The mathematician who doesn’t like mental arithmetic

Jakob Björnberg’s office looks just like you would expect a mathematician’s office to look. The room is dominated by a whiteboard covered in Greek symbols and equations. In response to the question of what it all means, he replies enthusiastically:

“This project is about the Heisenberg model. What I find so cool is that it combines probability theory and representation theory, which you might say is part of algebra. To understand the model, we can use both theories.”

Models related to phase transitions

Although he simplifies his explanation as far as possible, it’s still not easy to keep up with what it’s about. Jakob Björnberg’s work is on probability theory, which he applies to physical problems. In particular, he is interested in phase transition, such as when something freezes and transitions from a liquid to a solid form.

“I mostly look at models that are related to how phase transitions in magnets occur. For example, iron can become magnetic at a lower temperature. And this is not a gradual transition – it occurs at a critical temperature, just like when water freezes,” he says.

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This may not sound like a difficult phenomenon to explain in mathematical terms, but the fact is that we do not fully understand how it happens. For example, a piece of iron can be described mathematically by letting each atom be a point in a grid, a system of mathematical points in other words, where each point can be positive or negative. During a phase transition, which requires energy, the iron will then randomly select a state that lies close to the one that minimises energy consumption.

“The thing is that a piece of iron might contain 10²³ atoms, meaning the number of particles is basically infinite. It’s fairly easy to formulate the problem in terms of probability theory, but difficult to solve,” he says.

Mathematicians and physicists working together

At the same time, we know that phase transitions actually do occur, and it might seem a minor matter to get all the mathematical details to line up. But physics is perhaps the most basic of the sciences, and mathematics is the language of science. Basic research in mathematics in relation to questions in physics therefore has to be one of the most fundamental things you can do as a researcher, and there is always an interplay between mathematicians and physicists.

“In some cases, the physicists have worked out a lot of things, and then the mathematicians come along a few years later and fill in the details. In other cases, it is the mathematicians who develop new methods that the physicists then use to do new calculations that they didn’t know how to do before,” says Jakob Björnberg.

Mathematics seems to be fundamental to how reality works

When it comes to basic research, it can sometimes be tough to explain to those not involved in it why society should invest resources in things where we don’t know what the returns will be. Jakob Björnberg has no problem with this, and especially not when it comes to mathematics.

“I think basic research is really fascinating, and increasing our understanding of mathematics is so fundamental. When you think about issues like that, you soon arrive at the question ‘why do we do anything at all?’ What is humanity contributing to? We don’t know why, but mathematics seems to be fundamental to how reality works,” he says.

Struggled with maths until high school

In fact, it was actually the abstract and somewhat elusive aspect that persuaded him to become a mathematician to begin with. He comes from a family full of social scientists, being the child of a professor of sociology and a judge, and with two sisters who have both chosen the social sciences path.

“Despite the fact that I’m the member of the family who works in the natural sciences, I’m probably also the worst at mental arithmetic. I struggled quite a lot with maths until high school. It was only when it got more abstract that I began to find it fun,” he says.

In fact, he hated maths at the primary and intermediate levels of school. His view is that at those levels it’s too narrow and focuses solely on a small part of the subject.

“What we did was to study one special case – real numbers – and had it drilled into us exactly how they worked. It was only when maths began to broaden out that I got interested,” he says.

 

THE HEISENBERG AND ISING MODELS

THEN: The Heisenberg and Ising models derive from the early twentieth century when thermodynamics was new. It was known that phase transitions happened, but there was no way of describing them mathematically. Ising’s doctoral supervisor, Wilhelm Lenz, formulated the Ising model, and Ising showed that it was impossible to use it to describe phase transitions in a one-dimensional material. Werner Heisenberg introduced an alternative model that incorporated quantum theory, and it was later proven that the simpler Ising model works if you have two dimensions.

NOW: In purely mathematical terms, no one has yet shown how a phase transition with three dimensions in the Heisenberg model can occur. This is something that mathematician Jakob Björnberg is working on.

THE FUTURE: The holy grail is to prove mathematically how a phase transition can occur in the three dimensional Heisenberg model. If we can’t show this in concrete terms, then something is missing in our understanding of something fundamental. In the words of Jakob Björnberg, the real challenge is to “find the missing tools that will confirm what we believe to be true”.