His finalist year: 1977
His finalist project: Figuring out how many steps it takes to condense sequences of numbers based on their properties
What led to the project: When Long Island, New York, native Jim Propp was a kid, his parents took the family on a car trip through Israel. Bored in the backseat, Propp started amusing himself by taking a census of the letters that appeared on the (English) road signs. He would write down the number of times each letter appeared in a phrase. For instance, "Right turn only" contains the letters G, H, I, L, N, O, R, T, U, Y, and the sequence would be 1-1-1-1-2-1-2-2-1-1. Then he'd play another round of the game with that sequence, compressing it to 7-3 (since 1 appears 7 times, and 2 appears 3 times). That then compresses to 1-1, with each number appearing once. That, in turn, compresses to 2. He kept doing this with different signs, seeing how many rounds it would take to get down to a single entry.
"It was a silly thing to do but it passed the time," he says. And as often happens in math, "you start out looking at something, and then it gets interesting intrinsically. You want to understand it." So, over the years, he tried to figure out interesting properties of the game—for instance, what's the shortest message that still takes five rounds to get to a single entry? And, what happens if you play the game with infinite sequences instead of finite ones?
He later learned that other mathematicians had pondered similar questions (though probably not in the backseats of their parents' cars). "I was a little disappointed to find out I wasn't the first person to come up with this game," he says. But their discoveries helped him understand the theory behind his game better. When he entered his work in the 1977 Westinghouse Science Talent Search, he won 2nd place overall.
The effect on his career: Placing so high helped Propp make the decision to continue studying math. It was a tough call, because "for a long time I really loved music—I still really love music." In particular, he liked composing, and wanted to write musical comedies. But then he decided that "an academic mathematical career is much more tolerant of people at all levels of achievement." As for composers, "unless you're really good at it, I think you'll lead a less happy life."
Propp majored in math at Harvard, then earned his doctorate at the University of California, Berkeley. He started off studying more esoteric topics, but then, as a professor first at the Massachusetts Institute of Technology and then at the University of Wisconsin–Madison, came "back to my first love, which is combinatorics," a branch of mathematics that deals with the ways you can combine certain numbers of objects, such as how many combinations of five cards you can deal from a 52-card deck. "It is just something you can think about almost in a spirit of play."
For instance, some of his best-known work looks at a concept called "domino tiling," particularly of a shape called an Aztec diamond. Picture a square balanced on its corner on a grid, with edges that look like staircases. Now imagine trying to cover this diamond with two-square dominoes. In what's called the "Arctic Circle theorem," Propp showed that as the size of the diamond and the number of dominoes approaches infinity, the dominoes tend to line up very neatly in the corners; the center is random. The space of randomness in the center is actually a perfect circle inside this diamond. "It's the kind of thing that can easily get undergraduates interested in combinatorics," says Peter Winkler, a professor of mathematics at Dartmouth College. "Jim Propp is really good at that."
Indeed, Henry Cohn, now a researcher at Microsoft who met Propp when he was an undergraduate at MIT during the early 1990s, recounts that he saw the Arctic Circle theorem during a talk and "it was one of the most amazing things I had ever seen."
Winkler also notes that "Propp has done marvelous work helping people understand how to randomly sample things." Coming up with a completely random sample from a bunch of data points is not an easy thing to do, but some of his work has looked at just how researchers might generate such sets.
What he's doing now: After spending many years in Wisconsin, Propp and his wife, a research psychologist, decided that they wanted to move back to the east coast. He took a job at the University of Massachusetts at Lowell. The couple made another big lifestyle change along the way—they had a son two years ago, and they had a daughter a few weeks ago.
Both are getting early lessons in mathematics; when Propp was holding his daughter recently, he started moving grapes around on the counter in a way that relates to a combinatorics problem he's currently thinking about. ("There aren't many branches of math that you can play with while moving grapes around on a counter while comforting a 1-month-old baby," he notes.)
His son is showing early mathematical tendencies as he did. Not too long ago, when Propp was reading to him, he told his son to pick four books off the shelf, which the boy promptly did. Then he announced "Five books!" and pulled out one more. "I was just blown away," Propp says. He'll have more opportunities to watch these developing mathematicians this spring, when he plans to take the semester off and do a stint as a stay-at-home dad. "I'm very excited," he says, as is his son, who—as Propp was taking off to work one recent morning—yelled "No teach calculus! Stay!"