This talk will present a broad, unified generalization of much
recent work
on the geometric derivation of soliton equations
and their bi-Hamiltonian integrability structure from
curve flows in various kinds of Riemannian geometries and Klein
geometries.
In particular, it will be shown that for each symmetric space
G/H
there is a hierarchy of geometric curve flows in which the
components of
the principal normal vector along the curve satisfy a
group-invariant
bi-Hamiltonian soliton equation. The derivation is based on a
natural
construction of moving parallel frames and moving
covariantly-constant frames
in such spaces.
Examples of symmetric spaces include, among others,
constant curvature manifolds e.g. Sn, flat conformal
manifolds,
Kahler and quaternion manifolds e.g. CPn,
QPn,
compact Lie group manifolds e.g. SU(n), Sp(n).
These examples lead to a wide class of bi-Hamiltonian soliton
equations
describing multicomponent group-invariant systems of modified
KdV type,
nonlinear Schrodinger type, and sine-Gordon type.
The corresponding curve flows are related to geometric variants
of
mKdV maps, Schrodinger maps, and wave maps on G/H.
Finite element exterior calculus is a theoretical approach to the
design and understanding of discretizations for a wide variety of
systems of partial differential equations. This approach brings to
bear tools from differential geometry, algebraic topology, and
homological algebra to develop discretizations which are compatible
with the geometric, topological, and algebraic structures which
underlie well-posedness of the PDE problem being solved. In the finite
element exterior calculus, many finite element spaces are revealed as
spaces of piecewise polynomial differential forms. These spaces
connect to each other in discrete subcomplexes of elliptic differential
complexes, which are themselves connected to the continuous elliptic
complex through projections which commute with the complex
differential. This structure relates directly to the stability of
discretization methods based on the finite element spaces.
Applications include elliptic systems, electromagnetism, elasticity,
elliptic eigenvalue problems, and preconditioners.
Helga Baum (Humboldt-Universität)
Globally hyperbolic Lorentzian manifolds with special holonomy
The (connected) holonomy groups of Riemanninan manifolds are well known
for a long time. The (connected) holonomy groups of Lorentzian manifolds
were classified only recently by Thomas Leistner. Anton Galaev finished
this classification by describing local analytic metrics for all of these
Lorentzian holomomy groups (including the still missing coupled types).
The next step in this program is to describe Lorentzian metrics with
special holonomy and precribed global properties (geodesically complete,
globally hyperbolic,...).
In the talk I will explain a method to construct globally hyperbolic
Lorentzian manifolds with special holonomy using Riemannian spin manifolds
with Codazzi spinors. This is a joint work with Olaf Müller.
Gloria Mari Beffa (University of Wisconsin)
Projective-type differential invariants and geometric evolutions
of KdV-type
In this talk we will consider curves in any flat
homogeneous manifolds G/H with G semisimple. We will define
projective-type differential invariants for these curves
and we will prove that there exist curve evolutions invariant under
G such that, when written in terms of the differential invariant
of the curves, they become completely integrable equations of KdV-type
if appropriate initial conditions are chosen. We will also describe the
background Poisson Geometry that causes this relation.
Andreas Cap (Universität Wien)
Overdetermined systems, conformal differential geometry, and the BGG complex
The starting point of my lectures will be a way to rewrite
certain overdetermined systems on Riemannian manifolds in closed
form. The method is based on including the orthogonal group O(n) into
the pseudo-orthogonal group O(n+1,1) and analyzing the standard
representation of O(n+1,1) from the point of view of this subgroup.
Next, I will indicate how, replacing direct observations by tools from
representation theory, this method can be generalized to a large class
of systems.Then I will explain how the inclusion of O(n) into O(n+1,1) that we
started from is related the passage from Riemannian to conformal
geometry. Refining the methods slightly, one obtains a construction
for a large family of conformally invariant differential operators.In the end, I want to sketch how the ideas generalize further to a
large class of geometric structures called parabolic geometries.
Claudia Chanu (Università di Torino)
A geometric approach to Kalnins and Miller non-regular separation
A geometric interpretation of non-regular additive separation for a PDE,
as described by Kalnins and Miller is provided.
This general picture contains as special cases
both fixed energy separation and constrained separation of Helmoltz
equation (not necesarily orthogonal).
Moreover, the geometrical approach to non-regular separation allows you to
explain why there are some coordinates in Euclidean 3-space
in which a R-separable solutions of Helmoltz equation exist
(depending on a fewer number of parameters than in the regular case)
but which are apparently not related to classical Staeckel form.
The differential equations that characterize this kind of non-regular
R-separation on a general Riemannian manifold are given.
Moreover, for the Euclidean 3-dimensional space general conditions on
the form of the metric tensor in these coordinates are provided.
