Symmetries and Overdetermined Systems of Partial Differential Equations PDF
Preview Symmetries and Overdetermined Systems of Partial Differential Equations
The IMA Volumes in Mathematics and its Applications Volume 144 Series Editors Douglas N. Arnold Arnd Scheel Institute for Mathematics and its Applications (IMA) The Institute for Mathematics and its Applications was estab- lished by a grant from the National Science Foundation to the University of Minnesota in 1982. The primary mission of the IMA is to foster research of a truly interdisciplinary nature, establishing links between mathematics of the highest caliber and important scientific and technological problems from other disciplines and industries. To this end, the IMA organizes a wide variety of programs, ranging from short intense workshops in areas of ex- ceptional interest and opportunity to extensive thematic programs lasting a year. IMA Volumes are used to communicate results of these programs that we believe are of particular value to the broader scientific community. The full list of IMA books can be found at the Web site of the Institute for Mathematics and its Applications: http://www, ima. umn. edu / springer /volumes. ht ml Presentation materials from the IMA talks are available at http://www, ima. umn. edu / talks / Douglas N. Arnold, Director of the IMA IMA ANNUAL PROGRAMS 1982-1983 Statistical and Continuum Approaches to Phase Transition 1983-1984 Mathematical Models for the Economics of Decentralized Resource Allocation 1984-1985 Cont, inullm Physics and Partial Differential Eqllat, ions 1985-1986 Stochastic Differential Equations and Their Applications 1986-1987 Scientific Computation 1987-1988 Applied Combinatorics 1988-1989 Nonlinear Waves 1989-1990 Dynamical Systems and Their Applications 1990-1991 Phase Transitions and Free Boundaries 1991-1992 Applied Linear Algebra 1992-1993 Control Theory and its Applications 1993-1994 Emerging Applications of Probability 1994-1995 Waves and Scattering 1995-1996 Mathematical Methods in Material Science 1996-1997 Mathematics of High Performance Computing Continued at the back Michael Eastwood Willard Miller, Jr. Editors Symmetries and Overdetermined Systems of Partial Differential Equations regnirpS Michae] Eastwood Willatd Mi]Ier, Jr. School of Mathematical Sciences School of Mathematics The I hliversity of Adelaide University of Minnesota SA 5005, Australia 127 Vincent Hail, 206 Chmch St SE http://ww w.rr~hs.adelaide.edu .au/pure/staff/ Minneapolis, MN 55455 meastwood.hmfl ht ~p://w wwSirra.ulml.edu/- miller/ Series Edifors Douglas N, Arnold Arnd Scheel [nsfitnte for Matllematics and its Applications University of Minnesota Minneapolis, MN 55455 USA ISBN 978-0-387-73830-7 e-[SBN 978-0-387-73831-4 Librm'y of Congeass Comrol Number: 2007933698 Mathematics Suhject Classification 12000): 58J70, 35 06, 53 06, 70 06, 81 06 !: 2008 Springer Science+Bu sK.enis Media. I,[ ~2 AII riybts reserved This woik may not be translated or copied in whole or in pail without the written pcmtission of tie pubtMaer (Springer Sciencc+Bushless Media, LLC, 233 Spring Stl~ct, New York, N~' 10013, USA), except for brief excerpts in conneciion with i~views or sc holaiiy alialysis. Use in connection with any foma of information stol'age and retrieval, dectronic adaptation, computer soft~are, or by silrd]ar or dissitmlar methodology now known or hereafler developed is fodAdden. The u~ in this pubs of tl'ade names, tradcmal ks, service ma~'ks, and similar tcrms, even ff they are not identifed as such, is not to be taken as an expression of opinion ~a to whether or not the~ are subject to proprictar~ rights Camera:ready copy provided by tbc IMA. Pdntcd on acid flee paper 987654321 spfinger com FOREWORD This IMA Volume in Mathematics and its Applications Symmetries and Overdetermined Systems of Partial Differential Equations contains expository and research papers that were presented at the IMA Summer Program which was held July 17-August 4, 2006. This summer program was dedicated to the memory of Thomas P. Branson, who played a leading role in its conception and organization, but did not live to see its realization. We would like to thank Michael Eastwood (University of Adelaide) and Willard Miller, Jr. (University of Minnesota) for their superb role as workshop organizers and editors of the proceedings. We take this opportunity to thank the National Science Foundation for its support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Arnd Scheel, Deputy Director of the IMA IN MEMORY OF PROFESSOR THOMAS P. BRANSON OCTOBER 10, 1953- MARCH ~11 2006 iiv PREFACE The idea for this Summer Program at the Institute for Mathematics and its Applications came from Thomas Branson. In March 2006 prepara- tions were in full swing when Tom suddenly passed away. It was a great shock to all who knew him. These Proceedings are dedicated to his mem- ory. In 1979 Thomas Branson gained his PhD from MIT under the super- vision of Irving Segal. After holding various positions elsewhere, he worked at the University of Iowa since 1985. He has published over 70 substantial scientific papers with motivation coming from geometry, physics, and sym- metry. His work is distinguished as a highly original blend of these three elements. Tom's influence can be clearly seen in these Proceedings and the impact of his work will be always felt. His untimely death is deeply saddening and we shall all miss him sorely. The symmetries that were studied in the Summer Program naturally arise in several different ways. Firstly, there are the symmetries of a differ- ential geometric structure. By definition, these are the vector fields that preserve the structure in question-the Killing fields of Riemannian differ- ential geometry, for example. Secondly, the symmetries can be those of another differential operator. For example, the Riemannian Killing equa- tion itself is projectively invariant whilst the ordinary Euclidean Laplacian gives rise to conformal symmetries. In addition, there are higher symme- tries defined by higher order operators. Physics provides other natural sources of symmetries, especially through string theory and twistor theory. These