What it Takes to Solve Math’s Toughest Problems

This is a story about keeping the faith in a discipline not exactly known for its religiosity: mathematics.

The field of arithmetic statistics — the study of statistical patterns in arithmetic objects such as number fields, class groups, and elliptic curves — rests on sets of conjectures whose predictions have been only partially proven.

In 2001, mathematician Jordan Ellenberg ’93, Ph.D. ’98, and some colleagues set out on a humbling yearslong odyssey to disprove one of the field’s most famous sets of predictions, the Cohen-Lenstra conjectures.

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More recently, Aaron Landesman ’16 has proved new cases of Cohen-Lenstra’s predictions over function fields, staking a much stronger claim in the conjectures’ favor.

Landesman talks about Ellenberg with the enthusiasm of a devotee. He’s gone out of his way to attend Ellenberg’s lectures; he’s read his books. “He gives the most inspiring talks, and it’s a lot of fun talking to him,” Landesman said.

As a graduate student, Landesman’s favorite paper was the piece Ellenberg co-authored in 2009 on the Cohen-Lenstra conjectures. Was Landesman intimidated, then, when he set out to prove what they had failed to in the retracted 2012 paper? Not in the least.

“What’s important is to be aware of the past work and make sure you understand the error, because you don’t want to do the same thing,” he said.

Read the full story at the Harvard Gazette.

*Photo courtesy of Veasey Conway/Harvard Staff Photographer.