In this talk I shall discuss special polynomials associated with rational
solutions for the Painleve equations and of the soliton equations which
are solvable by the inverse scattering method, including the Korteweg-de
Vries, modified Korteweg-de Vries, classical Boussinesq and nonlinear
Schrodinger equations.
The Painleve equations (PI-PVI) are six nonlinear ordinary differential
equations that have been the subject of much interest in the past thirty
years, which have arisen in a variety of physical applications. Further
they may be thought of as nonlinear special functions. Rational solutions
of the Painleve equations are expressible in terms of the logarithmic
derivative of certain special polynomials. For PII these polynomials are
known as the Yablonskii-Vorob'ev polynomials, first derived in the 1960's
by Yablonskii and Vorob'ev. The locations of the roots of these
polynomials is shown to have a highly regular triangular structure in the
complex plane. The analogous special polynomials associated with rational
solutions of PIV are described and it is shown that their roots also have
a highly regular structure.It is well known that soliton equations have symmetry reductions which
reduce them to the Painleve equations. Hence rational solutions of
soliton equations arising from symmetry reductions of the Painleve
equation can be expressed in terms of the aforementioned special
polynomials. Also the motion of the poles of the rational solutions of
the Korteweg-de Vries equation is described by a constrained
Calogero-Moser system describes the motion of the poles of rational
solutions of the Korteweg-de Vries equation, as shown by Airault, McKean,
and Moser in 1977. The motion of the poles of more general rational
solutions of equations in the Korteweg-de Vries, modified Korteweg-de
Vries and classical Boussinesq equations, and the motion of zeroes and
poles of rational and new rational-oscillatory solutions of the nonlinear
Schrodinger equation will be discussed.
Sub-Riemannian geometry is modelled by Carnot groups in the
same way that Riemannian geometry is modelled by Euclidean space. So
the structure of contact/quasiconformal/conformal maps of a sub-Riemannian
space depends on that of the corresponding maps of a Carnot group. We say
that a Carnot group is rigid if the space of contact maps of a conneceted
open set is finite-dimensional, and the problem addressed in this talk is
when Carnot groups are rigid.
Luca Degiovanni (Università di Torino)
Complex variables for separation of Hamilton-Jacobi equation on real
pseudo-Riemannian manifolds
The geometric theory of separation of variables is extended to
include the case of Killing tensor on real pseudo-Riemannian manifolds with
complex eigenvalues. The manifold is not complexified, it is just necessary
to introduce complex-valued functions. The classical results on separation
of variables (including Levi-Civita criterion and Stäckel-Eisenhart
theory) can then be reformulated in a very natural way.
Boris Doubrov (Belarus State University)
Exterior differential systems for ordinary differential equations
We consider geometric structures associated with systems of ordinary
differential equations. In particular, we explore various exterior
differential systems defined by ODE's and show how to construct associated
absolute parallelisms and Cartan connections in a natural way.
The whole theory is split into a series of examples from the simplest ones
to the general case of arbitrary order ODEs. We also present practical
algorithms suitable for explicit calculation of invariants of ordinary
differential equations. In the case of a single ODE we list the generators
in the algebra of all contact invariants.As an application, we discuss the classes of trivializable equations and
Elie Cartan's C-classe equations, which can be solved without any
integration at all.
Projective differential geometry was initiated in the 1920s,
especially by Elie Cartan and Tracey Thomas. Nowadays, the subject is
not so well-known. These two lectures aim to remedy this deficit and
to present motivating reasons for a revival. The deeper underlying
reason is that projective differential geometry provides the most
basic application of what has come to be known as the "Bernstein-
Gelfand-Gelfand machinery". As such, it is completely parallel to
conformal differential geometry. On the other hand, there are direct
applications within Riemannian differential geometry. We shall see,
for example, a good geometric reason why the symmetries of the
Riemann curvature tensor constitute an irreducible representation of
SL(n,R) (rather than SO(n) as one might naively expect). Projective
differential geometry also provides the simplest setting in which
overdetermined systems of partial differential equations naturally
arise. The approach will be via connections on tensors (rather than
via frames).
The symmetry operators for the Laplacian in flat space were recently described
and here we consider the same question for the square of the Laplacian. Again,
there is a close connection with conformal geometry. There are three steps.
The first is to show that the symbol of a symmetry is constrained by an
overdetermined PDE. The second is to show existence of symmetries with
specified symbol (using the AdS/CFT correspondence). The third is to compute
the composition of two first order symmetry operators. There are one or two
interesting twists to the story. This is joint work with Thomas Leistner.
Tom Branson played a leading role in the conception and organization
of this Summer Program. Tragically, he passed away in March this year
and the Summer Program is now dedicated to his memory. This session
will be devoted to a discussion of his work. The format will be
decided in consultation with others during the earlier part of the
Program and anyone wishing to present material is asked to contact
the moderator.
