Quanta recently featured an article on the settlement of the Langlands conjecture, and our own Dima Arinkin was one of the people involved in the proof.

This problem, first presented by Robert Langlands, a professor at Princeton, in 1967. The Langlands conjecture 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.

Arinkin’s contribution involved working closely with Gaitsgory, including as the lead author of a six-author paper generated during the pandemic on eigensheaves. In the world of the geometric Langlands program, eigensheaves are supposed to play the role of sine waves. Gaitsgory and his collaborators had identified something called the Poincaré sheaf that seemed to be serving the role of white noise.

By early 2023, Gaitsgory and Raskin, together with Arinkin, Rozenblyum, Færgeman and four other researchers, had a complete proof of the work up until that point and it would take the team another year to write up the proof, which they posted online in February 2024.

The interconnectedness of the geometric program with number theory, and function field will undoubtedly create waves in those adjacent programs of the Langlands conjecture, as well as bleed over to other topics.

Link: https://www.quantamagazine.org/monumental-proof-settles-geometric-langlands-conjecture-20240719/