


Invariant Theory and Differential Geometry
Org: Ray McLenaghan (Waterloo) and Roman Smirnov (Dalhousie) [PDF]
 STEPHEN ANCO, Brock University, St. Catharines, ON
BiHamiltonian operators in Lie group geometry and wave maps
[PDF] 
In this talk I will show that the recent geometric derivation of
biHamiltonian operators for arclengthpreserving flows of curves on
constant curvature Riemannian manifolds (due to Sanders and Wang, and
Mari Beffa) extends to semisimple Lie group manifolds. In particular
the biHamiltonian operators are found to be directly encoded in the
Cartan structure equations of a leftinvariant moving frame associated
with a curve and its flow. These operators lead to a hierarchy of
commuting flows and conservation laws generated by a recursion
operator which has the form of a "squareroot" of the corresponding
operator known for constant curvature Riemannian manifolds. As one
main result, the 1 flow in this hierarchy is shown to be a wave map
equation (i.e., nonlinear sigma model) with the Lie group as
the target space.
 IAN ANDERSON, Utah State University, Logan, Utah
Symbolic Analysis of Lie's Theorem
[PDF] 
A fundamental theorem in Lie theory is the classical result (due to
Lie himself) which asserts that for each finite dimensional, real Lie
algebra g there exists a (local) Lie group G whose
associated Lie algebra is the given algebra g. In this
talk I will discuss the symbolic implementation of this theorem.
Surprisingly, the standard proof of this theorem (due to Cartan) does
not translate into a very useful symbolic algorithm. I will explain
why and then give another proof of Lie's theorem which is
computationally effective. Applications to invariant theory will be
presented.
 CLAUDIA CHANU, Università di Torino, via Carlo Alberto 10, 10123 Torino,
Italy
Conformal Killing tensors and fixed energy Rseparation for
the Schroedinger equation
[PDF] 
A general geometric framework for the separation of variables in a
null PDE of second (or higher) order is presented. The method is
applied to the case of the Rseparation of the Schrödinger
equation with a fixed value of the energy. An intrinsic
characterization of the fixed energy Rseparation involving
conformal Killing tensors is shown.
This is joint work with Giovanni Rastelli.
 ALAN COLEY, Dalhousie Univ
Spacetimes with vanishing curvature invariants
[PDF] 
All fourdimensional Lorentzian spacetimes with vanishing scalar
invariants constructed from the Riemann tensor and its covariant
derivatives (VSI spacetimes) are determined. A subclass of the Kundt
spacetimes results and the corresponding VSI metrics can be displayed
in local coordinates. Some potential applications of VSI spacetimes
are discussed. The algebraic classification of the Weyl tensor in
higher dimensional Lorentzian manifolds is then described, and higher
dimensional VSI spacetimes are discussed.
 ROBIN DEELEY, University of Victoria
Invariant Classification of Killing Tensors on the Sphere
[PDF] 
In recent years, a method for classifying the orthogonally separable
coordinate systems for the HamiltonJacobi equation has been
developed. This method uses invariants of the vector space of Killing
tensors under the action of the isometry group. It has been applied
to spaces of constant curvature, including E^{2}, E^{3}, M^{2} and
S^{2}. This talk focuses on the invariant classification for S^{2},
along with an alternative classification based on an eigenvalue
approach. In addition, if time permits, results for S^{3} and S^{n}
will also be discussed.
This is joint work with Ray McLenaghan and Roman Smirnov.
 LUCA DEGIOVANNI, Mathematics Departement, University of Torino, via Carlo
Alberto 10, 10123 Torino, Italy
Classification of Killing tensor on flat 2manifolds
[PDF] 
An alternative way to obtain the well known classification of Killing
tensors and separable coordinates systems, on Euclidean and Minkowski
planes, is given. The classification is obtained considering the whole
class of transformation that preserve the type of coordinates
associated to a given Killing tensor. On flat 2manifolds, the
infinitesimal generator of these transformations form an integrable
distribution with rank, in a generic point, equal to the dimension of
the vector space of Killing tensors. Thus the integral surfaces of
the distribution can be found just looking for the loci where the rank
decreases. The process is purely algebraic.