A multi-dimensional equation of the dispersionless Hirota type is said to be
integrable if it possesses infinitely many reductions to a family of
commuting (1+1)-dimensional systems.
The integrability conditions constitute a complicated overdetermined system
of PDEs, which is in involution. This system possesses a remarkable
Sp(6)-invariance, suggesting a connection with the theory of hypersurfaces
of the Lagrangian Grassmanian.
Sergey Golovin (Queen's University)
Partially invariant solutions to ideal magnetohydrodynamics
I will give two examples of partially invariant solutions to
(1+3)-dimensional ideal magnetohydrodynamics equations. The first
solution is generated by the admissible subgroup O(3) of rotations. The
second one is partially invariant with respect to a subgroup of two
translations along Oy and Oz axes and rotation about Ox axis. The
solutions generalize well-known one-dimensional flows with spherical and
planar waves. The complete analysis of the submodels will be given, from
the investigation of overdetermined systems for non-invariant functions
up to the description of physical properties of obtained fluid motions.
Sergey Golovin (Queen's University)
Differential invariants of Lie pseudogroups in mechanics of
fluids
We present bases of differential invariants for Lie pseudogroups
admitted by the main models of fluid mechanics. Among them
infinite-dimensional parts of symmetry groups of Navier-Stokes equations
in general (V.O. Bytev, 1972) and rotationally-symmetric (L.V.
Kapitanskij, 1979) cases; stationary gas dynamics equations (M. Munk, R.
Prim, 1947); stationary incompressible ideal magnetohydrodynamics (O.I.
Bogoyavlenskij, 2000). Applications of the obtained bases to
construction of differentially-invariant solutions and group foliations
of the differential equations are demonstrated.
A. Rod Gover (University of Auckland)
Overdetermined systems, invariant connections, and short detour complexes
With mild restrictions, each overdetermined differential operator is
equivalent to a (tractor-type) connection on a prolonged system, and
this connection depends only on the operator concerned. On the other
hand in Riemannian geometry (for example), natural conformally
invariant overdetermined operators may, given suitable curvature
restrictions, be extended to an elliptic conformally invariant complex
that we term a short detour complex. (These complexes yield an
approach to studying deformations of various structures, and these
complexes and their hyperbolic variants also have a role in gauge theory.)
These constructions are intimately related.
Robin Graham (University of Washington)
The ambient metric to all orders in even dimensions
The ambient metric associated to a conformal manifold is an
important object in conformal geometry. However, the basic construction
is obstructed at finite order in even dimensions. This talk will describe
how to complete the construction to all orders in even dimensions. One
obtains a family of smooth ambient metrics determined up to smooth
diffeomorphism. These ambient metrics arise as an invariantly defined
smooth part of inhomogeneous Ricci-flat metrics with asymptotic expansions
involving log terms. This is joint work with Kengo Hirachi.
Chong-Kyu Han (Seoul National University)
Symmetry algebras for even number of vector fields and for linearly perturbed complex structures
We discuss the existence of solutions and the dimension of the solution spaces for infinitesimal symmeries of the
following two cases: firstly, even number (2n) of vector fields in a manifold of dimension 2n+1, and secondly, almost complex
manifold with linearly perturbed structure.
We use the method of complete prolongation for thses overdetermined linear pde systems of first order and checking the
integrability of the associated Pfaffian systems.
Kengo Hirachi (University of Tokyo)
Ambient metric constructions in CR and conformal geometry
Ambient metric is a basic tool in CR and conformal geometry. It was
first introduced
by Fefferman in an attempt to write down the asymptotic expansion of
the Bergman kernel
and later was generalized to the case of conformal geometry by
Fefferman-Graham.
In these talks, I will start with the construction of the ambient
metric and
describe its applications, including the construction of local CR/
conformal invariants
(with an application to the Bergman kernel), invariant differential
operators and
Branson's Q-curvature.
We consider a rational group action on the affine space and propose a
construction of a finite set of rational invariants and a simple
algorithm to rewrite any rational invariant in terms of those
generators.
The construction is shown to be an algebraic analogue of the moving
frame construction of local invariants [Fels & Olver 1999]. We introduce
a finite set of replacement invariants that are algebraic functions of
the rational invariants. They are the algebraic analogues of the Cartan's
normalized invariants and give rise to a trivial rewriting.This is joint work with Irina Kogan, North Carolina State University.
This talk reviews a method that enables one to construct all discrete point symmetries of any given differential equation that has
nontrivial Lie point symmetries. The method extends to other classes of symmetries, including contact, internal and generalized
symmetries. It is based on the observation that the adjoint action of an arbitrary symmetry induces a Lie algebra automorphism. By
classifying all such automorphisms, it becomes possible to find discrete symmetries with little more effort than it takes to determine
the Lie symmetries. To this end, we present a fairly concise classification of all automorphisms of real Lie algebras of dimension five
or less.
