Geometric Langlands Conjecture Proved

“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 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.”

“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.”

“Gaitsgory has dedicated the past 30 years to proving the geometric Langlands conjecture. Over the decades, he and his collaborators have developed a massive body of work on which the new proof rests.”

“Vincent Lafforgue, of Grenoble Alps University, likened these advances to a “rising sea,” in the spirit of the preeminent 20th-century mathematician Alexander Grothendieck, who spoke of tackling hard problems by creating a gradually rising sea of ideas around them.”

“In classical Fourier analysis, a procedure called the Fourier transform creates a correspondence between two different ways of thinking about the graph of a wave”

“On one side of the correspondence are the waves themselves”

“On the other side of the correspondence is the spectrum of frequencies of the sine waves — that is, their pitches”

“The Fourier transform goes back and forth between these two sides. In one direction, it allows you to break down a wave into a collection of frequencies; in the other, it allows you to reconstruct the wave from its constituent frequencies.”

“The ability to move across this divide is central to a wide range of applications — without it, we wouldn’t have modern telecommunications, or signal processing, or magnetic resonance imaging, or numerous other essentials of modern life.”

“Langlands proposed that something similar occurs in the number theory and function field columns of the Rosetta stone, but with more complicated waves and frequencies.”

“Langlands’ program came to be seen, in the words of Edward Frenkel of the University of California, Berkeley, as a “grand unified theory of mathematics.””

“Right from the beginning of Langlands’ work, mathematicians had an idea of what the spectral side of a geometric Langlands correspondence should look like. This third column of Weil’s Rosetta stone concerns compact Riemann surfaces, which are spheres, doughnuts, and doughnuts with multiple holes. A given Riemann surface has a corresponding object called its fundamental group, which tracks the different ways that loops can wind about the surface. Mathematicians suspected that the spectral side of the geometric Langlands correspondence should consist of certain distillations of the fundamental group known as its “representations.””

“Riemann surfaces play a large role in physics, particularly in conformal field theory, which governs the behavior of subatomic particles in certain force fields.”

“Gaitsgory and Nick Rozenblyum of the University of Toronto wrote two books about sheaves totaling nearly 1,000 pages. Only once in the two-volume set is the geometric Langlands program even mentioned. “But its purpose was to lay the foundations, which we ultimately used very intensively,” Gaitsgory said.”

“A host of further challenges await mathematicians — exploring the connection to quantum physics more deeply, extending the result to Riemann surfaces with punctures, and figuring out the implications for the other columns of the Rosetta stone”

“Not everything can carry over — for instance, in the number theory and function field settings, there is no counterpart to the conformal field theory ideas that enabled researchers to construct special eigensheaves in the geometric setting”

“But many researchers are optimistic that the rising sea of ideas will eventually reach these other domains. “It’s going to seep through all the barriers between subjects,” Ben-Zvi said.”