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Rational points and new dimensions

How can you solve equations that define not ‘just’ curves, but also two-dimension surfaces or even higher-dimensional objects? That’s the big question that mathematician Martin Bright and his team will be trying to answer. They’ve received a NWO Science-XL grant of 2.8 million euros.

‘Ever since Descartes gave us x- and y-coordinates, it has been possible to use algebraic equations to describe geometric objects,’ says Bright. ‘When at school you write down the equation of a straight line or circle, that’s what you’re doing. However, in the past century the methods for applying “geometric” constructions to study these equations have vastly broadened, leading to many breakthroughs. So far, most progress has been around equations that define curves, like the equation of a circle. Our project is about equations that define surfaces or even higher-dimensional objects.’

‘Progress on the big ideas in maths often take place by studying very specific smaller problems’

Building up a research community

The research is divided into nine projects. According to Bright, progress on big ideas in maths often takes place by studying very specific smaller problems. ‘With this grant we will appoint a number of PhDs and postdocs,’ he says. ‘Their work will serve as important stepping-stones on the way to our ultimate goal. All of us will get together regularly to exchange ideas and share new developments.’ This way, the team also aims to build up an internationally leading Dutch research community.

Example of a multidimensional object that Bright’s team uses to study equations.

Three research directions

Bright and his team will study in three different directions: extending our knowledge about curves to higher-dimensional objects; comparing the world of rational numbers to that of modular arithmetic (see box) and looking at concepts between whole-number solutions and rational solutions, also known as fractions. ‘This last direction concerns so-called Campana points, a recent development. They are a generalisation of whole-number solutions on the one hand, and fraction solutions on the other hand. Not much is understood about them yet, but they offer promising insights.’

One equation, two different directions

Bright: ‘In the second direction we use tools from modular arithmetic. This is the kind of calculation you use when adding up clock times. For example 10 + 3 = 1 when you’re adding up on a clock face. .  Solving equations here is much easier than when using ‘normal’ whole numbers or fractions but is closely related and gives information on the more difficult problem. In modern mathematics we can even do geometry in the setting of modular arithmetic, and we can compare the geometry of the same equation in the two different settings.’

Blue-sky research

The research will mostly be fundamental. Bright: ‘One of the projects, however, has indirect applications in theoretical physics, namely string theory. Another project is very specifically concerned with the construction of better error-correcting codes, which are used all the time to ensure the integrity of data in transmission or storage.’ The rest of the results will help deepening our understanding of both the fields of algebra and geometry. And, hopefully, contribute to the breakthroughs of the future.

NWO Science-XL

NWO believes curiosity-driven, fundamental research is necessary for innovations that make society economically successful and socially resilient. The ENW XL-grant gives researchers the opportunity and freedom to start, strengthen or expand excellent, challenging and innovative lines of research. The NWO Domain Board Science has approved 21 grant applications in the Open Competition Domain Science-XL programme. The topics vary from studying the coronavirus Achilles’ heel to learning about danger through the experiences of others. Together, the projects have been granted about 60 million euros.

The NWO XL scheme is for consortia of scientists involving more than one institution. Martin Bright is the principal applicant on behalf of the Mathematical Institute at Leiden University. The other mathematicians are: Ronald van Luijk, also from Leiden; Valentijn Karemaker and Marta Pieropan from the University of Utrecht and Steffen Müller and Cecília Salgado from the University of Groningen.

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