Echoes of Electromagnetism Found in Number Theory

A new magnum opus posits the existence of a hidden mathematical link akin to the connection between electricity and magnetism.

by Kevin Hartnett at Quanta Magazine: In 2018, as he prepared to be awarded the Fields Medal, math’s highest honor, Akshay Venkatesh carried a piece of paper in his pocket. On it, he had written a table of mathematical expressions that for centuries have played a key role in number theory.

Though the expressions had featured prominently in Venkatesh’s own research over the last decade, he carried them around not as a memento of what he’d accomplished, but as a reminder of something he still didn’t understand.

The columns of the table were filled with cryptic-looking mathematical expressions: On the far left were objects called periods, and on the right, objects called L-functions, which might be the key to answering some of the most important questions in modern mathematics. The table suggested some kind of relationship between the two. In a 2012 book with Yiannis Sakellaridis of Johns Hopkins University, Venkatesh had worked out one direction of it: If they were handed a period, they could determine whether it had an associated L-function.

But they couldn’t yet understand the relationship in reverse. It was impossible to predict whether a given L-function had a matching period. When they looked at L-functions, they largely saw disorder.

That’s why Venkatesh kept the paper in his pocket. He hoped that if he stared at the list long enough, the common traits in this seemingly random collection of L-functions would become clear to him. After a year of toting it around, they hadn’t.

“I couldn’t understand what the principle behind this table was,” he said.

2018 was a big year for Venkatesh in more ways than one. In addition to receiving the Fields Medal, he also moved from Stanford University, where he’d been for the previous decade, to the Institute for Advanced Study in Princeton, New Jersey.

He and Sakellaridis also started talking with David Ben-Zvi, a mathematician at the University of Texas, Austin who was spending the semester at the institute. Ben-Zvi had built his career in a parallel area of math, investigating the same kinds of questions about numbers that Sakellaridis and Venkatesh were interested in, but from a geometric point of view. When he heard Venkatesh give a talk about this mystery table he carried with him everywhere, Ben-Zvi almost immediately began to see a new way to make periods and L-functions communicate with each other.

That moment of recognition instigated a multiyear collaboration that came to fruition this past July, when Ben-Zvi, Sakellaridis and Venkatesh posted a 451-page manuscript. The paper creates a two-way translation between periods and L-functions by recasting periods and L-functions in terms of a pair of geometric spaces used to study basic questions in physics.

More here.