educated guesswork, but can produce only answers that are thought to be "good enough" in many cases, and not necessarily optimal.

Can we forecast the weather?

The weather of the Earth is caused by the motion of a large fluid around it heated by the Sun. Sometimes, it is complicated and unpredictable; at other times, it seems to be much better behaved and predictable for days ahead. We can write down the equations representing the fluid flow relating the velocity, density and pressure in the fluid, but we have no general solutions mathematically. In fact, we don't know whether they can be solved with general starting conditions, we have only some special case solutions. Essentially the same equations are used in airplane design, car design, blood flow, power station design, environmental modelling and in refrigerator design and so are economically of great importance.

The problem occurs when the flow of the fluid becomes turbulent, a feature noted and commented on by Leonardo da Vinci; and is closely related to chaos.

Professor Budd then exhibited a double pendulum, which is a simple metal bar attached at one end to a fixed point and free to swing through 360°in a plane. At its other end it has attached another bar also free to swing in the same plane. When this was started by holding it vertically above its anchor point and allowed to fall, its motion quickly became chaotic and unpredictable. When the audience was challenged to indicate by a clap when the lower bar had turned over its attachment point for the last time in its motion, it was clear that this was a non-trivial exercise.

He ended on a note of encouragement: if you are young and mathematically inclined, go for it – look at Galois – fame and fortune await.

http://www.claymath.org/Millennium_Prize_Problems

Andy Pepperdine

2003-12-21