From Scientific American:

Topology is a branch of mathematics that studies properties that only change incrementally, in integer steps, rather than continuously. Thors Hans Hansson, a physicist at Stockholm University who served on this year’s Nobel Committee, explained the core concept of topology during the awards announcement by pulling a cinnamon bun, a bagel, and a Swedish pretzel from a bag. “I brought my lunch,” he joked, then explained that, to a topologist, the only difference between the three foods was the number of holes in them, rather than their taste. A cinnamon bun has no holes, while a bagel has one and a pretzel has two. To a topologist, then, the bun would fall in the same category as a saucer, while the bagel would be paired with a cup, and a pretzel with a pair of spectacles. Thouless, Kosterlitz and Haldane’s prize-winning insights revolve around the idea that these same sorts of “topological invariants” could also explain phase changes in matter, albeit not familiar ones such as a liquid freezing to a solid or sublimating to gas. Instead, the phase changes the theorists studied took place chiefly in thin two-dimensional films cooled to cryogenic temperatures.

The first insight came in the early 1970s, when Thouless and Kosterlitz worked together to overthrow the long-held consensus that phase transitions such as superconductivity (the flow of current without resistance) and superfluidity (a fluid possessing zero friction) simply cannot occur in two-dimensional systems due to thermal fluctuations, even at absolute zero. They found instead that cold two-dimensional systems could in fact undergo phase transitions through a totally unpredicted phenomenon, the formation of pairs of vortices at very low temperatures that then suddenly drift apart as the temperature rises past a certain thermal threshold. This “KT transition” (for “Kosterlitz-Thouless”) is universal, and has been used to study superconductivity in …

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