Topology: investigating the properties of blobs.

Nobel Prize in Physics 2016

On 4th October 2016, three physicists were awarded the Nobel Prize in Physics for their amazing insights into the quantum states on the surface of materials. Using topology, they explained the behaviour of semiconductors and superfluids – both of which behave in particularly odd ways.

Pretty weird stuff.

The bizarre behaviour of superconductors (materials that can carry a current without any electrical resistance, usually occurring when it is cooled down to close to absolute zero, or -273°C) includes its stepwise changes in its conductivity when a magnetic field around it changes continuously. Before Thouless, Haldane and Kosterlitz, physicists had no idea how to explain this behaviour.

However, Thouless realised that these sudden jumps could be explained using topology, a branch of maths that investigates the properties of blobs (i.e. shapes that can be twisted and deformed, but not glued together or torn apart).

Topology

Yum!

So imagine that you have a blob of play-doh in your hands. The main rule in this branch of maths (there are always rules in maths – makes it easier to handle) is that you can only make smooth deformations with it.

This means that you can only bend, mush and shape it; you are not allowed to tear, punch holes or stick other bits together. Shapes that can morph into each other continuously have the same topology.

So a bagel has the same topology as a picture frame – they both have one hole in the middle.

However, a bun does not have the same topology as a bagel or a picture frame, because no amount of bending or shaping of the bagel without sticking bits together or breaking it apart will get rid of the hole; similarly, you cannot create a hole in a bun without breaking the smooth deformations rule.

But a well-made waffle has the same topology as a bun because even though there are indents, there are no through holes so you can shape a waffle into a bun or a bun into a waffle.

Suppose now you have a figure of 8 – this has two holes, and cannot have the same topology as a bagel or a bun. Starting to get the hang of it?

Imagine a coffee mug. Does it have the same topology as a bun (0 holes), a bagel (1 hole), a figure of 8 (2 holes) or a pretzel (3 holes)?

A morphing coffee mug, by Graeme McRae

So topology is basically the study of how holes affect the property of the blob, and since you can only have an integer number of holes (0, 1, 2, 3 …) and never 1.5 or 3.2 holes, this can explain the stepwise changes in conductivity of a superconductor.

Now you can explain some very deep sciency stuff using a bun, a bagel and a pretzel. Not bad, huh?