This volume contains papers based on some of the talks given at the NSF-CBMS conference on "The Geometrical Study of Differential Equations" held at Howard University (Washington, DC). The collected papers present important recent developments in this area, including the treatment of nontransversal group actions in the theory of group invariant solutions of PDEs, a method for obtaining discrete symmetries of differential equations, the establishment of a group-invariant version of the variational complex based on a general moving frame construction, the introduction of a new variational complex for the calculus of difference equations and an original structural investigation of Lie-Bäcklund transformations. The book opens with a modern and illuminating overview of Lie's line-sphere correspondence and concludes with several interesting open problems arising from symmetry analysis of PDEs. It offers a rich source of inspiration for new or established researchers in the field.

This book can serve nicely as a companion volume to Selected Topics in the Geometrical Study of Differential Equations, by Niky Kamran, in the AMS series, CBMS Regional Conference Series in Mathematics.

Readership

Graduate students and research mathematicians.

Table of Contents

• R. Milson -- An overview of Lie's line-sphere correspondence
• V. Torrisi and M. C. Nucci -- Application of Lie group analysis to a mathematical model which describes HIV transmission
• R. Beals -- Geometry and PDE on the Heisenberg group: A case study
• G. Marí Beffa -- Invariant evolutions of curves and surfaces and completely integrable Hamiltonian systems
• B. A. Shipman -- On the fixed points of the toda hierarchy
• I. M. Anderson, M. E. Fels, and C. G. Torre -- Group invariant solutions in mathematical physics and differential geometry
• P. E. Hydon -- Discrete symmetries of differential equations
• T. A. Ivey -- Integrable geometric evolution equations for curves
• J. A. Sanders and J. P. Wang -- On integrability of evolution equations and representation theory
• M. Oberguggenberger -- Symmetry groups, nonlinear partial differential equations, and generalized functions
• R. H. Heredero -- Lie symmetries of differential-difference equations
• E. L. Mansfield and P. E. Hydon -- On a variational complex for difference equations
• I. A. Kogan and P. J. Olver -- The invariant variational bicomplex
• E. G. Reyes -- On geometrically integrable equations and hierarchies of pseudo-spherical type
• I. A. Kogan -- Inductive construction of moving frames
• V. Itskov -- Orbit reduction of contact ideals and group-invariant variational problems
• T. Robart -- About the local and formal geometry of PDE
• P. A. Clarkson and E. L. Mansfield -- Open problems in symmetry analysis