For a few months now, the mathematical world has been abuzz. Rumors abound of a new proof, monumental in length and virtually impenetrable even to the experts – and which, if correct, has the potential to reform the entire mathematical landscape from here on out.
Now, as the dust settles around the nearly 1000 pages of dense math offered up by a team of nine mathematicians, a consensus seems to be growing: it’s true. A key piece of the Langlands Program – a set of ideas so important it’s sometimes referred to as the “grand unified theory” of math – really has been toppled.
What is the Langlands Program?
Like so many foundational ideas in mathematics, the concept that is now known as the Langlands Program began as a somewhat hastily scribbled note to a pal about something that looked like it might be cool. You know, if it panned out.
“Dear Professor Weil,” the then fairly fledgling mathematician Robert Langlands wrote in a January 1967 letter to the mathematical legend that was André Weil. “While trying to formulate clearly the question I was asking you before Chern’s talk I was led to two more general questions.”
“Your opinion of these questions would be appreciated,” he continued. “I have not had a chance to think over these questions seriously and I would not ask them except as the continuation of a casual conversation.”
Now, perhaps this could be considered presumptuous – it would be kind of like your high school PE teacher asking LeBron James to weigh in on a new kind of offensive play he’s been thinking about lately – but as it turned out, that letter contained the germ of something monumental.
Known today as the Langlands Program, what he had sketched out would prove to be “a collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms, and representation theory,” mathematician Bill Casselman, now Professor Emeritus at the University of British Columbia, wrote in 1988. “These have formed the core of a program still being carried out, and have come to play a central role in all three subjects.”
So, what makes it so important? Well, let’s start with a simple example: can you work out the following math problem?
(X – VII) × III = ?
It’s not a particularly difficult one, but chances are extremely slim that you could work it out directly. More likely, you did it in three steps: first, you’d have translated it into a language you’re more confident in – namely, Arabic numerals; second, you’d have actually worked it out; and third, you’d have translated it back into the original notation to get, hopefully, an answer of IX.
Scale that process up by a few orders of magnitude, and you have the general idea of the Langlands Program. It’s “the ‘theory of everything’ in mathematics,” mathematics educator Judy Mendaglio wrote in 2018, just after Langlands had been awarded the prestigious Abel Prize in recognition of his work. “[A] set of conjectures that seek to unify knowledge from different branches of mathematics.”
“The idea is that a problem in one area of mathematics may be very difficult to analyze using the tools available in that area,” Mendaglio explained. “However, if the structures within the problem can be related to similar structures in a different field, where there are better analytical tools available, then the analysis may be conducted with less difficulty and the results related back to the original problem. In this way, even deeper structures in the original area of mathematics are revealed.”
So what’s the news?
At its core, the “Langlands Program” is actually a collection of closely related conjectures across a range of mathematical fields. “It is such a vast subject that few can really have an overview,” wrote theoretical physicist Edward Witten in 2007. “Despite all the hard work, I personally only understand a tiny bit of the Langlands program.”
“The deepest aspect of it, as far as we know, involves the number theory setting where Langlands started close to forty years ago,” he noted. “However, the Langlands program has all kinds of manifestations.”
And one of the major branches, especially in the past couple of decades, has been the “geometric” form of the Langlands Program – a corner of the problem in which “some of the ideas are converted from number theory into statements in geometry,” Witten explained.
It’s traditionally been one of the more fruitful routes of attack – but still, fiendishly difficult. So when claims surfaced this year of not just a breakthrough, but an entire dang proof of the geometric Langlands Conjecture – well, it definitely caught people’s attention.
“It’s the first time we have a really complete understanding of one corner of the Langlands program, and that’s inspiring,” David Ben-Zvi, Professor of Mathematics at the University of Texas at Austin, told New Scientist. “That kind of gives you confidence that we understand what its main issues are.”
“There are a lot of subtleties and bells and whistles and complications that appear,” he said, “and this is the first place where they’ve all been kind of systematically resolved.”
It’s certainly no small achievement – in any sense of the word. Taking up five papers across more than 900 pages, the proof is “really a tremendous amount of work,” Edward Frenkel, Professor of Mathematics at the University of California, Berkeley, told New Scientist. In fact, it’s so complex that even other mathematicians find it semi-bewildering – though many are nevertheless confident that it holds up.
“It is beautiful mathematics, the best of its kind,” Alexander Beilinson, one of the main figures behind the formulation of the geometric Langlands Program, told Quanta Magazine earlier this year.
What does this mean?
Okay, so it’s big news within the math community, but why should the average Joe care about this? Well, as you might expect from something casually referred to as a “theory of everything”, this result can affect much more than just abstract math.
“It wasn’t just that they went and proved it,” Ben-Zvi told Quanta. “They developed whole worlds around it […] It’s going to seep through all the barriers between subjects.”
It’s already old news, for example, that the geometric Langlands Program has strong connections with quantum and condensed matter physics, and the few mathematicians who understand the proof so far think it’s likely to attract attention in that area pretty soon.
Even more fundamentally, though, are the potential implications for the other two corners of the Program – those centered in number theory and function fields.
“It feels (at least to me) more like […] one piece of a big rock has been chipped off,” Dennis Gaitsgory, a researcher at the Max Planck Institute and one of the nine-person team behind the new mega-proof, told Quanta. “But we are still far from the core.”
That’s not for lack of trying, however. Along with fellow author Sam Raskin, Professor of Mathematics at Yale, Gaitsgory has already made some progress translating the proof over to the function fields corner of the Program.
And they’re not alone: “I’m definitely one of the people who are now trying to translate all this geometric Langlands stuff,” said Peter Scholze, a number theorist at the Max Planck Institute who was not involved in the proof – although he’s “currently a few papers behind,” he told Quanta, “trying to read what they did in around 2010.”
For others, though, the real reward is the proof itself – and what it reveals about the nature of math.
“A lot of the things that go into geometric Langlands were things I imprinted on as a student,” Raskin told Yale News. “It had a big impact on my mathematical tastes. It’s a set of questions I’ve always found interesting and rewarding to work on.”
“There’s this experience I have sometimes with mathematics where it seems strange how much there is to keep discovering and engaging with,” he added. “It doesn’t seem like there’s a reason for mathematics to be as complex and interesting as it is. It’s not just a random zoo of things. You gain an intuition in thinking about mathematical objects, even though you can’t always approach them.”
