L-functions and Number Theory

2010 — 2013
l-functions and number theory

Overview

Number theory is a subject as diverse as it is ancient, and this diversity is well represented in the mathematics departments of PIMS universities. These universities are home to academics with expertise in algebraic and analytic number theory, arithmetic algebraic geometry, computational number theory, number theoretic cryptography and information security, and representation theory, and have long-held reputations for producing cutting-edge research in these fields. Much has changed, though, in the arithmetic landscape of Western Canada since PIMS last funded a CRG in number theory during the period 2003-2005. An influx of new, young researchers (Bauer, Doran, Cunningham, Gordon, Greenberg, Kadiri, Lalín, Ng, Silberman) into PIMS universities, particularly in Alberta, has added to the region’s already impressive list of accomplished number theorists. We feel the time is ripe to bring all of these mathematicians together, again under the umbrella of a PIMS CRG, to encourage the development of promising young faculty and their work while continuing to promote the number theoretic excellence for which Western Canada is acknowledged worldwide.

Timeliness and importance

Canada is a world leader in number theoretic research. Recently, Western Canadian universities have invested in number theory by adding many strong young faculty members to a core of established researchers with world-class reputations. In parallel, important scientific developments have spurred exciting progress in the field and opened the doors to new avenues of research:

  1. Explosive growth in subjects like random matrix theory and mock theta functions is leading to impressive new results in analytic number theory.
  2. The field of Diophantine equations has recently undergone a significant change of perspective. Although the negative resolution of Hilbert's tenth problem renders experts generally pessimistic about the existence of an algorithm to decide solvability in general.  Recent computational work has inspired important heuristics that indicate that an algorithm to decide the existence of rational points on curves may in fact exist and might already be known.
  3. The rapid development of Darmon’s theory of modular points and advances Kudla’s program are increasing our understanding of the arithmetic nature of cycles on modular varieties and their connections with L-functions and representation theory.
  4. Recently, there have been developments in both local and global aspects of the Langlands program, e.g., the proof of the Fundamental Lemma by Ngo and the potential modularity results of Harris and Taylor. These developments have led to spectacular results, e.g., the proof by Taylor of the Sato-Tate conjecture, and suggest many promising avenues for further investigation of the nature of arithmetic statistics.

Our CRG looks forward to playing a major role in the advancement of the fundamental scientific theories listed above.

Participants

Coordinator
Matthew Greenberg, University of Calgary

Principal investigators
Amir Akbary, University of Lethbridge
Nils Bruin, Simon Fraser University
Clifton Cunningham, University of Calgary
Charles Doran, University of Alberta
Matthew Greenberg, University of Calgary
Greg Martin, University of British Columbia
Lior Silberman, University of British Columbia
Vinayak Vatsal, University of British Columbia

Participating faculty
Mark Bauer, University of Calgary
Michael Bennett,  University of British Columbia
Peter Borwein, Simon Fraser University
David Boyd,  University of British Columbia
Paul Buckingham, University of Alberta
William Casselman, University of British Columbia
Imin Chen, Simon Fraser University
Stephen Choi,  Simon Fraser University
Gerald Cliff, University of Alberta
Michael Jacobson, University of Calgary
Julia Gordon, University of British Columbia
Dragos Ghioca, University of Lethbridge
Habiba Kadiri, University of Lethbridge
Matilde Lalín, University of Alberta
Kristen Lauter, Microsoft Research
James Lewis, University of Alberta
Nathan Ng, University of Lethbridge
Renate Scheidler, University of Calgary
Hugh Williams,  University of Calgary/CSEC

Scientific advisory committee
William Casselman,  University of British Columbia
Stephen Kudla, University of Toronto
Ram Murty, Queens University
Bjorn Poonen,  MIT
Vinayak Vatsal,  University of British Columbia

 

Scientific Events

 

Seminars

Visitors

Distinguished chairs: TBA

  • Michael Stoll (Bayreuth, Germany), Host: Nils Bruin (SFU), May 1-17, 2010
  • Hugo Chapdelaine (Laval), December 14-19, 2010
  • David Roe (Harvard), January 19-22, 2011 (seminar on January 21)
  • Pierre Charollois (Jussieu), March 2011, (seminar on March 10)
  • Cameron Franc (McGill), April 2011, (seminar on April 4)
  • Paul Mezo (Ottawa), June 2011
  • Hadi Salmasian (Ottawa) June 2011
  • Pramod Achar (LSU) June 2011
  • Samit Dasgupta (UCSC) TBA

Post-doctoral Fellows

  • Johnson Jia, Supervisor: Vinayak Vatsal (UBC), September 2010-August 2012
  • Christopher Marks, Supervisor: Terry Gannon (UA), September 2011-August 2013
  • Ce Bian, Supervisor: Matthew Greenberg (UC), September 2011-August 2013
  • David Roe, Supervisor: Matthew Greenberg (UC), September 2011-August 2013

Organizers

Clifton Cunningham and Mathew Greenberg