Topology in the Limelight

Topology is having a moment. Maybe not as much as this never-ending election season or this Pringles “ringle” with 40,000 retweets and counting (seriously, you should go look—it’s a self-supporting ring of potato chips, need I say more?), but it’s been getting more recognition than usual for a field of theoretical math. Somewhat ironically, it’s all because of the Nobel prize, and there’s not even a Nobel for math. (And no, it’s not because a mathematician had an affair with Nobel’s wife; he was a bachelor.) Earlier this month, the Nobel prize in physics went to three physicists for their work on topological phase transitions and topological phases of matter.

Delicious topology. Image: Dave Crosby, via Flickr. CC BY-SA 2.0

As science news editors across the globe sighed and shelved their pre-written explainers about LIGO and gravitational waves, they got to work figuring out how to talk about the work that actually won the prize this year. The easiest part of the prize to explain ends up being the most off-putting word, topology, so the bagels and coffee cups were at the ready.

Nobel Committee member Thor Hans Hansson had an adorable illustration of the basic idea of a topological property, in this case genus. But his explanation and some of the other ones I saw sometimes gave people the impression that the physicists were studying literal, if miniature, bagel- and pretzel-shaped objects floating around in superconducting materials. That’s not quite right. I tried to make it a bit clearer in an article I wrote for Scientific American. I can also recommend Kevin Knudson’s article about it for The Conversation and Brian Handwerk’s piece for Smithsonian. Vasudevan Mukunth also has a nice article on The Wire reminding us that the value of these physics discoveries is not necessarily in their utility or applications.

For me, one of the most fun things to come out of the Nobel’s mathematical connection is a series of interviews Rachael Boyd did on the blog Picture This Maths. (I wrote about Picture This Maths this past July.) One of the interviewees, Ruben Verresen, complains about the usual description of topology so many of us give of topology, which tends to be of the donut-coffee mug variety. “The issue is that they seem very arbitrary: what’s so special about holes?” Even if a reader or listener understands what you’re describing, he thinks it’s not all that interesting. He writes, “if I explain topology by comparing a donut to a coffee mug, I can just see my listener slowly turning off.” Instead, he thinks we should emphasize the difference between properties that are local and those that aren’t. Local properties even include things like height, anything that can be assessed by looking at the property in a small region and then combining those observations over the entire object. He says that emphasizing this distinction can make it more clear why someone should care about topology in the first place.

Boyd’s interviews also made me aware that someone’s been writing a comic strip about topology and nobody told me! The strip, by Tom Hockenhull, is in Chalkdust Magazine.

I’ve seen topology around a few other places recently, so this post comes with a dessert course. Mathematician Jean-Luc Thiffeault used ideas from topology to analyze taffy-pullers, both modern and old-timey, and I was ON IT for Smithsonian. Candy and math? Yes, please. Then there’s this Nature News Q&A with Microsoft researcher Alex Bocharov about why Microsoft is investing to heavily in building a topological quantum computer. One of the most interesting things I learned when I was writing about the physics Nobel prize was that physicists are trying to figure out how to use topology to build a quantum computer. Once you’ve heard it, it makes sense: topological properties are more robust to small perturbations than other properties, so in theory, information would be less prone to degradation from outside noise.

Finally, behold the ultradonut topology of the nuclear envelope, a real paper title from the Proceedings of the National Academy of Sciences. Mmm, ultradonut.