By Kumar Murty

By any measure, Prof. Robert Langlands can be considered the Canadian living mathematician who has had, and is having, the greatest impact on our discipline. His far-reaching ideas and conjectures have mobilized mathematicians around the world, including Fields medallists Vladimir Drinfeld, Laurent Lafforgue and Ngo Bau Chau, all of whom are busily engaged in investigating, and in some cases proving, the phenomena to which he has drawn attention.

This past July, I had the honour of visiting Prof. Langlands, and his wife, Charlotte, in their Ottawa home. We spent several hours together, mostly reminiscing about the personal highlights of his career and the individuals who influenced his trajectory. Prof. Langlands' time teaching in Turkey remains a special memory in his life, so much so that a framed poster of a conference he gave there in 2007 is the first thing that greets visitors when they enter the Langlands apartment. When asked what he likes to spend his time doing these days, he brought out a novel he was reading a novel in Turkish – one of the numerous languages he taught himself. It was also very special to get a chance to talk with Charlotte, a formidable artistic talent whose sculptures punctuate the apartment. Together, they have shared an extraordinary life.

On the occasion of Prof. Langlands' 88th birthday, which takes place today on October 6, I wanted to share the remarks I made during our Forward From the Fields Medal event, where we presented him with the inaugural Fields Institute Lifetime Achievement Award.

A true original thinker 

Robert Langlands was born in New Westminister, a town south of Vancouver. He showed intellectual ability at an early age and in high school, he was encouraged by his teachers to pursue studies at the university level. He received his Bachelor’s and Master’s degrees from the University of British Columbia and his PhD was from Yale University on the representation theory of Lie groups. He stayed at Yale for only two years 1958-1960, and, in fact, completed his thesis in the first year. He used the second year to study Eisenstein series and proved his first results on their analytic continuation.

He was then appointed as a junior faculty member at Princeton University and spent a total of 7 years there, including a year at the Institute for Advanced Study. It was during this period that he became interested in automorphic forms and the work of Harish-Chandra and Atle Selberg, and wrote an important paper on the dimension of the spaces of automorphic forms. He also produced fundamental results on Eisenstein series.

After spending a year in Turkey (a visit that he says was at least partially inspired by Agatha Christie novels!), he moved to Yale University as a faculty member in 1967. That was the same year that at the age of 30, he wrote a famous letter to Andre Weil. Bill Casselman of UBC describes the content of this letter as “a collection of far-reaching and uncannily accurate conjectures relating number theory, automorphic forms and representation theory”. A press release on the occasion of Langlands winning the Abel Prize in 2018 remarked that this 17-page letter to Weil presented insights that “were so radical and so rich that the mechanisms he suggested to bridge these mathematical fields led to a project named the Langlands program.”

In 1972, he was appointed to the School of Mathematics of the Institute for Advanced Study in Princeton where he remained until his retirement in 2007. During the more than three decades that he spent at the Institute, he continued to develop his ideas and amongst his achievements, one might include enunciating the principle of functoriality and identifying the trace formula as a powerful tool to investigate functoriality developing the functorial properties of trace formulas associated to different groups including the theory of base change and of endoscopy enunciating the fundamental lemma as a crucial result needed for the further development of the theory establishing the connection with geometry through the theory of Shimura varieties.

Putting all the pieces together

Prof. Langlands’ work has been recognized with many awards, including the Shaw Prize (jointly with Richard Taylor), the Wolf Prize (jointly with Andrew Wiles) and most significantly, the Abel Prize. In reading the citations to these and many other prizes, one finds a common theme, namely a grand vision of synthesis amongst different parts of mathematics, namely number theory, representation theory, harmonic analysis and algebraic geometry. Taken together, this synthesis may be viewed as a general non-Abelian reciprocity law.

Reciprocity laws are well-known to lie at the heart of number theory. Starting from the law of quadratic reciprocity, that so fascinated Gauss that he gave numerous proofs of it, to higher degree reciprocity laws such as cubic, quartic and more generally cyclotomic reciprocity, a magical relationship is revealed between the factorization of prime ideals in Abelian extension fields and properties of the primes themselves in the base field.

The reciprocity law for Abelian extensions was established by Artin in the context of class field theory. This theory is a body of results that classifies Abelian extensions of a number field in terms of data from that field. It was developed in Princeton by Artin and Tate with tools of Galois cohomology. One of Artin’s insights was that the reciprocity law could be stated as the equality of two kinds of L-functions in the Abelian case, namely an L-function of the kind introduced by Hecke and another L-function introduced by Artin himself.

Langlands’ early exposure to class field theory was at Princeton when he was asked by Salomon Bochner to give a course on the subject. In his own words, Langlands says that he had to learn the subject quickly as the semester was about to begin, but as a result of teaching that course, the problem of a non-abelian generalization of this theory became fixed in his mathematical ambition.

 Beyond reciprocity

Going beyond the Abelian reciprocity law is considered one of the big challenges of number theory. In fact, Hilbert’s 9th problem asked for the general reciprocity law. Langlands’ proposed solution of this problem was to define a new kind of L-function, one associated to certain infinite dimensional representations, so-called automorphic representations. He outlined this construction in his famous Yale University lectures on Euler products. The Langlands L-functions may be considered a tremendous generalization of the L-functions defined by Hecke. Langlands then outlined a conjectural recipe for how to associate an automorphic representation to any Artin L-function, even the non-Abelian ones.

From the point of view of a reciprocity law being given by an equality of L-functions, we may view the Taniyama conjecture proved by Wiles as a reciprocity law. In that case, one asks that the L-function associated to an elliptic curve is the same as an L-function associated to a modular form (in particular an automorphic representation).

More generally, for L-functions that arise from algebraic geometry, or more generally from motives, one expects a version of the Hasse-Weil conjecture to hold about the analytic continuation and functional equation. In Langlands’ view, this conjecture will be proved by establishing a reciprocity law (which strictly speaking is a stronger statement) identifying geometric L-functions with automorphic L-functions. The first test case of this is of course for Shimura varieties, and in this setting the assertion has been proven in many cases. There is a striking speculation by Langlands that Shimura varieties may play a role in the general reciprocity law that Kummer extensions played in the proof of Artin reciprocity. In other words, he speculates that there may be some formal techniques by which the general case of the Hasse-Weil conjecture can be reduced to that of Shimura varieties.

I hope I have managed to give you at least a snippet of the panoramic and breathtaking nature of the ideas and conjectures put forth by Prof. Langlands. Please join me in congratulating him on his contributions and achievements to mathematics over a fruitful and extensive career spanning nearly five decades. We are truly fortunate in Canada to have produced such an extraordinary mathematician and fine human being.