Kosterlitz and Thouless described the topological phase transition that accompanies superconductivity, where the surface goes from the red state—unable to support tight vortex pairs—to the blue state, which allows current to flow without resistance.

Image Credit: Royal Swedish Academy of Sciences

Wikipedia has this helpful but brain-twisting graphic to illustrate the concept.

Watch it through a couple of times; I'll wait.

Image Credit: Wikipedia user JasonHise

In the 1970s, Kosterlitz and Thouless proved that superconductivity could occur in thin layers, a finding that ran contrary to the prevailing opinions of experts at the time. Their theory of the phenomenon invoked vortex pairs to explain the superconductivity—at suitably low temperatures, vortices can form pairs that balance each other out and prevent the dissipative loss of resistance.If you've ever been canoeing, there's a strong analogy there. When you dip your paddle into the water and draw it back, you can see water rush in to fill the space where the paddle was. This water often spirals back on itself, forming a small circular current of water—a vortex. These vortices usually dissipate quickly, losing angular momentum to the water around them. If you take a circular plate, though, and dragthrough the water, you can create aof counter-spinning vortices, which turns out to be more stable than solitary ones by an order of magnitude. There's a fantastic visual demonstration of these vortex pairs (technically called Falaco solitons ) below:One interesting thing about these vortex pairs is that they are joined by a "string" that extends into the third dimension and helps keep them stable—when that "string" is cut, the vortices lose their characteristic stability and dissipate rapidly. A similar rule appears to hold true in the physics of electromagnetic vortex pairs in a conductor. At sufficiently low temperatures, these pairs can form, enabling superconductivity, the flow of electricity without resistance or loss of energy as heat. When the temperature rises above a critical value, however, the ties are broken and the vortices drift apart from one another, following different rules than in their paired form—you can imagine it might be tougher to create these vortex pairs in a turbulent jacuzzi than a calm swimming pool.Thouless' contributions didn't end with 2D superconductivity, however; his understanding of topology led him to an explanation for one of the more mysterious properties to emerge from experimental physics in recent decades—the quantum Hall effect. Demonstrated in 1980, the effect is a phenomenon in which the electrical conductance of a thin material in a strong magnetic field changesin discrete jumps, doubling, then tripling, etc. as the strength of the magnetic field drops. This behavior puzzled scientists, but Thouless had the solution, postulating a solution focused on the topology of the material that accounted for the manifestation of a quantum effect at such a large scale.Thouless and Kosterlitz explored the weird world of 2D materials, but this year's third laureate, Duncan Haldane, took it down a dimension further. In addition to proving that the quantum Hall effect could arisean external magnetic field, Haldane used mathematics to examine one-dimensional chains of atoms, which also have the potential to display unusual topological properties. Haldane's work in the early 1980s showed that chains of magnetic atoms display different properties depending on the spin of the atoms involved—spin-1 particles showed topological effects, while spin-1/2 particles did not.Quantum mechanical spin can be hard to understand, but the quick-and-dirty explanation of it is this: imagine an arrow stuck through a ball, so that the arrow is pointing upward—this is the particle's spin vector, and the particle is "spin up". Now, if that particle is spin-1, rotating that particle by 180° will make it a "spin down" particle, while a 360° rotation will bring the spin vector back to where it started, pointing up. This is all nice and intuitive. Spin-1/2 is where it gets weird—if a particle has "half-integer" spin, rotating a "spin-up" particle by 360° will get you a "spin-down" particle, which has to be rotated by another 360° to be returned to its original state.The practical applications of physics this advanced are sometimes hard to discern, but spectators are optimistic. "Their theoretical innovations point the way to quantum computers, highly sensitive detectors, and new experiments that can provide insights into the behavior of exotic, and sometimes bizarre, states of matter," said Laura Greene, president-elect of the American Physical Society. "They have ignited a firestorm of research, and although applications are still yet to come, I believe it’s only a matter of time before their research leads to advances as unimaginable to us now as lasers and computer chips were a hundred years ago," . If we ever achieve long-sought-after technologies like room-temperature superconductivity, they'll probably rely on the advances made by Thouless, Kosterlitz, and Haldane.P.S. While the "popular vote" would certainly have gone to LIGO—the gravitational wave observatory that went online last year and made its first detection almost immediately thereafter—the Nobel committee clearly has its own mysterious process. Some have speculated that LIGO wasn't considered for this year's prize because of the February 1deadline for nominations: while the detection happened in Autumn of last year, the official announcement wasn't made until February 11of 2016. This doesn't mean the committee was in the dark, or that LIGO wasn't nominated—key figures in the physics community would have known about the result before February 1—but the deadline could still have played a role in their decision-making process.