The shapes of everyday things, like a tangle of string or a coffee mug, don't seem to require sophisticated math to understand. But there's an entire field of study, called topology, that examines how different shapes are related. Amazingly, some of this same math applies to quantum behavior that emerges near absolute zero. And this year's physics prize goes to three researchers that identified this relationship.
The basic concepts of topology are deceptively simple. Let's say you have a tangle of string. If you find the two ends and pull to remove any slack, how many knots will end up in the string? And how many different configurations of tangles will produce the same number of knots? Answering those questions mathematically is where topology comes in.
Similar math can be applied to three-dimensional items. For example, a bowl shape can be transformed into a variety of other different shapes, but not a coffee mug, since the latter has a single hole in it. Neither of those can be transformed into a traditional pretzel, which has three. Again, topology can help identify equivalent shapes and the means of transforming one into another.
All of that may seem perfectly reasonable—but also perfectly unrelated to the quantum phenomena that occur near absolute zero. It's precisely because those relationships are utterly unobvious that their recognition was a major breakthrough, one meriting a Nobel Prize.
The math applies to a variety of systems, but many of them share something in common: they involve a very thin layer of material. These thin layers confine phenomena to what's essentially two dimensions and confines things like electrons and magnetic fields, producing some unusual effects. One of these is called the Quantum Hall effect, where magnetic fields and electrons pair up, resulting in a situation where the possible levels of current being carried are quantized; they are all integer multiples of a base value.
David J. Thouless of the University of Washington, Duncan Haldane at Princeton, and Michael Kosterlitz at Brown made key insights that related this sort of behavior to topology. At low temperatures, pairs of vortexes (a vortex and an anti-vortex) can spontaneously form in a very thin surface. As long as the pairs don't get far apart, the energy difference between this setup and a vortex-free state is minimal. Similar work can then explain lots of quantum behavior beyond the Quantum Hall effect—superconductivity, for example, is dependent upon electrons forming pairs.
The theory also explains the phase transitions that occur as temperatures increase. This process provides enough energy to allow the pairs to separate, bringing an end to many of the quantum phenomena at higher temperatures.
It's really hard to describe the full impact of this work because it touched on so many phenomena. In some cases, it provided a theoretical understanding of things we'd been observing for decades. In others, further developments of the theoretical ideas led to predictions that were later confirmed by experiments, such as the existence of equivalent phenomena in three-dimensional materials. The theory and the math behind it have been applied to exotic systems like topological quantum liquids and topological insulators.
In many ways, that makes the achievement much harder to understand than something like the theory behind the Higgs boson, which has a specific consequence and led to a singular discovery. Here, the three researchers made developments that influenced countless fields. Not bad for working with the math that describes a tangle of string.