Symmetries and overdetermined systems of partial differential equations - Eastwood, Michael (The IMA Volumes in Mathematics and its Applications, 2008)
Symmetries in various forms pervade mathematics and physics. Globally, there are the symmetries of a homogenous space induced by the action of a Lie group. Locally, there are the infinitesimal symmetries induced by differential operators, including not only those of first order but of higher order too. This three-week summer program considered the symmetries preserving various natural geometric structures. Often these structures are themselves derived from partial differential equations whilst their symmetries turn out to be contrained by overdetermined systems. This leads to further topics including separation of variables, conserved quantities, superintegrability, parabolic geometry, represantation theory, the Bernstein-Gelfand-Gelfand complex, finite element schemes, exterior differential systems and moving frames. There are two parts to the Proceedings. The articles in the first part are expository but all contain significant new material. The articles in the second part are concerned with original research. All articles were thoroughly refereed and the range of interrelated work ensures that this will be an extremely useful collection. These Proceedings are dedicated to the memory of Thomas P. Branson who played a leading role in the conception and organization of this Summer Program but did not live to see its realization. . Expository Articles -- Overdetermined Systems, Conformal Differential Geometry, and the BGG Complex -- Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations -- Notes on Projective Differential Geometry -- Ambient Metric Construction of CR Invariant Differential Operators -- Fine Structure for Second Order Superintegrable Systems -- Differential Geometry of Submanifolds of Projective Space -- Pseudo-Groups, Moving Frames, and Differential Invariants -- Geometry of Linear Differential Systems Towards Contact Geometry of Second Order -- Research Article -- On Geometric Properties of Joint Invariants of Killing Tensors -- Hamiltonian Curve Flows in lie Groups G ? U(N) and Vector Nls, mKdV, Sine-Gordon Soliton Equations -- Conformal Killing Spinors and the Holonomy Problem in Lorentzian Geometry — A Survey of New Results — -- Projective-Type Differential Invariants for Curves and Their Associated Pdes of Kdv Type -- Algebraic Construction of the Quadratic First Integrals for a Special Class of Orthogonal Separable Systems -- Geometry of Non-Regular Separation -- Higher Symmetries of the Square of the Laplacian -- Metric Connections in Projective Differential Geometry -- Exterior Differential Systems with Symmetry -- Partially Invariant Solutions to Ideal Magnetohydrodynamics -- Invariant Prolongation and Detour Complexes -- Inhomogeneous Ambient Metrics -- Pfaffian Systems Of Frobenius Type And Solvability Of Generic Overdetermined Pde Systems -- Exact And Quasi — Exact Solvability of Second Order Superintegrable Quantum Systems -- A Remark On Unitary Conformal Holonomy -- Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion -- Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds -- Differential Systems Associated with Tableaux over Lie Algebras --