| The kingdom of mathematics can be divided into many disparate realms: number theory, geometry, algebra, topology, analysis, combinatorics. But the greatest mathematical achievements often happen when someone discovers an unexpected connection between two of these domains. This makes it possible to port fresh ideas from one field to attack problems in another. Mathematicians call it "bridge building."
The Langlands program, introduced by the Canadian mathematician Robert Langlands in 1967, is bridge building taken to its extreme. As Quanta explained in a comprehensive video we produced in 2022, the program comprises a sweeping suite of conjectures that act like an interstate highway system, intimately linking some of the most distant reaches of the mathematical universe.
Langlands' original conjectures posited a precise correspondence between very different mathematical entities in the fields of number theory (the study of arithmetic) and harmonic analysis (the study of how functions can be broken up into simpler pieces). In 1994, the mathematician Andrew Wiles proved one small instance of this correspondence: He showed that every elliptic curve, a type of equation that comes from number theory, can be associated to a highly symmetric function from harmonic analysis called a modular form. Doing so allowed him to prove Fermat's Last Theorem, one of math's biggest open problems at the time. It also opened up a rich and important dialogue between the two areas of study.
The Langlands program aims to build bridges between far more than elliptic curves and modular forms. It hypothesizes correspondences between more general objects in number theory and harmonic analysis, and hints at links to other fields entirely, including algebraic geometry, representation theory and quantum physics.
This is why the Langlands program is often referred to as the "grand unified theory of mathematics." The implications of such a theory are both philosophical and practical. What were once thought of as different mathematical worlds, each with its own language and customs, might actually all be reflections of the same underlying reality. And an understanding of this unity would give mathematicians the ability to leap across the subject's illusory boundaries — to attack a given problem in whichever setting suits it best and then bring the result back to its place of origin.
Although it often seems impossibly abstract and esoteric, the Langlands program is the broadest and most ambitious undertaking in mathematics today. For decades, in papers with staggering page counts, mathematicians have taken up Langlands' dream and carried it places he never imagined. And this expansion shows no signs of slowing.
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The bridge-building ambitions of the Langlands program date back to the French mathematician André Weil. In a jailhouse letter to his sister, the brilliant, unconventional philosopher Simone Weil, he dreamed of a "Rosetta stone" for math — a way to translate between the languages of several different fields. It was no coincidence, then, that when Langlands first sketched out his own vision of mathematical unity in a 17-page letter, he addressed it to Weil.
While Langlands' original vision connected number theory and harmonic analysis, mathematicians have since used Weil's Rosetta stone to translate his program into other settings. In the 1980s, Vladimir Drinfeld formulated a geometric interpretation of one of its key conjectures. For decades, it went unsolved. Then, earlier this year, in five papers totaling more than 800 pages, nine mathematicians finally proved the geometric Langlands conjecture. The work was a momentous achievement, and it lent credence to the Langlands program's many other fronts.
Mathematicians had to forge lots of connections to geometry just to get to that point. In 2021, Quanta reported on how two mathematicians, Peter Scholze (one of the youngest Fields medalists ever) and Laurent Fargues, offered "the most tangible evidence yet that earlier mathematicians weren't foolish to attempt the Langlands program by geometric means." The work involved using a geometric object called the Fargues-Fontaine curve to prove a particular Langlands correspondence.
In 2023, a trio of mathematicians uncovered yet another link to geometry when they returned to some of Langlands' early work and proved a correspondence — between so-called L-functions and periods — that he had thought about but hadn't considered important enough to study. It took years of effort and ultimately a 451-page paper, but by showing that geometry played a role in the problem, the three mathematicians proved Langlands wrong. The correspondence was important after all.
The Langlands program is only continuing to expand its reach. There are still many bridges left to build and explore — and no way to know what connections will be revealed next, or what feats they'll allow mathematicians to accomplish. By upending assumptions about the geography of mathematics, this unified theory of mathematics could someday change the way we conceive of the subject itself. | 
