Ebook: Symmetries and Overdetermined Systems of Partial Differential Equations

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.

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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.

 

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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.

 

Content:
Front Matter....Pages i-xv
Front Matter....Pages xvi-xvi
Overdetermined Systems, Conformal Differential Geometry, and the BGG Complex....Pages 1-24
Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations....Pages 25-40
Notes on Projective Differential Geometry....Pages 41-60
Ambient Metric Construction of CR Invariant Differential Operators....Pages 61-75
Fine Structure for Second Order Superintegrable Systems....Pages 77-103
Differential Geometry of Submanifolds of Projective Space....Pages 105-125
Pseudo-Groups, Moving Frames, and Differential Invariants....Pages 127-149
Geometry of Linear Differential Systems Towards Contact Geometry of Second Order....Pages 151-203
Front Matter....Pages 204-204
On Geometric Properties of Joint Invariants of Killing Tensors....Pages 205-221
Hamiltonian Curve Flows in lie Groups G ? U(N) and Vector Nls, mKdV, Sine-Gordon Soliton Equations....Pages 223-250
Conformal Killing Spinors and the Holonomy Problem in Lorentzian Geometry — A Survey of New Results —....Pages 251-264
Projective-Type Differential Invariants for Curves and Their Associated Pdes of Kdv Type....Pages 265-275
Algebraic Construction of the Quadratic First Integrals for a Special Class of Orthogonal Separable Systems....Pages 277-304
Geometry of Non-Regular Separation....Pages 305-317
Higher Symmetries of the Square of the Laplacian....Pages 319-338
Metric Connections in Projective Differential Geometry....Pages 339-350
Exterior Differential Systems with Symmetry....Pages 351-366
Partially Invariant Solutions to Ideal Magnetohydrodynamics....Pages 367-381
Invariant Prolongation and Detour Complexes....Pages 383-401
Inhomogeneous Ambient Metrics....Pages 403-420
Front Matter....Pages 204-204
Pfaffian Systems Of Frobenius Type And Solvability Of Generic Overdetermined Pde Systems....Pages 421-429
Exact And Quasi — Exact Solvability of Second Order Superintegrable Quantum Systems....Pages 431-444
A Remark On Unitary Conformal Holonomy....Pages 445-460
Polynomial Associative Algebras for Quantum Superintegrable Systems with a Third Order Integral of Motion....Pages 461-469
Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds....Pages 471-496
Differential Systems Associated with Tableaux over Lie Algebras....Pages 497-513
Deformations of Quadratic Algebras, the Joseph Ideal for Classical Lie Algebras, and Special Tensors....Pages 515-526
Analogues of the Dolbeault Complex and the Separation of Variables....Pages 527-536
Back Matter....Pages 537-550
....Pages 551-563