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"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.”
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). ..."
The Langlands program, introduced by the Canadian mathematician Robert Langlands in 1967, is bridge building taken to its extreme. ... 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.
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