Blenders mix milk and chocolate topping. Cereal makers add sultanas and puffs to bran flakes. We all mix things every day, but we know surprisingly little about the mathematics of mixing. One of this year’s Fields Medallists is working to bridge this gap. Many people can shuffle cards, but Artur Avila can shuffle a line.

Artur’s method for shuffling a line is quite simple. First, he cuts the line at several distances, to get several line segments. He rearranges these segments, and joins them together to make a new line. Artur then repeats this process with the new line – cutting at exactly the same distances, and rearranging the segments in exactly the same way. With enough repetitions, hopefully the line gets all mixed up.

Artur’s process doesn’t always shuffle the line very well. Following Artur’s rules, you could take the first quarter of a line and put it at the other end. If you repeat this three more times, you end up exactly where you started!

So if you want a good shuffle, how careful do you have to be? With some very tricky maths, Artur showed that you don’t need to be very careful at all. If you pick the cuts at random, your shuffle is almost certain to mix things up. Artur’s research proved that most shuffles were weakly mixing, which means they do a fair job of mixing up the original line.

Mathematician Artur Avila

2014 Fields Medal winner Artur Avila studies the maths of mixing.

Image: Oberwolfach Photo Collection/Schmid, Katrin CC:BY/SA

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