Langlands program

What was discovered

A web of conjectures relating Galois representations (number theory) to automorphic forms (analysis), generalising Gauss and Artin reciprocity. It is called the grand unified theory of mathematics.

Langlands wrote to Weil that if he was not interested, he had a wastebasket at hand.

What it led to

• Wiles's proof (modularity)
• the fundamental lemma (Ngô, 2008)
• the geometric Langlands programme (proof announced in 2024)
• connections with physics (Kapustin-Witten)

Built on

• Modular arithmetic and quadratic reciprocity1801
• Takagi's class field theory1920
• Weil conjectures and their proof1949-1974
• modular forms

Made possible

• Proof of the geometric Langlands conjecture2024

Links