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Winning the Abel with bubbles

By David, 28 March 2019 News

Picture of a woman standing infront of a blackboard.

The Abel Prize, won this year by Karen Uhlenbeck, is sometimes referred to as the Nobel Prize of mathematics. Image: Andrea Kane/Institute for Advanced Study

One of the biggest prizes in mathematics has just been announced! Karen Uhlenbeck has won the Abel Prize, and will receive her award from the King of Norway in May. She has answered questions in many areas of mathematics. Plus, one of her most famous results explores questions that have no answer.

How do you find the shortest path between two points? One way is to get a piece of elastic and stretch it between those points. The elastic will automatically find the shortest distance because that’s the path it stretches the least.

You can use this trick to solve other questions too. If you want to find the shortest distance between two places on a globe, elastic will find it.

Now what’s the smallest surface that encloses a cup of air? Here, you’ll need to replace your elastic with something flatter – like bubble mix. A bubble adjusts to be as small as possible, so it forms a sphere, the most efficient way to enclose volume.

Time for a question that is sometimes impossible to answer. What’s the smallest surface that connects two circles? We can try to solve it with bubble mix.

Two circular pipecleaner bubble wands with a bubble stretched between.

With two bubble wands you can make a cylindrical bubble.

Take two bubble wands, put them on top of each other and then pop any bubble mix stretching across the circles. Finally, pull the two wands apart slowly and a cylinder will start to appear. It’s not a perfect cylinder though. It’s curved, with a narrow waist and wider ends.

So far so good. But as you pull the circles apart, the waist gets narrower and narrower until it suddenly collapses. There is no more cylinder, only two circles.

Mathematically, you’ve found an impossible problem. When the circles are far enough apart, there is no smallest surface that connects them. You can join the circles with a surface but there’s always a surface that’s smaller. And if you keep shrinking and shrinking the surface to find the limit, the final result isn’t a surface at all!

Karen developed tools to help mathematicians understand what’s happening with these impossible problems. These techniques can be used on lots of different types of problems, not just soap bubbles.

Karen has worked on many other areas of mathematics too. Physicists are particularly interested in her work on Yang-Mills theory, the mathematical structure at the core of our understanding of subatomic particles. Her work is so important that, in a lot of areas, it’s just taught as the normal way of doing things. And that’s the mark of a true genius.

Need some bubble mix? We have a recipe for you!

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