The Cauchy-Riemann equations is an example of a system of partial differential equations that is equipped with a multiplication
(a bi-linear operation) on its solution set. This multiplication is an immediate consequence of the multiplication of holomorphic
functions in one complex variable.
Another, more sophisticated, example is the multiplication of cofactor
pair systems, that provides a method for generating large families of dynamical systems that can be solved through the method of
separation of variables.
Andreas Juhl (Uppsala University)
Q-curvature and holography
We present a formula for the Q-curvature of an even-dimensional
Riemannian manifold in terms of 1. all holographic coefficients of
the volume function which describes the volume of an associated
Poincare-Einstein metric and 2. the structure of harmonic functions of
the Laplacian of the Poincare-Einstein metrics. The formula refines
known results as, for instance, the relation between the integrated conformal
anomaly (the top degree holographic coefficient) and the total
Q-curvature (Graham-Zworski) and Branson's formula for the contribution
to Q with the maximum number of derivatives.
Ernie Kalnins (University of Waikato)
Topics in superintegrability and quasi-exact solvability
Arthemy V. Kiselev (Ivanovo State Power University)
, Thomas Wolf (Brock University)
The SsTools environment for classification of integrable super-equations
The new SsTools environment for REDUCE is presented; the software is
aimed at classification of evolutionary symmetry integrable
super-field systems of PDE with homogeneous differential polynomial
right-hand sides. Here we present the exhaustive list of these systems
that precede the translation invariance with respect to the scaling
weights and which admit arbitrary parities of the times along the
symmetry flows.
Arthemy V. Kiselev (Ivanovo State Power University)
Algebraic properties of Gardner's deformations
We consider the deformations of (1+1)-dimensional Hamiltonian
systems that preserve their integrability. We prove that Gardner's
deformations, being dual to the Baecklund transformations, are
inhomogeneous generalizations of the infinitesimal symmetries. We show
that the extensions of the Magri schemes generate new integrable
hierarchies using two methods.
Arthemy V. Kiselev (Ivanovo State Power University)
, Thomas Wolf (Brock University)
The SsTools environment for classification of integrable super-equations
Same abstract as of 7-17 poster session.
Irina Kogan (North Carolina State University)
Differential and variational calculus in invariant frames
The talk is devoted to theoretical and computational aspects of performing
differential and variational calculus relative to a group-invariant frame
on a jet bundle.
Many important systems of differential equations and variational problems,
arising in geometry and physics, admit a group of symmetries. S. Lie
recognized that symmetric problems can be expressed in terms of
group-invariant objects: differential invariants, invariant differential
forms, and invariant differential operators. It is desirable from both
computational and theoretical points of view to use a group-invariant
basis of differential operators (invariant frame) and the dual basis of
differential forms (invariant coframe) to perform further computations
with symmetric systems. Complexity of the structure equations for a
non-standard coframe and non-commutativity of differential operators
present, however, both theoretical and computational challenges. I will
present formulas and symbolic algorithms for vector fields prolongation,
integration by parts, Euler-Lagrange and Helmholtz operators, all relative
to an invariant frame. Applications
to the problem of finding group-invariant conservation laws and solving
invariant inverse problem of calculus of variations will be considered.
This talk is based on joint work with P. Olver and
I. Anderson.
Commutators of the symmetries of a superintegrable system do
not necessarily close to form a finite-dimensional Lie algebra, but
instead may be quadratic in a basic set symmetries. This is
known to be a generic property of non-degenerate superintegrable
systems. This talk will discuss the structure of these algebras
and their uses, for example, in the classification of
superintegrable systems.
I will describe the projective differential invariants
of submanifolds of projective space and give several examples of
their uses. Applications will include: Griffiths-Harris rigidity
of homogeneous varieties and studying the spaces of lines on a
projective variety. I will also explain relations with the
study of G-structures. These lectures will also serve as an
elementary introduction to moving frames.
For a conformal manifold we introduce the notion of an ambient
connection, an affine connection on an ambient manifold of the
conformal manifold, possibly with torsion, and with conditions relating
it to the conformal structure. The purpose of this construction is to
realise the normal conformal tractor holonomy as affine holonomy of
such a connection. We give an example of an ambient connection for
which this is the case, and which is torsion free if we start the
construction with a C-space, and in addition Ricci-flat if we start
with an Einstein manifold. Thus for a C-space this example leads to an
ambient metric in the weaker sense of Cap and Gover, and for an
Einstein space to a Ricci-flat ambient metric in the sense of Fefferman
and Graham. This is joint work with Stuart Armstrong (Oxford
University).
