"Langlands outlined the first version of the programme in 1967, when he was a young mathematician visiting the IAS. His starting point was the theory of algebraic equations (such as the quadratic, or second-degree, equations that children learn in school). In the 1800s, French mathematician Évariste Galois discovered that, in general, equations of higher degree can be solved only partially. ... Inspired by subsequent developments in Galois’s theory, Langlands’ approach allowed researchers to translate algebra problems into the ‘language’ of harmonic analysis, the branch of mathematics that breaks complex waveforms down into simpler, sinusoidal building blocks. In the 1980s, Vladimir Drinfel’d, a Ukrainian-born mathematician now at the University of Chicago in Illinois, and others proposed a similar connection between geometry and harmonic analysis. Although this idea seemed to be only loosely inspired by the Langlands programme, mathematicians subsequently found stronger evidence that the two fields are connected. (Drinfel’d received a Fields Medal in 1990.)"