Natalie Wolchover writes about the Langlands program, a decades-long mathematical project that began when Robert Langlands spotted a correspondence in 1967 between Galois representations (symmetries of polynomial equations in number theory) and modular forms (objects in harmonic analysis). The program has since expanded into a web of such "wormhole" correspondences across number theory, geometry over finite fields, and Riemann surfaces, earning it the nickname "grand unified theory of mathematics." Wolchover explains the original correspondence using the equation x³−2=0 as a concrete example, then explores what mathematicians think these connections might ultimately mean.
- The correspondence enabled Andrew Wiles to prove Fermat's Last Theorem in 1994 via the Taniyama-Shimura-Weil conjecture, a precursor to Langlands' general vision.
- André Weil had independently glimpsed the same kind of pattern in a 1940 letter to his sister Simone Weil, describing a "mathematical Rosetta stone" with three columns.
- Physicists Anton Kapustin and Edward Witten showed the geometric Langlands correspondence is a consequence of electric-magnetic duality in a specific quantum field theory, suggesting both sides may be dual views of a single underlying structure.
- Edward Frenkel compares the two sides of a Langlands correspondence to different shadows cast by the same object (like a coffee cup projecting a disk on a table and a rectangle on a wall), with the "true source" remaining undiscovered.
- The correspondence enabled Andrew Wiles to prove Fermat's Last Theorem in 1994 via the Taniyama-Shimura-Weil conjecture, a precursor to Langlands' general vision.
- André Weil had independently glimpsed the same kind of pattern in a 1940 letter to his sister Simone Weil, describing a "mathematical Rosetta stone" with three columns.
- Physicists Anton Kapustin and Edward Witten showed the geometric Langlands correspondence is a consequence of electric-magnetic duality in a specific quantum field theory, suggesting both sides may be dual views of a single underlying structure.
- Edward Frenkel compares the two sides of a Langlands correspondence to different shadows cast by the same object (like a coffee cup projecting a disk on a table and a rectangle on a wall), with the "true source" remaining undiscovered.
