Sam Raskin has wrapped his head around a math problem so complex it took five academic studies — and more than 900 pages — to solve.
The results are a sweeping, game-changing math proof that was decades in the making. Working with Dennis Gaitsgory of the Max Planck Institute and a team of seven other mathematicians, Raskin has solved a segment of the Langlands Conjectures, long considered a “Rosetta Stone” of mathematics.
The Langlands Conjectures, named after Canadian mathematician (and former Yale professor) Robert Langlands, suggested in the 1960s that deep, unproven connections exist between number theory, harmonic analysis, and geometry — three areas of math long considered distinctly separate. Proving these connections, mathematicians say, could suggest ways to translate certain areas of math that had seemed dissimilar.
Raskin, the James E. English Professor of Mathematics in Yale’s Faculty of Arts and Sciences, and Gaitsgory led a team that solved the geometry portion of Langlands.
“There’s definitely something buzzy about this one,” said Raskin, a soft-spoken man who will fill a blackboard with equations at the drop of a heptagon. “People always love to see an old problem fall.”
Alexander Goncharov, the Philip Schuyler Beebe Professor of Mathematics at Yale, calls the geometric Langlands solution a “landmark” achievement.
“The Langlands Conjecture is one of the most fundamental problems in mathematics, in terms of its reach and value. The full solution of the Geometric Langlands Conjecture is truly amazing. It had been considered totally out of reach before this,” said Goncharov, an expert on arithmetic algebraic geometry. “And on the way to the solution, Sam and his collaborators had to develop new mathematics tools just to make it possible to think about the problem.”
In a New Scientist article about the proof, David Ben-Zvi, a mathematician at the University of Texas at Austin, wrote, “It’s the first time we have a really complete understanding of one corner of the Langlands program, and that’s inspiring.”
Devotees of geometric Langlands are not the only observers interested in Raskin’s proof. There is also interest from theoretical physicists, who have noted that a geometrical symmetry group (a key feature in geometric Langlands) rests at the heart of explaining particle physics. In addition, physicists say Raskin’s methodology may shed light on unsolved connections between natural phenomena such as electricity and magnetism.
It all dovetails nicely with one of Raskin’s core beliefs about his work: “I think about mathematics as a part of nature, with interesting things to be discovered along the way,” he said.
The road to Langlands
In a sense, Raskin has been pointed toward this eventual success since he was a teenager.
As an undergraduate at the University of Chicago, many of his math mentors at both the faculty and graduate student levels were delving into geometric Langlands research; campus symposiums and seminars were devoted to it. As a graduate student, Raskin met Gaitsgory, who would become his Ph.D. advisor at Harvard.
“A lot of the things that go into geometric Langlands were things I imprinted on as a student,” Raskin said. “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 adds. “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.”
Intuition and informed instinct have certainly been part of geometric Langlands research, even before it went by that name.
Beginning in the 18th century, mathematicians started to discover weird patterns in prime numbers. Generations of scholars continued to delve into these patterns and what they might mean, well into the 20th century.
For example, scholars noticed there are symmetries in certain prime numbers that are consistently commutative, meaning that even if you changed their order in a binary operation, you’d get the same result. Normally, symmetries are not commutative.
“Everyone knew there was this question of what happens if the symmetry group is not commutative and mathematicians started studying examples of this,” Raskin explained. “They found examples they didn’t necessarily understand. And then Robert Langlands came out with a very systematic proposal for what the general story should be about these non-commutative symmetries in number theory.”
Then geometry joined the discussion. The same, unsolved behavior seen in prime numbers was also happening in geometry, in points on a curve. By the 1990s, mathematicians Alexander Beilinson and Vladimir Drinfeld (Raskin met both of them in his student days) had zeroed in on an approach to proving geometric Langlands using highly complex (and not yet fully developed) geometry constructs called “eigensheaves.”
It would take more than 20 years for another group of mathematicians — Raskin, Gaitsgory, and their team — to prove it.
“It’s very remarkable how, as you probe Langlands with various calculations, you develop this complex intuition for how things are going to behave,” Raskin said. “You begin to be able to predict outcomes to questions you haven’t studied, sometimes.”
‘The satisfaction of a proof’
A key step to the solution came in 2020.
Raskin, then a faculty member at the University of Texas at Austin, was part of a six-person team that published a paper on the harmonic analysis portion of Langlands — the idea that mathematical functions can be broken down into “waves.”
In a 2022 paper, Raskin and one of his graduate students, Joakim Faergeman (who is now a graduate student at Yale) expanded upon certain aspects of the 2020 paper and applied them specifically to geometric Langlands. Part of the work involved thinking through some rather thorny problems with math concepts known as “irreducible representations.”
Once Raskin accomplished that — at a time when he and his wife were expecting the birth of their second child — a clear proof for geometric Langlands was within reach.
He and his colleagues completed their five-study solution earlier this year.
“We have the only theorem that works for all groups,” Raskin said. “A very robust, satisfying theorem.
“In math, we have this luxury — you can actually say something is true,” he added. “It’s not a model. It’s math. Things are either true or false. This is a case where we can say that all of the intuition from theoretical studies was correct. It’s the satisfaction of a proof.”
