Math Magic

With a random approach, two young scholars solve a tricky problem in one night.

July 2022

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As a new college graduate in South Korea, Jinyoung Park needed a job, and she wanted to be a mathematician. But she didn’t know how to become one, so she taught middle school math for seven years. 

Now, as an assistant professor of mathematics at Stanford, she is drawing accolades from around the world for using a clever randomized process to solve a math problem that has stumped other mathematicians for more than 15 years.

“People are very excited that I was a middle school teacher,” says Park, who, along with Stanford PhD student Huy Tuan Pham, ’18, MS ’18, in a single night in March, co-authored a proof that solved the Kahn-Kalai conjecture.

“I would call their proof magical,” says math professor Jacob Fox, Pham’s adviser. 

First posed in 2006 by two math professors, the conjecture makes a claim about the “threshold” at which random sets or graphs might result in a useful structure—say, a triangle. Mathematicians use threshold levels—the computational equivalent of a freezing or boiling point—to determine when these structures will appear. However, exact thresholds are hard to compute. The conjecture posited that the “expectation threshold,” which is less precise but much easier to calculate, is actually very close to the true threshold. 

For months, Park and Pham had been working on a different problem that gave them the idea that they needed. Their remarkably short proof (six pages) is partly dedicated to constructing “covers,” or collections of sets, that show the Kahn-Kalai conjecture to be true. Their technique could be applied to other tough math problems. Well, tough for most of us. “After we had that idea,” Pham says, “it took us one night to confirm that it really worked.”

“It was easy,” Park says, with a laugh.

Tracie White is a senior writer at Stanford. Email her at traciew@stanford.edu.

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