 MICHAEL EASTWOOD, University of Adelaide, South Australia 5005
Projective Invariance and Killing Fields
[PDF] 
The infinitesimal symmetries on a Riemannian manifold satisfy the
Killing equation.
This equation sees only the
projective geometry of the underlying metric, i.e., the
geometry of the unparameterised geodesics. I shall explain what this
means, its consequences, and how this observation generalises to other
Killing equations.
 RYAD GHANAM, University of Pittsburgh at Greensburg
Representations for lowdimensional Lie algebras
[PDF] 
In this talk we will report on progress in the problem of finding
linear representations for lowdimensional real Lie algebras. For
each Lie algebra g of dimension less than or equal to 6, we will
give a matrix Lie group whose Lie algebra is the given algebra in the
list. We will also give a representation of the Lie algebra in terms
of vector fields.
 SIGBJORN HERVIK, Dalhousie University, Halifax, NS
Spacetimes with Constant Scalar Invariants
[PDF] 
In this talk we will discuss spacetimes with constant scalar
invariants. There are many examples of such spacetimes, among them
spacetimes with vanishing curvature invariants and homogeneous spaces.
A certain class of spacetimes to which all known examples belong, as
well as their mathematical and physical properties, will be
discussed.
 JOSHUA HORWOOD, University of Cambridge, Department of Applied Mathematics
and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA,
United Kingdom
Classification of orthogonal coordinate webs in
threedimensional Minkowski space
[PDF] 
The use of isometry group invariants to classify orthogonally
separable Hamiltonian systems and their associated orthogonal
coordinate webs in spaces of constant curvature has been remarkably
successful on the Euclidean and Minkowski planes. Recently, Horwood,
McLenaghan and Smirnov derived an invariantbased classification for
the eleven orthogonal coordinate webs in threedimensional Euclidean
space. In this talk, I will focus on a substantially harder problem,
namely threedimensional Minkowski space, for which there are fifty
distinct coordinate systems which permit orthogonal separation of the
associated HamiltonJacobi and Helmholtz equations. I will outline an
invariant classification scheme for the corresponding thirtyeight
orthogonal coordinate webs, emphasizing not only the role of the group
invariants in its development, but also the importance of group
covariants, reduced invariants and conformal symmetries.
 NIKY KAMRAN, McGill University
Null surfaces and contact geometry
[PDF] 
We will start with a survey of the null surface formulation of the
Einstein field equations of gravitation, which has been developed over
many years by Newman and his collaborators. We will then show how
this makes it possible to rediscover the isomorphism between
threedimensional conformal Lorentzian geometry and and the contact
geometry of third order ordinary differential equations, first brought
to light by Cartan and Chern. We will also show that fourdimensional
conformal Lorentzian geometry corresponds to the contact geometry of a
class of overdetermined system of secondorder pdes in two independent
variables.
 IRINA KOGAN, North Carolina State University
Rational and Algebraic Invariants and the Moving Frame
Method
[PDF] 
We consider rational actions of the connected algebraic groups on an
affine space, and provide algorithms for constructing finite
generating sets of rational and algebraic invariants, together with
the algorithms for rewriting any rational invariant in terms of the
generators. The construction of algebraic invariants, we propose, can
be seen as an algebraic counterpart of the Fels and Olver moving frame
construction for local smooth invariants on a differential manifold.
In particular, we provide an algebraic equivalent for the notions of
crosssection and invariantization. The algebraic formulation reduces
all algorithms to Groebner bases computations, and can be easily
implemented in any computeralgebra system. A generating set of
rational invariants is obtained as a side product of our algorithm for
constructing a generating set of algebraic invariants.
This is a joint work with E. Hubert, INRIA, France.
 WILLARD MILLER, JR., University of Minnesota, Minneapolis, Minnesota
Secondorder superintegrable systems
[PDF] 
A Schrödinger operator with potential on a Riemannian space is
2ndorder superintegrable if there are 2n1 (classically)
functionally independent 2nd order symmetry operators. (The 2n1 is
the maximum possible number of such symmetries.) These completely
integrable Hamiltonian systems are of special interest because they
are multiintegrable, even multiseparable, i.e., variables
separate in several coordinate systems, and the systems are explicitly
solvable in terms of special functions.
