‘Wild’ might not be the first word that springs to mind when you think about mathematics. But there is, most definitely, a wild side to this research area, consisting of problems that are extremely elusive and difficult to ‘tame’. Martin Bright works on one of these problems. His aim is to tame the wild side of the Brauer group, a mathematical instrument enabling the study of whole-number solutions to equations.
Unleashing beautiful tools to understand ugly solutions
Almost everyone will have heard of Pythagoras’ theorem in school. In a right-angled triangle – one that has an angle of 90 degrees – the square of the side opposite the right angle is equal to the sum of the squares of the other two sides. Or, to put it in a nice equation: a2 + b2 = c2. In this equation a, b and c can take on a wide range of values. These may be both fractions and whole numbers. One whole-number solution is 32 + 42 = 52 (9 + 16 = 25). Equations that can be solved using whole numbers are called Diophantine equations, after the Greek mathematician Diophantus of Alexandria, who studied these types of equations in the third century AD.
‘If you are a mathematician and you see this, you think to yourself: well, this is interesting. Are there more of these solutions?’ Bright wonders. In the case of Pythagoras’ theorem, the simple answer is: yes. 52 + 122 = 132, for example. But then it gets more complicated. How many of these answers are there? Could you go on to produce new ones forever or is there a finite number? ‘In general, this turns out to be a very difficult problem,’ says Bright. ‘Not so much for Pythagoras’ theorem itself, but for a lot of other equations. For example, you only need to change this famous equation very subtly to get one that has no whole-number solutions at all. Then you start to wonder: if you give me an equation, how can I decide whether or not it has any whole-number solutions? And if it does: can I say something about how many there are?’
This is where the Brauer group comes in. In mathematics, you can associate equations with geometric objects. These objects may be one dimensional (a line), two dimensional (a surface), three dimensional (a volume) or even more. Solutions to Diophantine equations always correspond to points on these objects. The Brauer group describes certain unchangeable properties of these objects. These properties can help you to tell these objects apart. Bright: ‘Say you have two lines: one that is shaped like a circle and one that is shaped like the number eight. One way to identify these shapes is to count the number of holes. We call this number an “invariant”. The Brauer group arises in a similar way. If you have a geometric object, you can attach this invariant to it. In this case, it is not a number, but something slightly more complicated. Using the Brauer group, you can sometimes prove whether or not an equation has whole-number solutions; it tells you where to look for them.’
‘It was the Mathematics Olympiad that really converted me to being interested in pure mathematics.’
Taming the wild
Some elements of the Brauer group are better understood than others. Elements that behave in a much more complicated way than others are said to be part of its ‘wild side’. In his research, Bright was able to tame the wild side of the Brauer group to some extent. ‘Before, we could do some calculations with it, but the solutions it gave were kind of ugly,’ he says. ‘We wondered if we could find some geometric understanding to help us get a grip on how these ugly solutions actually work. We found a tool that was developed in a geometric context which happened, somewhat surprisingly, to be exactly what we needed to understand this wild Brauer group a bit more.’
Some specific Diophantine equations turn out to have applications in cryptography, the study of techniques to secure information. For Bright, though, the joy of working with these equations lies more in developing a fundamental understanding of the mathematics involved. ‘The notions I work with may be very abstract, but they are also very beautiful,’ he says. ‘There are surprisingly simple solutions to what at first glance appear to be very difficult problems. On the other hand, the problem of trying to solve equations is something anybody can relate to. The idea is simple: here you have an equation, now try to find whole-number solutions to it. The tools we use for that purpose end up being very abstract, but they are also very natural and beautiful to work with.’
In high school, mathematics was not the only subject Martin Bright (Bedford, 1976) was interested in. He also loved practical subjects like physics and engineering. ‘It was the Mathematics Olympiad that really converted me to being interested in pure mathematics,’ he says. ‘You don’t actually know what “pure” mathematics is until you arrive at university, though. Here, maths is suddenly a lot more abstract and requires a lot more independent thinking. I thoroughly enjoy that.’ All this culminated in his current appointment as assistant professor at the Mathematical Institute at Leiden University. Here, he teaches courses like Linear Algebra and works on research in the area of Diophantine equations and arithmetic geometry.