- This year's Abel Prize for mathematics goes to two influential probabilistic mathematicians.
- Probability, the redheaded stepchild of mathematics, isn't just for oddsmaking.
- Proofs that use different approaches can shed light on nuance and special cases.

Two retired professors are sharing the __mathematics version of the Nobel Prize__ for their lifelong contributions to the changing nature of math in the computing age. Both mathematicians spent decades applying ideas from probability theory to different kinds of discrete mathematics in order to shake loose new ways to solve seemingly intractable problems. The Abel Prize, awarded since just 2003, honors career mathematical accomplishments with a prize of about $700,000.

Wait—there’s not a __Nobel Prize__ for mathematics? It’s true, and although you may have heard a __lascivious story to explain why__, no one really knows for sure. In that debunking, Snopes speculates that Alfred Nobel didn’t think mathematics could make the kind of practical breakthroughs he saw in other fields. (We love literature, but that’s not exactly an applied science either.) But what’s interesting about this year’s Abel Prize winners is how their work brought theoretical math into the world of applications.

Probability has long been the redheaded stepchild of mathematics fields, because “pure” mathematicians believe it’s descriptive and observational instead of abstract and objective. That’s not true—probability also isn’t *statistics*, though the two are often linked in the public imagination—but pure mathematicians often won’t listen until you offer ... proof. (That’s a math class joke.)

The *New York Times* spoke with mathematician François Labourie, who was on this year’s Abel Prize committee. “They were some of the first persons to show that probabilistic methods are central to mathematics,” Labourie explained. “Now it is totally obvious.”

Furstenberg studied and published on ergodic theory for much of his career, including both his doctoral dissertation at Princeton and a __landmark 1981 textbook__ in the field. In ergodic theory, mathematicians use observed points and trajectories to form educated hypotheses about what entire systems are doing.

The *Times *explains it this way:

“Imagine a drunkard stumbling around a room and bouncing off the walls. By noting how often the drunkard passes a given spot, one might be able to infer the shape and size of the room. The general idea of using the trajectory of an object to reveal information about the space it is moving through is called ergodic theory.”

That idea might sound abstract and high level, but it’s a practical thing that can streamline a lot of other, thornier problems in mathematics. Sometimes, a theorem is first proven by a long and brutal series of steps, and of course the proof itself is as valid as a more elegant one. But there’s more than one way to skin a math problem, as they say. Szemerédi's theorem in particular is a fine cipher for different math fields to show how they like to prove things.

Furstenberg made waves when he published a far simpler proof for Szemerédi's theorem, which says in any set of integers that has a quality called “positive density,” you can pick any value and find a regularly spaced subset of integers of at least that same length. The subset is made by addition—adding 3 over and over, for example, to form 3, 6, 9, 12, and on and on—and the new area of study that Furstenberg made with his initial proof is called __ergodic Ramsey theory__.

By making a new proof that gently branched off of pure arithmetic combinatorics and introduced ideas from his field of ergodics, Furstenberg added himself to a long existing chain of the way the theorem had come to be in the first place. Mathematicians begin with what they know and then start to prove new ideas, often literally one thing at a time. Someone proved the theorem for lengths of 1 and 2, then someone else proved it for 3.

Like an individual proof by induction, the cases continue to grow until we can generalize from them. Furstenberg likely couldn’t have jumped from the individual proof for the special case of value 3 to his general proof using ergodics. He needed the original general proof from Szemerédi himself.

It’s easy to see how this entire scenario underlines how much the different branches of mathematics not only harmonize, but often require each other. But it's easy, in part, because of Furstenberg.

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