Felipe Leitner (Universität Stuttgart)
About conformal SU(p,q)-holonomy
If the conformal holonomy group \$Hol(\mathcal{T})\$ of a simply
connected space with conformal structure of signature \$(2p-1,2q-1)\$ is
reduced to \$\U(p,q)\$
then the conformal holonomy is already contained in the special unitary
group \$\SU(p,q)\$.
We present two different proofs of this statement, one using conformal
tractor
calculus and an alternative proof using Sparling's characterisation of
Fefferman metrics.
The global trivializations of the tangent and cotangent bundles of Lie groups
significantly simplifies the analysis of variational problems, including
Lagrangian mechanics and optimal control problems, and Hamiltonian systems.
In numerical simulations of such systems, these trivializations and the
exponential map or its analogs (e.g. the Cayley transform) provide natural
mechanisms for translating traditional algorithms into geometric methods
respecting the nonlinear structure of the groups and bundles. The interaction
of some elementary aspects of geometric mechanics (e.g. non-commutativity
and isotropy) with traditional methods for vector spaces yields new and
potentially valuable results.
Elizabeth L. Mansfield (University of Kent at Canterbury)
One of the ways overdetermined systems have been studied is via
Spencer cohomology of the symbol of the system. This machinery can seem
rather forbidding but nevertheless intriguing as to what it might offer,
as it is intrinsically co-ordinate independent. In this talk we
"deconstruct" the key definitions and prove a relationship between a
system being a characteristic set and being involutive. In fact, we turn
the concepts around so that we can use the now familiar concepts of
syzygies (a.k.a. compatibility conditions) to investigate involutivity.
Ian Marquette (University of Montreal)
Polynomial Poisson and Associative Algebras for Classical and Quantum Superintegrable Systems with a Third Order Integral of Motion
We consider a general superintegrable Hamiltonian system in a
two-dimensional space with a
scalar potential. It allows one quadratic and one cubic integral of motion.
We construct the most
general cubic Poisson algebra generated by these integrals for the classical
case. For the quantum
case we construct the associative cubic algebra and we present specific
realizations. We use them to calculate the energy spectrum. All classical
and quantum superintegrable potentials separable in cartesian coordinates
with a third order integral were found. The general formalism is applied to
these potentials.
We present a new natural (meaning that the
Hamiltonian
is the sum of kinetic and potential energy) integrable
Hamiltonian system on the two dimensional sphere such that
the
integral is polynomial in velocities of third degree.
Projectively equivalent metrics on closed manifolds
There actually will be two subposters: first
deals with
projective Lichnerowich-Obata conjecture and second gives
a
complete answer on the topological question what manifolds
can
carry two different metrics sharing the same geodesics.
I present a solution of a classical problem posed by
Sophus Lie in 1882. One of the main ingredients comes from
superintegrable systems. Another ingredient is a study of
the
following question and its generalizations: when there
exists a
Riemannian metric with a given a projective connection.
Ray McLenaghan (University of Waterloo)
Separation of variables theory for the Hamilton-Jacobi equation from
the perspective of the invariant theory of Killing tensors
The theory of algebraic invariants of Killing tensors defined
on pseudo-Riemannian spaces of constant curvature under the action of the
isometry group is described. The theory is illustrated by the computation of
bases for the invariants and reduced invariants on three dimensional Euclidean
and Minkowski spaces. The invariants are employed to characterize the
orthogonally separable coordinate webs for the Hamilton-Jacobi equation for the
geodesics and the Laplace and wave equations.
Ray McLenaghan (University of Waterloo)
Transformation to pseudo-Cartesian coordinates in
locally flat pseudo-Riemannian spaces
A tractable method is presented for obtaining transformations
to
pseudo-Cartesian coordinates in locally flat
pseudo-Riemannian spaces.
The procedure is based on the properties of parallel vector
fields.
As an illustration, the method is applied to obtain certain
transformations
that arise in the Hamilton-Jacobi theory of separation of
variables.
(Joint work with Joshua Horwood)
A primary aim of this Summer Program is to promote fruitful
interaction between various research groups and individuals currently
working, perhaps unwittingly, on overlapping themes. This session
will be devoted to a public discussion of problems and possible
directions for future research and collaboration. The format will be
decided in consultation with others during the earlier part of the
Program and anyone wishing to present material is asked to contact
the moderator.
Joint work with E.G.Kalnins, J.R. Kress and G.S. Pogosyan.
A classical (or quantum) superintegrable system is an integrable
n-dimensional Hamiltonian system with potential that
admits 2n-1 functionally independent constants of the motion
polynomial in the momenta, the maximum possible.
