Robert P. Langlands
Author's comments: Problems of endoscopy first arose as I began the study of Shimura varieties in Bonn during the academic year 1970/71. I reflected on them for a long time, in part in collaboration with Labesse, in part in collaboration with Shelstad. I presented a fairly mature form of my reflections in the Paris lectures, Les débuts d'une formule des traces stable, in which the presence of a major obstacle, overcome considerably later through the efforts of a number of mathematicians, in particular Waldspurger and Ngô, was clearly described.
Author's comments (Apr. 6, 2014): The concept ``Langlands program'' appears in the title of an article by Stephen Gelbart in the BAMS of April, 1984, but Gelbart himself assured me that it was already current, at least orally, before then. He also drew my attention to a phrase of Armand Borel in his Bourbaki seminar of June, 1975, ``plutôt un vaste programme, élaboré par R. P. Langlands depuis environ 1967.'' I do not recall that I was uneasy with the phrase ``Langlands program'' in 1984, but it then referred principally to matters on which I myself had long reflected.
Author's comments: These two letters were, in fact, electronic messages and were written very recently, in December, 2013. They were inspired by a conversation with Julia Mueller and Michael Volpato, in which I tried to explain to them the origins of the general definition of automorphic \(L\)-functions.
Author's comments. Although these notes were written as a foreword to, or an appreciation of, a book by Qing Zou that is to appear soon, the primary purpose for me was to describe some possibilities in the theory of automorphic representations upon which I believe it is important for specialists to reflect. I was grateful to Qing Zou for the somewhat unexpected request to write an appreciation. The book itself will appear in Chinese; the translated title is ``From Kummer to Langlands—The history of the Langlands Program''
Author's comments: These notes are a very first draft of the very first part of a continuing series of lectures that will be held at the Yildiz Teknik Universitesi in Istanbul and may ultimately become an informal essay on various simple aspects of mathematical history and related matters. As they now stand, they are no more than a tentative beginning both linguistically and conceptually. They are posted primarily for use by the audience at the lectures. I apologize in advance for all their failings, grammatical and mathematical.
Authors's comments: These are my responses to questions of Farzin Barekat, a graduate student at the University of British Columbia, where I was an undergraduate student for four years and a graduate student for one year. The questions and my responses were transmitted electronically. An abbreviated version of my responses will be published in a newsletter of the mathematics department of the university.
Author's comments: The following article appeared in novembre, 2007 in the popular scientific review, Pour la Science, but in a version slightly revised by the editors and their consultants for expository purposes and with diagrams added. I am sure the revised form is indeed easier for a layman to understand, but some assessments were added that are not mine. Rather than interfere with the editors' difficult task of turning arcane material into something meaningful to their readers, I let the revised version stand.
Author's comments: This review comes with a supplement (footnote) that contains the comments of several leading specialists and will be much more useful to the potential reader of the book, whether a novice or a specialist, than the review itself.
Author's comments: The article is an exercise in the reading of mathematics from earlier times. An explanation of Descartes's solution of the problem of Pappus as included in the appendix "La géométrie" to "Discours de la méthode" and an explanation of a solution to another form of the same problem by Fermat, described briefly in a letter included in his collected works, are taken as an occasion to compare the mathematical styles of the two men and to observe their mutual debt to Apollonius as well as the differences in their depth of understanding of his work.