words from Persi Diaconis

This is a delightful article about Persi Diaconis,who is now working on the mathematics of "smooshing" cards. https://www.quantamagazine.org/20150414-for-persi-diaconis-next-magic-trick/ But the most memorable take-away for me is the following:

When Diaconis returned to college after a decade as a professional magician, his first three grades in advanced calculus were C, C and D. “I didn’t know you were supposed to study,” he said. His teacher told him that he should write down the proofs and practice them as if they were French verbs. “I said, ‘Oh, you’re allowed to do that?’” Diaconis said. “I thought you were just supposed to see it.”

Write down the proofs and practice them as if they were French verbs! I love that analogy. And it makes so much sense. Math is a language and a certain amount of rote practice is helpful for those of us who are learning it in later life. (Persi Diaconis had a somewhat unorthodox and nontraditional educational trajectory. I guess mine may be more unorthodox still.) The following paragraphs from the article were also fascinating:

When it came to smooshing, instead of just trying to “see it,” Diaconis devoured the literature on fluid mixing. “When we started talking about the connections between cards and fluid mixing, he read the whole 200 pages of my Ph.D. thesis,” said Emmanuelle Gouillart, a researcher who studies glass melting at Saint-Gobain, a glass and construction materials company founded in Paris in 1665. “I was really impressed.”

While Diaconis grew more conversant in fluid mechanics, Gouillart benefited from his unique insight into card shuffling. “It turned out that we were studying very similar systems, but with different descriptions and different tools,” Gouillart said. The collaboration led her to develop a better way to measure correlations between neighboring particles in the fluids she studies.

Diaconis, meanwhile, has developed a mathematical model for what he calls “the sound of one hand smooshing.” In his model, the cards are represented by points scattered in a square, and the “hand” is a small disk that moves around the square while rotating the points under it by random angles. (It would be easy, Diaconis noted, to extend this to a two-handed smooshing model, simply by adding a second disk.)

Diaconis has been able to show — not just for a 52-card deck but for any number of points — that if you run this smooshing model forever, the arrangement of points will eventually become random. This might seem obvious, but some shuffling methods fail to randomize a deck no matter how long you shuffle, and Diaconis worried that smooshing might be one of them. After all, he reasoned, some cards might get stuck at the edges of the table, in much the same way that, when you mix cake batter, a little flour inevitably gets stranded at the edges of the bowl and never mixes in. But by drawing on 50 years of mathematics on the behavior of random flows, Diaconis proved that if you smoosh long enough, even cards at the edge will get mixed in.

His theoretical result says that the smooshing model will eventually mix the cards, but doesn’t say how long it will take. The model does provide a framework for relating the size of the deck to the amount of mixing time needed, but pinning down this relationship precisely requires ideas from a mathematical field still in its infancy, called the quantitative theory of differential equations. “Most studies of differential equations focus on what happens if you run the equation for a long time,” Diaconis said. “People are just now starting to study how the equation behaves if you run it for, say, a tenth of a second. So I have some careful work to do.”

Diaconis is optimistic that the work will lead him not just to an answer to the smooshing question, but to deeper discoveries. “The other shuffles have led to very rich mathematical consequences, and maybe this one will too,” he said.