The 2018 Abel Prize has been awarded to Robert Langlands, an excellent choice. The so-called “Langlands program” has been a huge influence on modern mathematics, providing deep insight into the structure of number theory while linking together disparate fields of mathematics, as well as quantum field theories and physics.

The Abel Prize site provides a wealth of information about Langlands and his work. Davide Castelvecchi at Nature appropriately describes the Langlands program as a “grand unified theory of mathematics” (Edward Frenkel’s Love and Math popularized this description).

Many blog posts here have discussed the Langlands program and ideas that have developed out of it. For a good example of how wide the impact of these ideas has been, this week the Perimeter Institute will be hosting a conference discussing the latest work on the geometric version of the Langlands program, as well as connections to gauge theory and conformal field theory.

For the original work of Langlands himself, besides the material at the Abel site, the AMS Bulletin has recently published a long article by Julia Mueller. For the original sources and a wealth of other material written by Langlands himself, see the IAS site that collects his writings.

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Wonderful! I assume this breaks the long-standing tradition of rewarding mathematicians for theorems but not for conjectures (which can sometimes be much more important).

Peter Shor,

Langlands did prove some special cases of his conjectures (e.g. Jacquet-Langlands, Langlands-Tunnell, local Langlands at

RandCfor GL(n)), and his work was much, much more than just formulating conjectures, containing all sorts of important detailed results. It is interesting to speculate what the math community would think of someone who only formulated conjectures of this importance, but never proved anything.The conjectures were built on some serious theorems – classification of irreducible representations of real groups (standing on Harish-Chandra), meromorphic continuation of Eisenstein series, etc etc ….

But yes, re-contextualising Satake’s isomorphism, that was more influential than either of the above.

And he deserves the prize for his prose style alone.

I enjoyed Lafforgue’s geometrically-oriented survey of recent work

https://arxiv.org/pdf/1803.03791.pdf

For those of us who need some help, Kevin Hartnett provides a brief but nice bit of exposition in Quanta.

https://www.quantamagazine.org/robert-langlands-mathematical-visionary-wins-the-abel-prize-20180320/

I second Peter Shor’s comment! I am old enough to remember a joke that went around math departments in the 1990s that with Bill Thurston you could now get a Fields Medal just for conjecturing something — and that with Edward Witten you didn’t even need your conjecture to be clearly stated. Although that was of course a bit unfair in both cases ðŸ™‚