If the constants are all quadratic the system is second order
superintegrable. Such systems have remarkable properties:
multi-integrability and multi-separability, a quadratic algebra of symmetries whoserepresentation theory yields spectral information about the
Schrödinger operator, deep connections with special functions and
with QES systems. For n=2 (and n=3 on conformally flat spaces with nondegenerate potentials) we have worked out the structure and classified the possible spaces and potentials. The quadratic algebra closes at order 6 and there is a 1-1 classical-quantum relationship. All such systems are Stäckel
transforms of systems on complex Euclidean space or the complex 3-sphere.
I will give a brief review of separation of variables theory
and its connection to symmetries of the equations of mathematical physics.
The distinction between regular and nonregular separation will be
discussed, as well as the intrinsic characterization of separable
systems for Hamilton-Jacobi and Schrödinger equations on Riemannian
manifolds. In the last part of the talk I will describe how these tools
can apply to the study and classification of second order
superintegrable systems.
Lorenzo Nicolodi (Università di Parma)
Involutive differential systems and tableaux over Lie algebras
I will outline some recent work (joint with E. Musso) on the construction of
involutive differential systems based on the concept of a tableau over a Lie algebra.
Particular cases of this scheme lead to differential systems describing various familiar
classes of submanifolds in homogeneous spaces which constitute integrable systems.
This offers another perspective for better understanding the geometry of these
submanifolds.
Anatoly Nikitin (National Academy of Sciences of Ukraine)
Low-dimensional Lie algebras
The latest results on low-dimensional Lie algebras, which were
obtained in the Department of Applied Research of Institute of
Mathematics (Kiev, Ukraine), are overviewed. A wide programme of
investigation was carried out. At first, existing classifications of
low-dimensional Lie algebras were tested, compared and enhanced.
Different properties of low-dimensional Lie algebras were studied and
a number of characteristics, values and objects concerning them,
including automorphisms, differentiations, ideals, subalgebras etc
were found. Using a new powerful technique based on the notion of
megaideal, we constructed a complete set of inequivalent realizations
of real Lie algebras of dimension no greater than four in vector
fields on a space of an arbitrary (finite) number of variables. This
classification amended and essentially generalized earlier works on
the subject. Bases of generalized Casimir operators were calculated by
means of the moving frames approach. Effectiveness of the proposed
technique was demonstrated by its application to computation of
invariants of solvable Lie algebras of general dimension restricted
only by a required structure of the nilradical. Contractions of
low-dimensional Lie algebras were described exhaustively with usage of
wide range of continuous and semi-continuous characteristics of Lie
algebras.
Anatoly Nikitin (National Academy of Sciences of Ukraine)
Galilean vector fields: tensor products and invariants with
using moving frames approach
All indecomposable finite-dimensional representations of the
homogeneous Galilei group which when restricted to the rotation
subgroup are decomposed to spin 0, 1/2 and 1
representations are constructed and classified. Tensor products and joint invariants
for such representations are found with using moving frames approach.
We provide five examples of conformal geometries which are naturally
associated with ordinary differential equations (ODEs). The first example
describes a one-to-one correspondence between the Wuenschmann class of
3rd order ODEs considered modulo contact transformations of variables
and (local) 3-dimensional conformal Lorentzian geometries. The second
example shows that every point equivalent class of 3rd order ODEs
satisfying the Wuenschmann and the Cartan conditions define a
3-dimensional Lorentzian Einstein-Weyl geometry. The third example
associates to each point equivalence class of 3rd order ODEs a
6-dimensional conformal geometry of neutral signature. The fourth
example exhibits the one-to-one correspondence between point
equivalent classes of 2nd order ODEs and 4-dimensional conformal
Fefferman-like metrics of neutral signature. The fifth example shows
the correspondence between undetermined ODEs of the Monge type and
conformal geometries of signature \$(3,2)\$. The Cartan normal
conformal connection for these geometries
is reducible to the Cartan connection with values
in the Lie algebra of the noncompact form of the exceptional group
\$G_2\$. All the examples are deeply rooted in Elie Cartan's works on
exterior differential systems.
I will present the basics of the equivariant method of moving
frames. First, the relevant constructions for finite-dimensional
Lie group actions will be presented. Applications include
the classification of differential invariants, invariant
differential equations and variational problems, symmetry and
equivalence problems, and the design of invariant numerical
algorithms. Then I will introduce infinite-dimensional Lie
pseudo-groups and discuss how to extend the moving frame
methods. The lectures will include a self-contained
introduction to the variational bicomplex.
Many aspects of parabolic geometries are by now well understood,
especially those related to differential geometry and the symmetries
of natural differential operators associated with these geometries.
In this talk we shall see how some aspects of geometric analysis may be
generalized from the best-known cases, namely Riemannian and conformal
geometry, resp. CR geometry, to more general geometries. In particular
we shall give results about Sobolev spaces and inequalities, and
also mention results about unitary representations of the natural
symmetry groups.
