The International Congress of Mathematicians, the quadrennial meeting of the International Mathematical Union, starts later this week. This year, it is taking place in Rio de Janeiro, Brazil. The ICM is a huge meeting of thousands of mathematicians attending hundreds of lectures and other events, but the best-known part of the meeting is probably the awarding of the Fields medal International Medal for Outstanding Discoveries in Mathematics. In 2014, shortly before the last ICM, I gave some suggestions for how to make cocktail party conversation about the award, often described as the Nobel Prize of mathematics. Add 4 to most numbers in the article, and it will work for this year as well!

I will not be attending the ICM, but the meeting has me thinking about math, the Fields medal, and Rio. That means I’ve got Stephen Smale on the brain. Smale, who rather famously did math on the beaches of Rio, is part of the story of how the Fields medal became what it is today.

I heard people joke about doing math on the beaches of Rio starting in grad school, but I didn’t know the reference. I gathered that a horseshoe was involved, but as far as I was concerned, it might as well have been a horseshoe crab. (It turns out there are no horseshoe crabs in Brazil, so that was unlikely.) In fact, Smale describes discovering the horseshoe map Jana Rodriguez Hertz talked about on a recent episode of the My Favorite Theorem podcast on the beach in Copacabana. Eventually that horseshoe map led to some of his most important work, a proof of the Poincaré conjecture in all dimensions greater than or equal to 4. He wrote about those discoveries in two articles for the Mathematical Intelligencer, from 1990 and 1998.

Why does it matter that he did the work on the beaches of Rio? Math is fairly portable, and mathematicians routinely travel around the world to work together and share ideas. Smale was in Brazil in 1960 visiting the relatively new Instituto Nacional de Matemática Pura e Aplicada, also known as IMPA. It probably wouldn’t have become a tidbit in mathematical lore if it weren't for the fact that the science advisor to President Johnson wrote pointedly about mathematicians who “seriously propose that the common man who pays the taxes ought to feel that mathematical creation should be supported with public funds on the beaches of Rio de Janeiro or in the Aegean Islands.” Smale’s articles about the “beaches of Rio” are in part a defensive response to this comment.

The work that started on the beaches of Rio eventually yielded a Fields medal. The comparison of the Fields to the Nobel has always irked me, for reasons I describe in my 2014 post, so I was eager to read about how this inapt comparison got started. Luckily for me, math historian Michael Barany had just written an article for the New York Times about the genesis of the Fields-Nobel comparison, followed up and expanded on in an article for the Notices of the American Mathematical Society (pdf). I hadn't realized Smale was part of that story.

In 1966, Smale was in Moscow when he was subpoenaed by the House Un-American Activities Committee. The committee did not have the power it had under McCarthy, but being in Moscow during the Cold War after speaking out against the United States when you had been subpoenaed by the HUAC wasn’t exactly a good look. A headline in the San Francisco Examiner declared “UC Prof Dodges Supbpena, Skips U.S. for Moscow.” Mathematicians coming to Smale’s defense played up the prestige of the award to the press, comparing it to the Nobel Prize. The comparison stuck.

This year, Barany published another interesting article and gave a talk I attended at the Joint Mathematics Meetings about the early history and intent of the Fields medal. For an illustrated interview with Barany on the topic, check out this post at Math with Bad Drawings. Fields wanted a prize that would support mathematics without provoking rivalries between nations, so he established an award for promising mathematicians. Over time, in part to whittle down the number of candidates, that mandate led to the codification of an age limit of 40. At this point, it's hard to imagine changes to the rules or how mathematicians understand the significance of the medal, but Barany asks mathematicians to reimagine the the award.

To the mathematicians who will be at the ICM, I hope you can take some time to enjoy the beaches of Rio, whether they lead to any mathematical insights or not. I will be staying at home, looking for insight in the bushes outside my office.