A Major Step Toward a Grand Unified Theory of Mathematics

There are many kinds of mathematics. So many, in fact, that before we go any further, it might help to take an 11 minute video tour of the map of mathematics, which is a lovely general overview to an endeavor that only begins with counting numbers.

Within each of the fields on the theoretical side of the map, mathematicians have developed specialized tools for solving the problems they’ve tackled. But the tools developed to work in one theoretical field have often only proven to be useful within that field. Mathematicians working in other fields haven’t been able to deploy them to solve the problems they are working upon.

Which has been disappointing because some of those tools are very powerful. What if it were possible to translate problems from other fields into ones that those powerful tools can solve?

That has been the challenge and promise of the Langlands project, which seeks to connect different fields of math by uncovering a shared language between them. Much as archaeology’s Rosetta Stone made it possible to translate ancient Egyptian hieroglyphs and demotic scripts by connecting them to written Greek.

In 1967, mathematician Robert Langlands conjectured it would be possible to connect several different fields in mathematics, which if successful, would make it possible to used the tools developed in each to gain insights in the others.

From here, lets turn to Quanta Magazine‘s Erica Klarreich’s reporting on a mathematical breakthrough:

A group of nine mathematicians has proved the geometric Langlands conjecture, a key component of one of the most sweeping paradigms in modern mathematics.

The proof represents the culmination of three decades of effort, said Peter Scholze, a prominent mathematician at the Max Planck Institute for Mathematics who was not involved in the proof. “It’s wonderful to see it resolved.”

The Langlands program, originated by Robert Langlands in the 1960s, is a vast generalization of Fourier analysis, a far-reaching framework in which complex waves are expressed in terms of smoothly oscillating sine waves. The Langlands program holds sway in three separate areas of mathematics: number theory, geometry and something called function fields. These three settings are connected by a web of analogies commonly called mathematics’ Rosetta stone.

Now, a new set of papers has settled the Langlands conjecture in the geometric column of the Rosetta stone. “In none of the [other] settings has a result as comprehensive and as powerful been proved,” said David Ben-Zvi of the University of Texas, Austin.

“It is beautiful mathematics, the best of its kind,” said Alexander Beilinson, one of the main progenitors of the geometric version of the Langlands program.

The proof involves more than 800 pages spread over five papers. It was written by a team led by Dennis Gaitsgory (Scholze’s colleague at the Max Planck Institute) and Sam Raskin of Yale University.

Please do click over to Quanta to find out more. Their coverage is an excellent entry point for understanding Langlands conjecture and how the mathematicians who published their proof of it some 57 years later tackled the challenge.

It’s a massive proof and if it holds, a massive achievement. It is also a strong candidate for being the biggest math story of 2024, which is just a little over half over. What other stories might join it by the end of the year?