The work of Robert Langlands

TL;DR

This paper reviews Robert Langlands' influential contributions to mathematics, highlighting the unifying Langlands Program across representation theory, automorphic forms, number theory, and arithmetic geometry, following his 2018 Abel Prize.

Contribution

It provides a comprehensive overview of Langlands' work, emphasizing the continuity and unifying themes across multiple fundamental mathematical areas.

Findings

Unification of key mathematical areas through the Langlands Program

Insights into the deep connections in representation theory and number theory

Accessible explanations for some complex aspects of Langlands' work

Abstract

This is a report on the work of Robert Langlands, following his award of the Abel Prize in 2018. It includes his contributions to the general areas of Representation Theory, Automorphic Forms, Number Theory and Arithmetic Geometry. We have tried to communicate the remarkable continuity that runs throughout all of his work, with its roots in several fundamental areas of mathematics. What is now known as the Langlands Program represents a unification of some of the deepest parts of these areas. We hope that at least some parts of the report will be accessible to a broader mathematical audience. Other parts will inevitably be more difficult. However, we can also hope that the report is presented in such a way as to lead to a better understanding of all sides of Langlands' work.