The million dollar proof
PhD student Raymond van Bommel decided to pore upon one of the most complicated mathematical problems of our time; the solution of it is worth 1 million dollars! Van Bommel did not get that far yet. However, he wrote his own computer program to make new calculations. ‘It could just be that someone else gets a new insight through these calculations'. Van Bommel will be awarded his PhD title on 31 October.
Van Bommel's research concerns the so-called 'conjecture of Birch and Swinnerton-Dyer'. 'This suspicion links two mathematical phenomena that apparently have nothing to do with each other,' he says. It describes a deep mathematical relationship between number theoretic and geometrical properties of elliptic curves. With the help of these curves, cryptographs can quickly and easily encrypt information. They use the curves for example for securing mobile phones.
The conjecture came about through computer stimulations, but has never been fully proven. According to Van Bommel, it is one of the most important conjectures in his field. If it is true and can be proven, it therefore will have many consequences. For example, for a description of the possible lateral lengths of a right triangle from which all side lengths are integers.
Except perhaps for future applications in cryptography, it is still difficult to say what the Birch and Swinnterton-Dyer conjecture can be used for later. 'This is often a difficult question for mathematicians. This research and many other research does not exist with the aim to come up with direct applications. The goal is the development of general mathematical theory. The applications often appear much later. For example, a lot of new physics is based on mathematics that was invented 50, 100 or 200 years ago.'
In 2000, the Clay Mathematics Institute drew up the 'millennium problems': a list of the seven most important, unsolved problems in mathematics. For the solution of each of these problems, the institute awards an amount of one million dollars. Of the seven millennium problems, meanwhile one has now been solved, the other six issues remain unanswered until today. The Birch and Swinnterton – Dyer conjecture is also on this list. 'For the suspicion of Birch and Swinnerton - Dyer, a number of special cases have been solved so far, but there is currently no prospect of a solution to the general problem,' says Van Bommel.
To further unravel the conjecture, Van Bommel wrote a new computer program with which he could test various new parts of it. 'With my computer program, mathematicians can check the conjecture for a whole new series of curves, which are a lot more complicated than the curves for which this was previously possible.' According to Van Bommel it could therefore be possible that someone else could use this calculations to gain new insights and that it brings mathematicians closer to the final proof of this conjecture. ‘It is often the case in mathematics that new examples help you to come up with new theory’.
Although Van Bommel is trained as a mathematician, he also has a passion for programming. He successfully participated in programming competitions several times. ‘Because I had the mathematical knowledge to understand the theory behind all of this, as well as the programming knowledge to write a program, this project suited my interests very well.’ The PhD student even discovered a few mistakes in the existing code: ‘Because I had my own program next to it, I saw that something was probably wrong. This turned out to be the case indeed. These mistakes have now largely been rectified.’ Unfortunately, the million dollars have not yet been won. 'There is still much more to be calculated and proved for this suspicion.'