symmetries are usually highly constrained-viewed as differen- tial operators, they themselves are overdetermined or have symbols that are subject to overdetermined differential equations. As a typical example, the symbol of a symmetry of the Laplacian must be a conformal Killing field (or a conformal Killing tensor for a higher order symmetry). The Sum- mer Program considered the consequences of overdeterminacy and partial differential equations of finite type. The question of what it means to be able to solve explicitly a clas- sical or quantum mechanical system, or to solve it in multiple ways, is the subject matter of the integrability theory and superintegrability theory of Hamiltonian systems. Closely related is the theory of exactly solvable and quasi-exactly solvable systems. All of these approaches are associated with the structure of the spaces of higher dimensional symmetries of these systems. Symmetries of classical equations are intimately connected with special coordinate systems, separation of variables, conservation laws, and integra- bility. But only the simplest equations are currently understood from these points of view. The Summer Program provided an opportunity for com- xi x ECAFERP parison and consolidation, especially in relation to the Dirac equation and massless fields of higher helicity. Parabolic differential geometry provides a synthesis and generalization of various classical geometries including conformal, projective, and CR. It also provides a very rich geometrical source of overdetermined partial G/P, differential operators. Even in the fiat model for G a semisimple Lie group with P a parabolic subgroup, there is much to be gleaned from the representation theory of G. In particular, the Bernstein-Gelfand-Gelfand (BGG) complex is a series of G-invariant differential operators, the first of which is overdetermined. Conformal Killing tensors, for example, may be viewed in this way. Exterior differential systems provide the classical approach to such overdetermined operators. But there are also tools from representation theory and especially the cohomology of Lie algebras that can be used. The Summer Program explored these various approaches. There are many areas of application. In particular, there are direct links with physics and especially conformal field theory. The AdS/CFT cor- respondence in physics (or Fefferman-Graham ambient metric construction in mathematics) provides an especially natural route to conformal symme- try operators. There are direct links with string theory and twistor theory. Also, there are numerical schemes based on finite element methods via the BGG complex, moving frames, and other symmetry based methods. The BGG complex arises in many areas of mathematics, both pure and applied. When it is recognized as such, there are immediate consequences. There are close connections and even overlapping work being done in several areas of current research related to the topics above. The main idea of this Summer Program was to bring together relevant research groups for the purpose of intense discussion, interaction, and fruitful collaboration. In summary, the topics considered in the Summer Program included: (cid:12)9 Symmetries of geometric structures and differential operators, (cid:12)9 Overdetermined systems of partial differential equations, (cid:12)9 Separation of variables and conserved quantities, (cid:12)9 Integrability, superintegrability and solvable systems, (cid:12)9 Parabolic geometry and the Bernstein-Gelfand-Gelfand complex, (cid:12)9 Interaction with representation theory, (cid:12)9 Exterior differential systems, (cid:12)9 Finite element schemes, discrete symmetries, moving frames, and numerical analysis, (cid:12)9 Interaction with string theory and twistor theory. The first week was devoted to expository/overview sessions. The ar- ticles derived from these lectures form the first half of this collection. We recommend these articles as particularly valuable for people new to the subject seeking an overall understanding of overdetermined systems and their applications. ECAFERP xi The following two weeks focussed on more specialized research talks with one or two themes each day. The second half of this volume consists of articles derived from these talks. It remains to thank the IMA for sponsoring and hosting this event. We believe that the participants found it extremely valuable and hope that these Proceedings will preserve and convey some of the spirit of the meeting itself. Michael Eastwood Willard Miller, Jr. School of Mathematical Sciences School of Mathematics University of Adelaide University of Minnesota CONTENTS Foreword ............................................................. v In memory of Professor Thomas P. Branson .......................... vii Preface .............................................................. ix PART I: EXPOSITORY ARTICLES Overdetermined systems, conformal differential geometry, and the BGG complex ................................................. 1 Andreas paC Generalized Wilczynski invariants for non-linear ordinary differential equations ................................................. 25 Boris Doubrov Notes on projective differential geometry ............................. 41 Michael Eastwood Ambient metric construction of CR invariant differential operators ............................................................. 61 Kengo Hirachi Fine structure for second order superintegrable systems ............... 77 Ernie .G Kalnins, Jonathan M. Kress, and WiUard Miller, Jr. Differential geometry of submanifolds of projective space ............ 105 Joseph M. Landsberg Pseudo-groups, moving frames, and differential invariants ............ 127 Peter .J Olver and Juha Pohjanpelto Geometry of linear differential systems towards contact geometry of second order ............................................ 151 Keizo Yamaguchi PART II: RESEARCH ARTICLES On geometric properties of joint invariants of Killing tensors .............................................................. 205 Caroline M. Adlam, Raymond .G McLenaghan, and Roman .G Smirnov iiix
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