In a forthcoming issue of the AMS Bulletin, there will be a long review by Robert Langlands of the book p-adic Automorphic Forms on Shimura Varieties by Haruzo Hida. The book itself is on a very technical subject, but the review includes long sections by Langlands that are much more generally about the current state of the so-called “Langlands Program”. While this inspired the “Geometric Langlands Program” that I’ve written about here recently, it’s a quite different subject, one that is very much central to research in number theory. Basically it deals with number fields (extensions of the field of rational numbers), and the function fields of geometric Langlands involve very different issues.
At the same time as making available his review, Langlands also made available commented copies of his correspondence with various experts in the subject about a draft of the review that he had sent them. Much of the review itself is likely to only be accessible to experts, and this is even more true of the correspondence. Casselman comments:
I also have the impression that you have edited this review for the pleasure of experts, and that therefore the cutting-room floor is filled with the sort of stuff The Naive Reader might appreciate.
The response to this from Langlands is:
I had in mind explaining more, but the editing was not a matter of choice but of necessity. I did not understand enough to say more.
I suspect few people will be able to follow the discussion here, but it gives a good idea of what is going on in an active but very difficult area of mathematics.
Both of these documents are from a fantastic resource, a web-site set up by Bill Casselman which contains pretty much the complete works of Langlands on-line. If you want to know more about the Langlands program and where it comes from, there’s lots of material worth reading on the site. One of the more readable sources for a beginner is the 1989 Gibbs symposium lecture on Representation theory- its rise and its role in number theory.
For a lower form of entertainment, there’s another book review, of Leonard Mlodinow’s Euclid’s Window, which appeared in the AMS Notices. The review is pretty much completely over the top, beginning with the sentence:
This is a shallow book on deep matters, about which the author knows next to nothing.
Update: I should also have mentioned that last month there was a small conference at the IAS on The L-group at 40, in honor Langlands’ 70th birthday.