Number theory is as old as human thought, if not older. The ancient civilizations were preoccupied with many fundamental questions of number theory. Rightly so. Civilization as we know it today would not be if it were not for the concept of zero, for example. Indeed many of the unsolved problems of number theory have the fertile quality of generating new fundamental concepts of mathematics. Over the centuries, this discipline has grown into a mighty banyan tree with extensive branches into other areas of mathematics: algebraic geometry, representation theory, group theory, harmonic analysis, theoretical physics and computer algorithms to name only a few.
The group at McGill, consisting of Henri Darmon, Eyal Goren, Jayce Getz, Heekyoung Hahn and John Labute has been focusing on arithmetic algebraic geometry and number theory with special emphasis on elliptic curves, L-functions, Shimura varieties and automorphic forms. The McGill group is part of an active Montreal-wide number theory network which organises, among other scientific activities, the world-famous Quebec-Vermont Number Theory Seminar. This seminar brings together all the members of the Montreal number theory community, as well as members of Université Laval and the University of Vermont. This number theory seminar also enjoys the active participation of some of the leading figures who come to Montreal on a regular basis and give short courses suitable for graduate students. Centre Interuniversitare en Calcul Mathématique et Algébrique (CICMA) is the basis of the number theory seminar. As such, course notes and research papers (in preprint form) are produced and circulated on a regular basis by the participants. Students at McGill can take advantage of the expertise grouped around CICMA and conduct research under the supervision of any of its members.
The number theory group teaches on a regular basis fundamental courses in number theory, algebra and algebraic geometry. Topics include: Elliptic curves, Riemann surfaces and theta functions, introductory number theory, algebraic number theory, analytic number theory, Introduction to algebraic geometry, modular forms, cyclotomic fields. In addition special courses may be offered depending on the interests of students. In recent years this included courses on Hilbert modular forms and varieties, Modular forms and the Birch-Swinnerton Dyer conjecture, Cryptography, Vector bundles on curves.
Students specializing in number theory are expected to fulfil first the basic requirements in algebra and analysis. Each year, advanced graduate courses are given both at McGill and Concordia. Students would be expected to enrol in these courses as well as participate in the number theory seminar. These courses and seminars are a source of possible topics for graduate research. They also provide interaction with some of the leading researchers in the field. One expects that the Masters program will normally take a maximum of two years to complete whereas the PhD program should not take more than four years to complete.