Teoman Ozer (Istanbul Technical University)
Symmetry groups of the integro-differential equations
and an approach for solutions of nonlocal determining equations
In this study we introduce the general theory of Lie group analysis of
integro-differential equations. A generalized version of the direct methods of
determination of symmetry group of the point transformations is presented for
the equations with nonlocal structure. First, the symmetry group definition of
point transformations for the integro-differential equations is discussed and
then a new approach for solving of nonlocal determining equations is
presented.
George Pogosyan (Yerevan State University)
Exact and quasi-exact solvability superintegrability in Euclidean
space
We show that separation of variables for second-order superintegrable
systems in two- and three-dimensional Euclidean space generates both exactly
solvable and quasi-exactly solvable problems in quantum mechanics. A principal
advantage of our analysis using nondegenerate superintegrable systems is that
they are multiseparable. Most past separation of variables treatments of
quasi-exactly solvable problems via partial differential equations have only
incorporated separability, not multiseparability.
We also propose another definition of exactly and quasi-exactly
solvability. The quantum mechanical problem is called exactly solvable
if the solution of Schroedinger equation, can be expressed in terms of
hypergeometrical functions and is quasi-exactly solvable if the Schroedinger
equation admit polynomial solutions with the coefficients necessarily
satisfying the three-term or higher order of recurrence relations.
In three dimensions we give an example of a system that is quasi-exactly
solvable in one set of separable coordinates, but is not exactly
solvable in any other separable coordinates.
The work done with colloboration with E.Kalnins and W.Miller Jr.
I will report on my ongoing joint work with Peter Olver on
developing systematic and constructive algorithms for analyzing
the structure of continuous pseudogroups and identifying various
invariants for their action.
Unlike in the finite dimensional case, there is no generally
accepted abstract object to play the role of an infinite dimensional
pseudogroup. In our approach we employ the bundle of jets of
group transformations to parametrize a pseudogroup, and we realize
Maurer-Cartan forms for the pseudogroup as suitably invariant forms
on this pseudogroup jet bundle. Remarkably, the structure equations
for the Maurer-Cartan forms can then be derived from the determining
equations for the infinitesimal generators of the pseudogroup
action solely by means of linear algebra.A moving frames for general pseudogroup actions is defined as
equivariant mappings from the space of jets of submanifolds into the
pseudogroup jet bundle. The existence of a moving frame requires local
freeness of the action in a suitable sense and, as in the finite
dimensional case, moving frames can be used to systematically produce
complete sets of differential invariants and invariant coframes for the
pseudogroup action and to effectively analyze their algebraic structure.Our constructions are equally applicable to finite dimensional Lie group
actions and provide a slight generalization of the classical moving frame
methods in this case.
Giovanni Rastelli (Università di Torino)
Separation of variables for systems of first-order Partial
Differential Equations: the Dirac equation in two-dimensional manifolds
The problem of solving Dirac equation on two-dimensional manifolds is
approached from the separation of variables point of vue, with the aim of
setting the basis for the analysis in higher dimensions. Beginning from a
sound definition of multiplicative separation for systems of two
first-order PDE of "eigenvalue problem"-type and the characterization of
those systems admitting multiplicatively separated solutions in some
arbitrarily given coordinate system, more structure is step by step added
to the problem by requiring the separation constants are associated with
differential operators and commuting differential operators. Finally, the
requirement that the original system coincides with the Dirac equation on a
two-dimensional manifold allows the characterization of those metric
tensors admitting separation of variables for the same Dirac equation and
of the symmetries associated with the separated coordinates. The research
is done in collaboration with R.G. McLenaghan.
1. Characterization of separation of variables by higher order
symmetries (80's)
2. Point and non-local symmetries (80's)
3. Computation of the structure of finite and infinite Lie
pseudo-groups of symmetries (90's)
4. Numerical Jet Geometry and Numerical Algebraic Geometry
(00's)
5. New problems in deformation of PDE systems (10's?)The talk will have a retrospective feel, while looking forward
to new problems and describing some links with a forth-coming
special year on Algebraic
Geometry and its Applications at the IMA (06-07).
From my earliest work, on the connection between symmetries
and separation
of variables with Kalnins and Miller, I focused on the
extraction of structural information
using computer algebra. After leaving separation of variables,
I developed the algorithmic
analysis and associated theory of overdetermined systems of
PDE.
Linear systems are always shadowed by non-linear ones. Poor
underlying complexity
means that modern tools such as Numerical Algebraic Geometry
(essentially computing with generic
points on the jet components of over-determined systems) are
needed.
Unifying analysis and algebraic techniques via deformations of
PDE pose intriguing open problems.
