Explorations in Computational Number Theory

Stephen Choi

Abstract

Number theory is one of the oldest, deepest and most vibrant branches of modern mathematics. It centrally incorporates some of the most sophisticated and profound mathematical ideas that have been developed (witness the proof of Fermat's Last Theorem) and yet remains broadly useful in many areas of pure and applied mathematics. It is remarkable how often number theory comes to bear both in other areas of mathematics and in applications. A notable recent example is internet security whose protocols are based on number theoretic problems. One of the applications is internet security whose protocols are based on number theoretic problems. One of the applications is in electronic communication, such as online banking and other internet communications. It uses protocols that are based on number theoretic concepts and new research results can give important insights into its security. Number theory has historically been motivated by the study of properties of integers and solutions to equations in integers, but now includes many other aspects, each with its own flavour and viewpoints. Broadly speaking, these can be divided into Analytic, Algebraic, Diophantine, and Geometric aspects of Number Theory. Research in Number theory today often involves knowledge and expertise from areas such as Algebra, Algebraic Geometry, Analysis, Combinatorics, Probability Theory, Representation Theory, Topology. Connections to applicable fields include Coding Theory and Cryptography. At Simon Fraser University, we have a strong group in Number Theory which covers the spectrum of Number Theory. Together with the groups at the University of British Columbia and the University of Washington at Seattle, we form one of the largest groups of Number Theory Researchers in North America. We maintain two active research seminars with UBC and UW which highlight current developments in Number Theory. These are the SFU - UBC Number Theory Seminar ?via video-conferencing at IRMACS) and Pacific Northwest Number Theory Conference. We also run a regular SFU Number Theory Study Seminar. Some projects in our group involve large scale computational data gathering and rely heavily on the use of WestGrid/Compute Canada High-Performance Computing (HPC) facilities in IRMACS. Our members are also active participants in the programs and initiatives of PIMS and MITACS.