The Royal Swedish Academy of Sciences announced today that the 2016 Nobel Prize in Physics, and the accompanying 8 million kronor (£727,000) cash prize, has been awarded to David J. Thouless (50%), F. Duncan M. Haldane (25%), and J. Michael Kosterlitz (25%), “for theoretical discoveries of topological phase transitions and topological phases of matter”.

The Academy explained the significance of the trio’s work in a press release:

This year’s Laureates opened the door on an unknown world where matter can assume strange states. They have used advanced mathematical methods to study unusual phases, or states, of matter, such as superconductors, superfluids or thin magnetic films. Thanks to their pioneering work, the hunt is now on for new and exotic phases of matter. Many people are hopeful of future applications in both materials science and electronics.
Royal Swedish Academy of Sciences

The Academy has made available a document explaining the scientific background and the importance of the work, in which they explain how topology—a field of mathematics focusing on space that can be deformed by stretching and bending but not by tearing or sticking, and in which properties only change step-wise—can explain strange behaviour in exotic forms of matter.

An example of a type of phase transition in a thin layer of very cold matter, as described by Kosterlitz and Thouless. In a cold material, vortices form in tight pairs, but at higher temperatures they separate and "sail" off in different directions. This changing behaviour

An example of a type of phase transition in a thin layer of very cold matter, as described by Kosterlitz and Thouless. In a cold material, vortices form in tight pairs, but at higher temperatures they separate and move apart. This changing behaviour is a key feature of the exotic matter described in the prize-winning work.
Source: Royal Swedish Academy of Sciences

The work for which the prize was awarded was carried out in the ’70s and ’80s, but research has boomed in recent years, and the work is being recognised now due to the hope that it can soon form the basis of the next generation of electronics or superconductors, or future quantum computers.

All three men were born in the UK, but now live and work in the United States. David J. Thouless, from Bearsden, is an Emeritus Professor at the University of Washington; F. Duncan M. Haldane, from London, is Eugene Higgins Professor of Physics at Princeton University; and J. Michael Kosterlitz, from Aberdeen, is Harrison E. Farnsworth Professor of Physics at Brown University.


For an example of the work being recognised, one may wish to refer to the two explanatory documents produced by the Academy:

Readers already well-versed in the scientific and mathematical background required may benefit from reading
Kosterlitz J M and Thouless D J 1973 J. Phys. C 6 1181 (doi:10.1088/0022-3719/6/7/010)

Abstract
A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.

Whatever your background or level of expertise, it is well worth familiarising yourself with this research. It was recognised by the Academy for its importance to future technologies, as well as being a fascinating application of abstract mathematics to important real-world systems.