Back in 2016, we covered a math story involving perfectoid spaces, which we described as the "up and coming math". In 2021, it appears to have arrived at a significant milestone, featuring prominently in what is one of the biggest math stories of the year to date.
That story involves the application of perfectoid spaces to deliver a blockbuster result in the mathematics world, a proof by Laurent Fargues and Peter Scholze connecting geometric objects to numbers. Or more specifically, doing so in a way that achieves the goal of mathematics' Langlands program "to find a geometric object that encoded answers to questions in number theory", directly linking two separate disciplines within mathematics in the process. In terms of significance, the achievement is akin to the invention of a mathematical Rosetta Stone.
To do that, Fargues and Scholze developed a new way to define a geometric object:
Imagine that you start with an unorganized collection of points — a “cloud of dust,” in Scholze’s words — that you want to glue together in just the right way to assemble the object you’re looking for. The theory Fargues and Scholze developed provides exact mathematical directions for performing that gluing and certifies that, in the end, you will get the Fargues-Fontaine curve. And this time, it’s defined in just the right way for the task at hand — addressing the local Langlands correspondence.
“That’s technically the only way we can get our hands on it,” said Scholze. “You have to rebuild a lot of foundations of geometry in this kind of framework, and it was very surprising to me that it is possible.”
After they’d defined the Fargues-Fontaine curve, Fargues and Scholze embarked on the next stage of their journey: equipping it with the features necessary to prove a correspondence between representations of Galois groups and representations of p-adic groups.
For background, here is a presentation by Laurent Fargues on the Fargues-Fontaine curve, which is where we found the accompanying illustration. Here is an outstanding introduction into the math of p-adic groups and how the structure of their numbering systems can lead to more powerful tools for answering mathematical questions. Meanwhile, there's really no such thing as a simple introduction to Galois groups, but here's the most gentle one we could find.
Turning back to how Fargues and Scholze defined the geometry for the Fargues-Fontaine curve to develop their proof, using the established building blocks of "sheaves":
Sheaves were substantially developed in the 1950s by Alexander Grothendieck, and they keep track of how algebraic and geometric features of the underlying geometric object interact with each other. For decades, mathematicians have suspected they might be the best objects to focus on in the geometric Langlands program.
“You reinterpret the theory of representations of Galois groups in terms of sheaves,” said Conrad.
There are local and global versions of the geometric Langlands program, just as there are for the original one. Questions about sheaves relate to the global geometric program, which Fargues suspected could connect to the local Langlands correspondence. The issue was that mathematicians didn’t have the right kinds of sheaves defined on the right kind of geometric object to carry the day. Now Fargues and Scholze have provided them, via the Fargues-Fontaine curve.
That passage may understate Fargues and Scholze's achievement, because it is a long way from suspecting sheaves might be a good objects to use for their purpose to identifying what kinds it would take and actually doing it in a solid proof.
... they came up with two different kinds: Coherent sheaves correspond to representations of p-adic groups, and étale sheaves to representations of Galois groups. In their new paper, Fargues and Scholze prove that there’s always a way to match a coherent sheaf with an étale sheaf, and as a result there’s always a way to match a representation of a p-adic group with a representation of a Galois group.
In this way, they finally proved one direction of the local Langlands correspondence. But the other direction remains an open question.
“It gives you one direction, how to go from a representation of a p-adic group to a representation of a Galois group, but doesn’t tell you how to go back,” said Scholze.
That puts the up and coming math at something of a halfway point in terms of the grander Langlands project, where getting to such a halfway point represents a major accomplishment for an endeavor that began in the late 1960s. Conceivably, Fargues and Scholze's one-way proof could be enough provide a new means for cracking several of the Clay Institute's Millennium problems, the solutions to which carry million dollar prizes for the mathematicians who succeed.
The bigger reward however would be to complete Langlands' conjecture and demonstrate reciprocity, the ability to start from Galois groups and go back to a p-adic number system. Succeeding in that would unleash new tools for mathematicians to develop and demonstrate new proofs. It's an exciting time for the potential to develop a unified field of mathematics!