Automorphic form

Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups.

Quotes

• He liked putting different pieces of mathematics together: geometry, analysis, topology… so automorphic forms should have appealed to him. But for some reason he didn’t get interested in that at the time. I think the junction between Grothendieck and Langlands was realized only in 1972 at Antwerp. Serre had given a course on Weil’s theorem in 1967–1968. But after 1968 Grothendieck had other interests. And before 1967 things were not ripe. I’m not sure.
  • Reminiscences of Grothendieck and His School
• Automorphic forms can be thought of as fundamental particles in harmonic analysis, which deals in part with waves and frequencies. In a symphony orchestra, automorphic forms issue instructions and work with eigenvalues — the different speeds a violin string moves when struck, for instance — to produce the notes played.
  • Contenta, Sandro. "The Canadian who reinvented mathematics". Toronto Star.