Number Theory

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 Patrick Allen, Henri Darmon, Eyal Goren, and Michael Lipnowski and their research groups has been focusing on arithmetic algebraic geometry and number theory with special emphasis on elliptic curves, Galois representations, L-functions, Shimura varieties and automorphic forms. The McGill group is part of an active Montreal-wide number theory network that organizes, 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. The Montreal Number Theory group is part of the Centre Interuniversitare en Calcul Mathématique et Algébrique (CICMA). 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: introductory number theory, analytic and algebraic number theory, algebraic geometry, elliptic curves and modular forms. The courses may be offered in the various home universities of our members, but students at McGill can take these courses as part of the graduate program. In addition, special courses, that change from year to year, are given in areas of additive combinatorics, automorphic forms, Galois representations, algebraic geometry, Shimura varieties, symmetric spaces, rational points, p-adic Hodge theory and p-adic modular forms. For more details, see here.

Students specializing in number theory are expected to fulfil first the basic requirements in algebra, topology and analysis. Each year, advanced graduate courses are given at McGill, Concordia and Université de Montréal. Students would be expected to enroll 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.

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