Analyzing the Mystery of Numbers
First developed by the ancient Greeks and Hindus, number theory holds the answer to many contemporary, real-world concerns, including: public key cryptography to protect information transmitted on the Internet; CDs that can withstand minor scratches without harm to the digital information they hold; and satellite positioning systems that allow shipments to be tracked.
Number theory also has increasingly important applications in theoretical physics and computer science. The research undertaken by Dr. Andrew Granville as Canada Research Chair in Number Theory is aimed at better understanding basic questions in number theory, particularly by analyzing the distribution of special types of numbers.
His main focus is on the distribution of prime numbers and other important number-theoretic sets. To this end, Granville has worked with Stanford University’s Dr. K. Soundararajan to understand what happens when things go wrong.
Their research has shown that, in many questions of interest, things only go wrong when a certain number theory function is pretentious—that is, when the function pretends to be something it is not. This understanding has allowed them to make several improvements for dealing with character sums and for proving the prime number theorem.
Their work has also proven that mass cannot “escape to infinity” in the case of automorphic forms. During this chair, Granville hopes to show that most of the main results of analytic number theory can be re-proven “pretentiously,” not only giving a new, easier perspective on the heart of this central subject, but, on occasion, proving results that were unapproachable by more classical means.