Table Of ContentThe 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
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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
