After breakfast one morning in August, the mathematician Terence Tao opened an email from three physicists he didn't know. The trio explained that they'd stumbled across a simple formula that, if true, established an unexpected relationship between some of the most basic and important objects in linear algebra.
The formula "looked too good to be true," said Tao, who is a professor at the University of California, Los Angeles, a Fields medalist, and one of the world's leading mathematicians. "Something this short and simple — it should have been in textbooks already," he said. "So my first thought was, no, this can't be true."
Then he thought about it some more.
The physicists — Stephen Parke of Fermi National Accelerator Laboratory, Xining Zhang of the University of Chicago and Peter Denton of Brookhaven National Laboratory — had arrived at the mathematical identity about two months earlier while grappling with the strange behavior of particles called neutrinos.
They'd noticed that hard-to-compute terms called "eigenvectors," describing, in this case, the ways that neutrinos propagate through matter, were equal to combinations of terms called "eigenvalues," which are far easier to compute. Moreover, they realized that the relationship between eigenvectors and eigenvalues — ubiquitous objects in math, physics and engineering that have been studied since the 18th century — seemed to hold more generally.
Although the physicists could hardly believe they'd discovered a new fact about such bedrock math, they couldn't find the relationship in any books or papers. So they took a chance and contacted Tao, despite a note on his website warning against such entreaties.
"To our surprise, he replied in under two hours saying he'd never seen this before," Parke said. Tao's reply also included three independent proofs of the identity.