The Letter
Robert Langlands and the Dictionary of Worlds
94 + 223 p.
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In January 1967, at Princeton, the Canadian mathematician Robert Langlands wrote his senior colleague André Weil seventeen handwritten pages sketching a precise correspondence between two mathematical worlds that seemed to have nothing in common. That letter became, over the decades, one of the most ambitious research programs of the twentieth century. This book tells what was in it, and what it took for the letter to be right — through to the proof of 2024.
Contents
The symmetries of numbers · Representations · Automorphic forms · L-functions
Princeton, January 1967 · The dictionary · Functoriality, local, global
Class field theory: the first entry · Modularity and Fermat · The trace formula and the fundamental lemma · Geometric Langlands · 2024