# Number Theory

### Overview

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 recent proof of Fermat's Last Theorem) and yet remains broadly useful in many areas of pure and applied mathematics. Indeed, it is remarkable how often number theory comes to bear both in other areas of mathematics and in applications. A notable recent example is cryptography and internet security whose protocols are based on number theoretic problems.

Number theory is particularly strong in Canada with the PIMS Number Theory Group featuring prominently. This group is large and well distributed across the PIMS Universities. It has a number of prominent senior world-class researchers leading a group of richly talented young mathematicians. The recent influx of new number theorists into several PIMS universities has created an exciting working group (S. Choi, I. Chen, and P. Lisoneck at SFU plus M. Bennett, G. Martin, and V. Vastal at UBC). Both UBC and SFU groups have been involved in modern computational number theory (D. Boyd, P. Borwein, W. Casselman). In addition to hiring two new number theorists in the past year, UBC has additionally supplemented its faculty in cognate areas such as harmonic analysis (I. Laba) and algebraic geometry (J. Bryan and Z. Reichstein). This CRG already organizes several ongoing joint activities including Pacific Northwest Number Theory seminars and mini-conferences, and supervises many successful PIMS PDFs. A multi-year period of concentration will allow coordination and integration between those activities that are already being organized and new opportunities (BIRS, PIMS Distinguished Visitors) in a single framework to maximize impact.

### PIMS Distinguished Chair

Professor Jeffrey D. Vaaler (University of Texas at Austin) delivered a lecture series in the second and third weeks of June 2003 at SFU.

Professor Bjorn Poonen (University of California, Berkeley) delivered a lecture series in July 2004 at SFU.

Professor Sergei Konyagin (Moscow State University) delivered a lecture series in March 2004 at UBC. Winner of the Salem Prize in 1990, Prof. Konyagin has made numerous significant contributions in number theory, approximation theory and harmonic analysis.

### Faculty

CRG Leaders:

- Peter Borwein (SFU)
- David Boyd (UBC)

UBC:

SFU:

U. Alberta:

U. Calgary:

U. Washington:

Other institutions:

- Amir Akbary (U. Lethbridge)
- Edward Dobrowolski (College of New Caledonia),
- Matt Klassen (DigiPen Inst. of Tech.)
- Kristin Lauter (Microsoft)

**Postdoctoral Fellows**

- Ben Green, PIMS PDF (2003-04) at UBC
- Friedrich Littman, PIMS PDF (2003-05) at UBC
- Christopher Rowe, PIMS PDF (2003-05) at UBC
- Ron Ferguson, MITACS PDF (2002-03) at SFU
- William Galway, PIMS PDF (2002-03) at SFU
- Alexa van der Waall, MITACS PDF (2002-03) at SFU