June 14, 2018

Francis Su's Favorite Theorem

The Harvey Mudd College mathematician tells us why he loves playing with Brouwer's fixed-point theorem

Mathematician Francis Su explains the seven-color map theorem on the torus, which is not his favorite theorem.

Francis Su

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

On this episode of our podcast My Favorite Theorem, my cohost Kevin Knudson and I were pleased to have Francis Su on the show. Dr. Su is a math professor at Harvey Mudd College, but when we recorded, he was at the Mathematical Sciences Research Institute (MSRI) in Berkeley, California. You can listen to the episode here or at kpknudson.com, where there is a transcript.

Dr. Su chose the Brouwer fixed-point theorem for the podcast. This theorem states that given a blob (more rigorously, a connected region in the plane or in higher-dimensional space with no holes, but “blob” works pretty well), if you look at a function from the blob to itself, there is always some point that doesn’t move. An example of what we mean by “function from the blob to itself” would be stirring or swirling tea in a mug. Every point in the mug moves somewhere else in the mug, so we could write down what happens by saying that point x moved to point y, point y moved to point z, and so on. Brouwer’s fixed point theorem says that if you do stir the tea in your mug, there must be at least one molecule of tea that ends up in the same place it started in. For more information on the theorem, you can start with Dr. Su’s page explaining it or Tai-Danae Bradley’s video about it for the PBS Infinite Series channel.

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In each episode of the podcast, we ask our guest to pair their theorem with food, beverage, art, music, or any other delight in life. Dr. Su chose parlor games and talked a little bit about some surprising applications of the Brouwer fixed-point theorem to games.

A Hex game board with a winning path for the blue player highlighted in white. Credit: Jean-Luc W *Wikimedia*(CC BY-SA 3.0)

For example, the game Hex is played by two people on a hexagonal grid. In each turn, a player places a hexagonal tile of their color on the grid. The goal is to have an unbroken chain from one side to the other by the end. Dr. Su mentioned a paper by David Gale that shows that the fact that Hex can never end in a tie is equivalent to the Brouwer fixed-point theorem. Kevin mentioned the book Five Golden Rules by John Casti, which includes an application of the Brouwer fixed-point theorem to football scheduling. For a deeper look at applications of the Brouwer and other fixed-point theorems, Dr. Su recommends the book Fixed Point Theorems with Applications to Economics and Game Theory by Kim C. Border.

You can find Dr. Su at his website, his blog The Mathematical Yawp, and his Twitter account. He also created the MathFeed Twitter account and iPhone/iPad app, which aggregate math news, and writes a Math Fun Facts page full of rabbit holes to go down. He was the president of the Mathematical Association of America from 2015 to 2016, and his retiring presidential address in January 2017 was beautiful. See also this Quanta Magazine interview with him.

You can find more information about the mathematicians and theorems featured in this podcast, along with other delightful mathematical treats, at kpknudson.com and here at Roots of Unity. A transcript is available here. You can subscribe to and review the podcast on iTunes and other podcast delivery systems. We love to hear from our listeners, so please drop us a line at myfavoritetheorem@gmail.com. Kevin Knudson’s handle on Twitter is @niveknosdunk, and mine is @evelynjlamb. The show itself also has a Twitter feed: @myfavethm and a Facebook page. Join us next time to learn another fascinating piece of mathematics.

Previously on My Favorite Theorem:

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