How abstract maths can help physicists better understand the universe

Dr Brown does not believe in artificial barriers. Indeed, the focus of his ERC project on “Periods in Algebraic Geometry and Physics” (PAGAP) has been very much on taking abstract mathematical ideas and applying them to particle physics. This branch of physics has attracted a great deal of public exposure thanks to the discovery in 2012 of the Higgs boson (helping to confirm the standard model of how the universe works).

“The Collider works by smashing particles together and looking at the pieces,” explains Dr Brown. “This involves a huge practical effort, but you also need theory. You need to predict accurately what you expect to see, using the known laws of physics, and then run experiments to find any differences.”

The mathematics needed for these calculations is incredibly complex, and this is where Dr Brown comes in. By coincidence, the numbers required were in fact discovered some 300 years ago by Leonhard Euler, a Swiss mathematician and physicist, and prized for their aesthetic appeal rather than their utility. “But it turns out that these are exactly the numbers needed to describe what we think is going on,” he says. From these numbers – called zeta-values – Dr Brown has developed an entire theory that enables theoretical predictions to be done faster and more effectively.

“While it is unusual to combine both the theoretical and the practical in one project, this is what makes PAGAP interesting” says Dr Brown. “It also reflects a change in mathematics in its relationship to other sciences.”

The changing face of mathematics



Indeed, Dr Brown, who was awarded an ERC grant of just over EUR 1 million, is helping to reduce the historical division between mathematicians and physicists that still to some extent persists. “This harks back to the days of Newton, when a lot of these guys would have made no distinction between the two”, says Dr Brown. The recent cross-fertilisation between maths and physics can also be seen in the fact that the term ‘quantum’ – which usually refers to sub-atomic physics – is now also applied to several branches of mathematics.

Dr Brown notes that Europe has recently seen an impressive increase in the number of mathematicians. The caricature of the solitary mathematician scribbling equations on a blackboard to an empty room is no longer true; this is a field bursting with new ideas.

Maths can also widen horizons. Dr Brown’s own career has taken him from the UK, where he grew up and where he completed his first degree, to the École Normale Supérieure (ENS) in Paris, where he obtained his PhD. “My mother is French, and I always wanted to go to France,” says Dr Brown. “By happy coincidence, Paris was the best place to study what I wanted to do.” He traces his initial interest to the lack of decent computers in school, which obliged him to learn how to programme at the age of 11 in order to play games.

While the passing of time introduces new ideas, fads and fashions, there is something irreducible about mathematics. Without a basic understanding, researchers cannot do anything quantitative. This brings us back to Dr Brown’s point about the futility of dividing maths into the purely abstract and the purely practical. Through this ERC project, he has been able to show that even the most obscure theoretical models from centuries past can help modern scientists answer some of our most pressing questions about the very nature of the universe.

Project details



Research area: Mathematics

Principal investigator: Dr Francis Brown

Host institution: Institut des Hautes Etudes Scientifiques (IHÉS), France

ERC project: Periods in Algebraic Geometry and Physics (PAGAP)

ERC call: Starting grant 2010

ERC funding: € 1 million for five years

Links:

Researcher’s webpage

Dr Francis Brown: Video on aspects of geometric algebra