Partial Differential Equations

Overview

Partial Differential Equations is a large subject with a history that dates back to Newton and Leibniz. They form the basis for many mathematical models in the sciences and in economics, yielding such famous equations as the Euler and Navier Stokes equations and Schrodinger's and Einstein's equations. Moreover the subject is far more that just a mathematical tool to address physical and economic phenomena; PDE's have guided and created the fields of nonlinear functional analysis, harmonic analysis, optimization and the modern calculus of variations, and have had a major impact on the field of geometry. The later, was recently highlighted by the role of Ricci flow in the eventual proof of the celebrated Poincare Conjecture.

This diversity is reflected in the research of many faculty members across the PIMS universities and this Collaborative Research Group will provide the means for:

• Bringing together faculty from different PIMS universities with both clearly defined, and not so clearly defined common research interests, and encouraging collaborations between these groups.
• Bringing in several postdoctoral fellows, creating critical masses at the various PIMS sites.
• Sponsoring graduate and post doctoral courses and minicourses which will provide access to a wide audience at PIMS universities. Currently there is not the critical mass to offer such course at the majority of PIMS universities.

Focus

The CRG on PDE will focus on several key areas (both for training and research):

• Geometry and analysis of dispersive equations
• Regularity for solutions of certain fundamental equations (Navier-Stokes, harmonic measures, the infinity Laplacian)
• The role of hyperbolic problems in traffic flow, kinetic theory and the material sciences
• Modern approaches to asymptotic analysis in the calculus of variations and applications
• general variational principles
• Universal inequalities in relevant function spaces
• The role of game theory and stochastic methods in certain elliptic and parabolic equations

Faculty

Participating Faculty from PIMS Universities

• UBC - Ivar Ekeland, Antoine Mellet, Nassif Ghoussoub, Stephen Gustafson, Tai-Peng Tsai, Richard Froese
• SFU - Rustum Choksi, Adam Oberman, Razvan Fetecau, JF Williams
• UVic - Reinhard Illner Martial Agueh Boualem Khouider Chris Bose
• UCalgary - Christian Rios
• UW - Gunther Uhlmann, Hart Smith

Activities

• Mar 30-Apr 04, 2008 at BIRS, Recent Developments in Elliptic and Degenerate Elliptic Partial Differential Equations, Systems and Geometric Measure Theory Organizers: David Cruz-Uribe (Trinity College), Steven Hofmann (University of Missouri-Columbia), Marius Mitrea (University of Missouri, Columbia), Salvador Perez Esteva (Universidad Nacional Au- tonoma de Mexico), Cristian Rios (University of Calgary), Eric Sawyer (McMaster University).
• June 2008, UBC Workshop on the Nash-Moser implicit function theorem and its applications in the calculus of variations Organizers: I. Ekeland (UBC), Florin Diacu (UVic)
• April 27-May 2, 2008 at BIRS, Nonlocal operators and applications. Organizers: Cyril Imbert (University Paris-Dauphine, CEREMADE), Antoine Mellet (University of British Columbia), Regis Monneau (ENPC).
• Aug 24-Aug 29, 2008 at BIRS, Asymptotics and Singularities in Nonlinear and Geometric Dispersive Equations Organizers: Zeng Chongchun (Georgia Tech.), Stephen Gustafson (University of British Columbia), Kenji Nakanishi (Kyoto University), Tai-Peng Tsai (University of British Columbia).
• Mini-symposium in PDE at PIMS-UBC, November 12-13, 2009
• Pacific Northwest Conferences

  • Victoria (September 2008) Organizers: R. Illner. M. Agueh.

  • UBC (January 17, 2009) Organizers: R. Froese, N. Ghoussoub.

• Seminars

  • PDE/Geometry/Math Physics seminar at UBC (S. Gustafson)

  • PDE seminar at SFU (R. Choksi)

  • PDE seminar at UVic (R. Illner)

  • PDE seminar at Calgary (C. Rios)

  • PDE seminar at Alberta (W. Allegretto)

  • PDE seminar at Washington (G. Uhlmann)

Visitors and Special Lectures

PIMS Postdoctoral Fellows:

• Yves van Gennip (Simon Fraser University), 2008-2010
• Mohammad El Smaily (University of British Columbia), 2008-2010
• Vianney Combet (University of British Columbia), 2010-2012
• Diego Ribeiro Moreira (University of British Columbia), 2010-2012
• Ian Zwiers (University of British Columbia), 2010-2012

PIMS Distinguished Visitors:

• Selim Esedoglu (Michigan) visiting SFU and UBC on January 16-24, 2009
• Wilfrid Gangbo (Georgia Tech) visiting UBC on January 15-18, 2009
• Changfeng Gui (Connecticut) visiting UBC on January 8-19, 2009
• Nader Masmoudi (Courant Institute) visiting UBC
• David Kinderlehrer, Carnegie Mellon University visiting UVic from Sept 24, 2008 to Sept 27, 2008 and then UBC until Oct 1, 2008.

Prof. Chang-Shou Lin, Department of Mathematics, National Taiwan University, visiting UBC from Sept 11, 2008 to Sept 15, 2008.