We first give examples of superintegrable systems and then we present
very recent results giving the general structure of superintegrable
systems in all 2D, and 3D conformally flat spaces, and a complete list
of such spaces and potentials in 2D. The results reported here were
obtained in collaboration with E. G. Kalnins, G. S. Pogosyan and
J. Kress.
 ROBERT MILSON, Dalhousie University
Killing tensors as irreducible representations of the general
linear group
[PDF] 
We show that the vector space of fixed valence Killing tensors on a
space of constant curvature is naturally isomorphic to a certain
irreducible representation of the general linear group. The
isomorphism is equivariant in the sense that the natural action of the
isometry group corresponds to the restriction of the linear action to
the appropriate subgroup. As an application, we deduce the
DelongTakeuchiThompson formula on the dimension of the vector
space of Killing tensors from the classical Weyl dimension formula.
 ANATOLY NIKITIN, Institute of Mathematics, Kiev, Ukraine
On the Galilean vector covariants
[PDF] 
I will present a complete list of the Galilean vector covariants and
describe the Galilei invariant wave equations for vector and spinor
fields.
 PETER OLVER, University of Minnesota
Lie pseudogroups
[PDF] 
A theory of moving frames is developed for Lie pseudogroups, leading
to new, explicit computational algorithms for determining their
structure and the structure of their differential invariant algebra.
The talk will focus on symmetry (pseudo)groups of differential
equations and variational problems.
 DENNIS THE, McGill University, Dept. of Math & Stats, 805 Sherbrooke
St. West, Montreal, QC H3A 2K6
Symmetries, conservation laws, and cohomology of Maxwell's
equations using potentials
[PDF] 
We discuss the symmetry and conservation law structure for the
freespace Maxwell's equations in Minkowski space and two of its
potential systems:
(1) the standard Lagrangian potential system using F=dA, and
(2) a natural potential system obtained by introducing joint
covariant vector potentials on both F and its (Hodge) dual *F.
In the absence of gauge constraints, the local symmetry,
adjointsymmetry and conservation law structure of these systems
follows as a consequence of their local 1form and 2form cohomology,
together with a general theorem describing how local symmetries of a
potential system with gauge freedom project to local symmetries of the
original system.
With Lorentz gauge imposed, the standard potential system is well
known to inherit the Killing symmetries of Maxwell's equations, but
not the inversion (conformal) symmetries. In contrast, the joint
potential system with Lorentz gauges imposed admits inversiontype
symmetries, as we show by a classification of firstorder symmetries
(of a certain geometric form) for this system. Finally, we derive new
nonlocal classes of symmetries and conservation laws of Maxwell's
equations as a result of this classification.
This talk is based on joint work with Stephen Anco.
 THOMAS WOLF, Brock University, St. Catharines, ON
Integrable Quadratic Hamiltonians on so(4) and so(3,1)
[PDF] 
In the talk a special class of quadratic Hamiltonians on so(4) and
so(3,1) is discussed. Results include a Hamiltonian with a 6th
degree first integral and a superintegrable Hamiltonian together with
inhomogeneous generalizations.
This work was done in collaboration with Vladimir Sokolov.
If time permits, a vector formalism will be introduced which allows us
to reach the same results more efficiently.
 ISMET YURDUSEN, Centre de Recherches Mathématiques (CRM), Université de
Montréal P.O. Box 6128, Centreville Station, Montréal,
Québec H3C 3J7
Prolongation Structure and Integrability of the coupled
KdVmKdV system
[PDF] 
Recently, Kersten and Krasil'shchik constructed the recursion operator
for a coupled KdVmKdV system, which arises as the classical part of
one of superextensions of the KdV equation. In this work, we study
the integrability of this system using the Painlevé test. Then, we
use the DoddFordy algorithm for the WahlquistEstabrook
prolongation technique in order to obtain the Lax pair. We find a
3×3 matrix spectral problem for the Kersten and Krasil'shchik
system. We also show that the Lax pair obtained is a true Lax pair
since the spectral parameter cannot be removed by a gauge
transformation, as can be proven by a gaugeinvariant technique.
This is a joint work with Ayse Karasu and Sergei Yu. Sakovich.

