The Langlands Program, Part I
The Langlands program is the name generally given to a highly influential vision unifying two different areas of mathematics, referring to a set of ideas first brought to light by Robert Langlands at the end of the 1960s. Langlands never won a Fields medal or Abel prize (perhaps this second circumstance will get remedied in the next few years), but the work of two of the Laureates speaking here in Heidelberg yesterday depends crucially on his. In this posting I’ll try and explain a little bit about this and how the work of Ngô Bảo Châu fits into it, leaving that of Sir Andrew Wiles for next time.
To state exactly what the Langlands vision is requires invoking a wide array of complex and sophisticated mathematical formalisms, so I certainly won’t do that here. An oversimplified caricature though is that Langlands posits an identification between the representations of two very different kinds of groups: Galois groups and matrix groups. It is this surprising identification of two quite different mathematical problems from different fields that provides a dramatic unifying vision.
Representations of Galois groups are central objects in modern number theory. Instead of looking explicitly for solutions in a number field of some polynomial equation, one can instead think of such solutions as a representation of the Galois group of the number field, and ask about the representation. Wiles was able to prove Fermat’s Last Theorem by showing that Galois representations with certain properties did not exist.
The hypothesis of “Langlands reciprocity” is that such Galois representations correspond to representations of matrix groups, such as the group of invertible real matrices of a fixed dimension. Complexity here arises due to the fact that the irreducible representations are infinite-dimensional, and the need to consider the cases of all places of a number field. For example, in the case of the rational numbers, one considers invertible matrices with entries p-adic numbers for each prime p, as well as matrices with real number entries. The “Langlands functoriality” hypothesis says that such representations of matrices for different groups will share the same relations that one expects for Galois representations (thought of as homomorphisms from the Galois group to these different groups).
A general proof of Langlands functoriality remains a long ways off, but Langlands had early on given an outline of what needed to be done to prove a special case (known as the “endoscopic” case). Some steps in this outline were completed relatively quickly, but one turned out be the hardest to crack: the “Fundamental Lemma”. In mathematics a “lemma” is generally a simple technical step needed as part of an argument, with the term implying that it is a minor issue. For many year when I would ask my Columbia colleague (and Langlands collaborator) Herve Jacquet what was new in the Langlands program, he would half-jokingly say “well, there’s progress on this lemma…”.
The Fundamental Lemma was finally proven in 2009 by Ngô, and it was for this that he won the Fields Medal in 2010. In his talk here he surveyed the subject of the Langlands program, from the point of view not of representations, but of L-functions, functions that characterize the representations.
Ngô grew up in Vietnam, where he says his serious interest in mathematics was spurred by doing badly in a mathematics competition, which caused him to get interested in working seriously on the subject. After high school he went to Hungary to study combinatorics, but this was disrupted by the fall of the Soviet system there, causing him to move to Paris for his graduate studies. There he ended up studying with Gerard Laumon, who set him to work on questions related to the Fundamental Lemma.
Ngô, now at the University of Chicago, gives much of the credit to Laumon for his success. He also credits other mathematicians working in the area (especially Kottwitz and Waldspurger) with helping him a lot. Work on this problem (like in many areas of mathematics), has been characterized not by cut-throat competition, but by cooperation and generous provision of help from researchers to each other.
In recent years Ngô has been regularly spending time back in Vietnam, trying to improve mathematics education there, as well as the infrastructure for mathematics research. His success has been a significant inspiration for Vietnamese young people, encouraging them to study mathematics and consider careers in mathematical research.