Robert Langlands has been awarded the 2018 Abel Prize, one of the chief honours in the world of maths, for opening up a research program which has dominated large areas of maths for the last 50 years, and still has many unsolved problems.
Langlands is Canadian by origin, but has lived and worked in the USA for many years. Another honour he has received is being allocated Einstein’s old office in the Institute for Advanced Study near Princeton University.
Ed Frenkel’s book “Love and Math”, alongside being an autobiography, tries to explain the Langlands program. Despite its breadth, the Langlands program is essentially about connections between different areas of maths at a level not studied at school, or even in most first-degree maths course, so a quick explanation is difficult. (Click here for more.)
This report by Rachel Thomas from Plus magazine does a better job than any other short explanation I’ve seen.
In 1967 a 30-year-old mathematician, still in the early stages of his career, wrote a 17 page letter to the eminent French mathematician André Weil. The covering note that he sent with the letter said:
“After I wrote [this letter] I realized there was hardly a statement in it of which I was certain. If you are willing to read it as pure speculation I would appreciate that; if not – I am sure you have a waste basket handy.”
The letter may have contained many unproved statements, but these turned out to be asking questions that would create a whole area of mathematical research, unmatched in modern mathematics for its scope, deep results and sheer size in terms of the number of mathematicians it has enticed. The author of the letter – Robert Langlands – has been awarded the 2018 Abel Prize for his “visionary program” that bridges previously unconnected areas of mathematics, and is now frequently described as a “grand unified theory of mathematics”.
The insights contained in Langlands’ letter to Weil, and more fully in his 1970 lecture Problems in the Theory of Automorphic Forms, set out a programme of research that is now referred to as the Langlands program. Langlands made a number of conjectures about the connections between two previously unrelated areas: number theory and harmonic analysis.
Harmonic analysis explores how functions and signals can be represented as a sum of waves. For example, the sound wave of the middle A on a tuning fork is a perfect example of a sine wave, written mathematically as sin(x)… Automorphic forms are a more generalised version of periodic waves, like the familiar sine wave, which operate in more complicated geometric settings….