Symmetries and Overdetermined Systems of Partial Differential Equations
July 17-August 4, 2006
This summer program is 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.
The symmetries to be studied in the this Summer Program naturally arise in
several different ways. Firstly, there are the symmetries of a differential
geometric structure. By definition, these are the vector fields that preserve
the structure in question—the Killing fields of Riemannian differential
geometry, for example. Secondly, the symmetries can be those of another
differential operator. For example, the Riemannian Killing equation itself
is projectively invariant whilst the ordinary Euclidean Laplacian gives rise
to conformal symmetries. In addition, there are higher symmetries 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
differential 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 Summer Program will consider the
consequences of
overdeterminacy and partial differential equations of finite
type.
The question of what it means to be able to solve explicitly a classical 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 integrability. But only the simplest equations are currently understood from these points of view. The Summer Program will provide an opportunity for
comparison 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 differential
operators. Even in the flat model G/P, 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 will explore 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
correspondence
in physics (or Fefferman-Graham ambient metric construction in mathematics)
provides an especially natural route to conformal symmetry 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 is to bring together relevant research groups for the purpose of intense discussion, interaction, and fruitful collaboration.In summary, topics to be considered in the Summer Program include:
Symmetries of geometric structures and differential
operators.
Overdetermined systems of partial differential
equations.
Separation of variables and conserved quantities.
Integrability, superintegrability and solvable
systems.
Parabolic geometry and the Bernstein-Gelfand-Gelfand
complex.
Interaction with representation theory.
Exterior differential systems.
Finite element schemes, discrete symmetries,
moving frames, and numerical analysis.
Interaction with string theory and twistor theory.
Organization:
The first week will be devoted to expository/overview sessions. This week will
be particularly valuable for people new to the subject to get an overall
understanding of overdetermined systems and their applications. The following
two weeks will focus on more specialized research talks with one or two themes
each day. These talks will be intermixed with discussions, aimed at assessing
what has been learned and pointing out the most promising areas for further
research. We will overlap themes in several areas to stimulate interaction
between different groups where we expect this interaction to be most
productive. We will publish an IMA volume of refereed papers, consisting of
expository/survey talks for the entire field, followed by more specialized
research contributions.
Proceedings:
Proceedings of the Summer Program will be published in the IMA Volume Series
and we now invite participants to submit articles. Though there is no absolute
limit on the length of a submission, in order to make this the quality volume
that it should be, please try to make your contribution as concise as possible.
The volume will include expository lectures from the first week and we are
aiming for a particularly useful volume. The volume will be dedicated to the
memory of Thomas Branson. All submissions will be refereed and the deadline for
submission is Friday, October 20, 2006. Please use LaTeX or similar (TeX,
AMS-TeX, etc.). See here
for further details. Submissions should be sent by email directly to one of
Michael Eastwood (
) or
Willard Miller (
),
who will edit the proceedings.
Schedule
Monday, July 17
8:15a-9:10a
Registration and Coffee
EE/CS 3-176
9:10a-9:20a
Welcome to the IMA
Douglas N. Arnold (University of Minnesota Twin Cities)
EE/CS 3-180
9:20a-9:30a
Introduction by the Organizers
Michael Eastwood (University of Adelaide), Willard Miller Jr. (University of Minnesota Twin Cities)
