## Topology and the 2016 Nobel Prize in Physics

The Nobel Prize in Physics was awarded this year to David Thouless, Duncan Haldane and Michael Kosterlitz (incidentally 3 Britons working at US universities) for discoveries in topological phase transitions. What does that mean? Phase transitions happen when material changes form, like frozen ice melting into liquid water at 32 Fahrenheit, or solid carbon dioxide sublimating into a gas to provide cheap and cheesy special effects. Phase transitions have been known about for a long time, so why the prize now? That has to do with topology, which is easier to explain with shapes than mathematics.

Take a soccer ball and a long piece of string. No matter how you tie the string around the ball, it's pretty easy to slip the string off. Now try the same with a coffee mug. Some ways you tie the string around the mug it can still be slipped off easily, but not if it's tied through the handle. In fact you can pull and shove but unless you cut the string, it's not coming off. That's the basic difference in *topology* between the ball and the mug. Those aren't the only possibilities: with a pretzel there are lots of ways a string can be tied around and through it without slipping off. It's a property of shapes that doesn't have to do with their size. A ball, a bowl and a banana all have the same property that "strings-fall-off", while a mug, a bagel and a hula hoop can all be tied up.

That's not the only kind of topology. For example if you walk in a circle around the South Pole holding out a magnetic compass, the needle keeps changing direction and maps out its own circle. This is a *vortex* - it's a location where everything around it changes as you move. Not only can you have vortices, you can also have *anti-vortices* where the effect is opposite (think of water flushing down a drain in the northern or southern hemispheres, circling either clockwise or anti-clockwise). Another topological property you can make with a long rectangular strip of paper. Bend it so you can glue the short ends together to make a loop, but put in a twist like in the picture. You've just made a *Mobius strip*, a shape that has no inside or outside. Just imagine walking along the surface - you'll cross all of it! In fact there's a very big sculpture of a Mobius strip on campus in the Redwood Grove near the McCaffrey Center. Can't you undo the twist? No, not without cutting the paper. Where does the twist "belong"? It's not in any one place, it belongs to the entire object. This is a feature of topology - it's a *global* effect.

What's the connection to phase transitions? Usually you can spot a phase transition by looking at something where you are, say the density of a material. When water boils, the density changes very quickly, and that happens everywhere in the water. So wherever you look, you'll notice a dramatic jump. The prize winners showed that you can have phase transitions where the effect is global because it's caused by pairs of vortices and anti-vortices locking together or falling apart. Only by examining the whole object can you detect the change, since it's related to topology - if you look in just one location, you don't see anything special happening. They showed this for two-dimensional materials, which seems very mathematical and theoretical since the world we live in has three spatial dimensions. How is this connected to physics? With modern technology you can build materials which are practically two-dimensional, where the third spatial dimension might only be a few atoms in size. Now the mathematical results about topology become relevant. In fact topology can make materials more sturdy and robust - a small defect in one place is not going to change the global topology. Storing electronic information gets easier if you don't have to worry about small glitches in the material. Instead of being just mathematics it has real physical benefits!

The work that was recognized by the award was done in the 1970s and 1980s, and back then might have been viewed as just a theoretical curiosity. The amazing thing is that decades later it could help create new kinds of technology, like topological insulators. Like all good research, you can never know where it's going to lead. The ideas are also very general: there's a lot of overlap between material physics (like superconductors) and particle physics (like quarks) where topology also plays a big role. In fact Prof Hetrick's PhD thesis was on topology in two-dimensional systems (ask him about the homotopy class of the Lie group U(1) on the manifold S_{1} for example). Some of us in the physics department were pretty sure the prize would go to the gravitational waves discovery by LIGO, but there are many amazing discoveries in physics deserving of the honor.