In recent years, physics has been swept by ideas from a branch of mathematics called topology. Topology is the study of objects that deform continuously without tearing, for example through stretching or twisting. But it is now proving crucial to understanding the shapes of quantum waves formed by the electrons inside matter. These waves can form shapes such as vortices, knots and braids that give materials a variety of exotic properties. In 1983, Barry Simon was the first person to make the link between strange phenomena in materials and topology.

Simon’s work explained the quantum Hall effect, first described by German physicist Klaus von Klitzing 40 years ago this month¹. Von Klitzing had seen electrons behave in a surprisingly orderly way when confined to a 2D layer of a semiconductor kept just above absolute zero and exposed to a strong magnetic field. When the voltage on the semiconductor ramped up, the electrical resistance did not change continuously. Instead, it jumped between values that were predictable. And this wasn’t affected by temperature fluctuations, say, or by impurities in the material.

Von Klitzing won a Nobel prize for the effect’s discovery in 1985. But it took several breakthroughs by theoretical physicists to begin to understand the phenomenon. And it took Simon — a mathematical physicist who uses mathematical tools to solve theoretical problems that emerge from nature — alongside collaborators, to recognize that equations created to describe the quantum Hall effect were a manifestation of topology^2,3. It was topology that was making the material’s resistance robust to small changes, allowing it to change in only discrete jumps.

Researchers have since brought ever-more-sophisticated ideas from topology into studies of matter and used them to predict a plethora of physical phenomena. Many of these have subsequently been found in the lab⁴, and physicists hope that one day they might have applications in fields such as quantum computing.

Nature caught up with Simon, who is at the California Institute of Technology in Pasadena, to ask how it all started, and about the relationship between mathematics and physics.

What made you think there was a connection between the quantum Hall effect and topology?

The thing that’s surprising about the quantum Hall effect is that something that appears to be continuous is quantized — it comes in discrete units. When I saw [theoretical physicist] David Thouless’s formula, I immediately thought of the topological concept of homotopy.

The simplest example to think of is how a circle can continuously map onto itself. In the case of the circle to a circle, there is a key issue: one circle winds around the other an integer number of times. And if you continuously deform the map, you’re not going to change that number.

So in your papers you showed that this topological effect, called a ‘winding number’, made the resistance jump between discrete values. Did you imagine that the discovery would be so successful?

I knew it would make a splash because it would appeal to high-energy physicists, who were already accustomed to ideas from topology. I didn’t realize it would have this long-lasting impact in solid-state physics.

As a mathematician, do you think in a different way from theoretical physicists? It seems that often, the two communities look at the same problems but have different standards for what constitutes a rigorous solution.

There is a sharp dividing line between physicists and mathematicians: whether you really ‘prove’ things in the mathematical sense of proving things. It’s the difference between demonstration and proof. There really is a very different style.

How would you describe the relationship between the two communities?

It really depends on the subfields. The condensed-matter physicists were so used to being looked down upon by the high-energy physics community — particle physicist Murray Gell-Mann described condensed matter as “squalid-state physics” — that they didn’t look down on other people. There’s a tradition among high-energy physicists and string theorists, that really goes back to Enrico Fermi, that’s not very positive towards maths. Sometimes there’s a lack of mutual respect.

Is that bad for business, in the sense of hampering research?

It’s certainly bad for life — obviously, it makes life less pleasant. Is it bad for business? Would science progress more without it? I don’t know. To the extent that these cultural things prevent collaboration, it’s very bad. Although sometimes it’s not clear, even if people were more accepting of each other, that they could successfully collaborate.

Have interactions between the two communities improved since the 1980s?

There are still separate camps, but the landscape has changed enormously. There is much more attention in both directions now than there was 40 years ago. It amazes me what has happened to the use of topological ideas in condensed-matter physics. It’s really, really striking.