Topology is a form of mathematics that describes the properties of objects that do not change during continuous deformations. The old joke is that a topologist struggles to tell the difference between a coffee mug and a doughnut. Both shapes have a single hole and if the doughnut was made of a sufficiently pliable material it could be reshaped into a coffee mug without being torn. 

Physicists have been particularly excited about topology since the mid 2000s, when the first experiments verified types of materials known as topological insulators. These exotic materials are insulators in their bulk but their surfaces can conduct like metals.

This video introduces topological materials and why they hold so much promise for emerging technologies such as quantum computing. But today topological research is certainly not limited to solid-state physics. Scientists are also starting to apply some of the mathematical concepts to understand complex fluid dynamic systems in earth sciences, biology, and the way plasma is controlled by magnetic fields inside a tokamak in the quest for practical nuclear fusion.

For a more in depth look at the latest developments in topological research see this feature article by science writer Jon Cartwright, originally published in the June 2021 issue of Physics World.