Chan Roath (Ministry of Education, Youth and Sport) www.moevs.gov.kh
Resolution on n-order functional differential equations with operator coefficients and delay in Hilbert spaces
I will discuss recent results on the (extrinsic) rigidity of the adjoint varieties.
Consider the adjoint action of a simple Lie group G on its Lie algebra g. This induces an action on the projective space P(g). The
action of G on P(g) has a unique closed orbit (the orbit of a highest weight space), and this orbit is an algebraic variety X. For
example, when G=Sp(2n) is symplectic, X is the Veronese embedding of projective n-space. When G=SL(n+1) is the special linear group, X
is the space of trace-free, rank=1 matrices.In order to study the rigidity of a projective variety Y, we look at the set C(k,y) of lines having contact to order k at y in Y. Note
that C(1,y) is just the (projectivized) tangent space. So we say any two varieties (of the same dimension) are identical to first-order.The contact sets C(k,y) arise as the zero sets of ideals I(k,y) generated by differential invariants. In general, we say two varieties X
and Y agree to order k at x and y if (1) I(j,x) = I(j,y) for all j = 1, 2, ..., k. We say X is rigid to order k if this condition forces Y to be (projectively equivalent to) X.
The various subjects in the title are connected by a common
strand!
In my talk, which is introductory in nature, I will give an
overview of the subjects, and describe this fascinating
connection.
Cartan's method of moving frames has been successfully applied to
the study of CR-manifolds, their mappings and invariants. For some types
of CR-manifolds there is a close relation to the point-wise or contact
geometry of differential equations. This can be used to find
CR-manifolds with special symmetries.
The recently introduced notion of multicontact structures provides a
general framework comprising certain geometries of differential
equations and CR-manifolds which in turn give examples with many
symmetries.
Astri Sjoberg (University of Johannesburg)
Symmetries, Associated Conservation Laws and Double Reductions of PDEs
When a differential equation admits a Noether symmetry, a conservation law is
associated with this symmetry, and a double reduction can be achieved as a result
of this association. The association of conservation laws with Noether symmetries
was extended to Lie Backlund symmetries and nonlocal symmetries recently.
This opened the door to the extension of the theory on double reductions to
partial differential equations (PDEs) that do not have a Lagrangian and
therefore do not possess Noether symmetries.
We present a theorem to effect a double reduction of PDEs with two independent
variables. Such a double reduction is possible when a PDE (or system of PDEs)
admits a symmetry which is associated with a conservation law.
Some examples are given.
The conformally invariant objects were always understood as affine
invariants of the underlying Riemannian connections which did not depend on
the choice within the conformal class. Although this definition is so easy
to understand, the description of such invariants is a difficult task and
many mathematicians devoted deep papers to this problem in the last 80
years. The classical approach coined already by Veblen and Schouten was to
elaborate special tensorial objects out of the curvatures, designed to
eliminate the transformation rules of the Riemannian connections under
conformal rescaling. The most complete treatment of such a procedure was
given in a series of papers by Günther and Wünsch in 1986. They provide a
version of calculus which allows to list all invariants in low homogeneities
explicitly.
The aim of this talk is to present a concise version of a similar calculus
for all parabolic geometries, relying on the canonical normal Cartan
connections.
New superintegrable potentials in Euclidean 2D and 3D spaces
We will survey new superintegrable potentials that have
appeared in the
literature recently to discuss directions for possible new
developments in the
area. This is joint work with Caroline Adlam and Ray
McLenaghan.
I will review the Hamilton-Jacobi theory of orthogonal separation of variables
in the context of the Cartan geometry, in particular, its most valuable asset,
- the method of moving frames. The central concept in this setting is that of
frames of eigenvectors (eigenforms) of Killing two-tensors which provides a
natural presentation of the theory in terms of principal fiber bundles.
Eisenhart (implicitly) employed this idea in 1934 to study orthogonal
separation of variables in Euclidean 3-space for geodesic Hamiltonians. I will
show how the corresponding problem for natural Hamiltonians can be solved with
the aid of a more general version of the moving frames method than the
one used
by Eisenhart (joint work with J.T. Horwood and R.G. McLenaghan). As an
application, the approach outlined above together with symmetry methods
will be
used to determine a new class of maximally superintegrable and multi-separable
potentials in Euclidean 3-space. These potentials given by a formula depending
on an arbitrary function do not appear in Evans' classification of 1990. A
particular example of such a potential is the potential of the Calogero-Moser
system (joint work with P. Winternitz).
Presented on behalf of Joshua T. Horwood (University of Cambridge).
We will describe the KillingTensor package and demonstrate its
features including the ability to study (multi-)separable (super-)integrable
potentials defined in Euclidean space. The algorithm is based on an orbit
analysis of the isometry group action on the 20-dimensional vector space of
valence two Killing tensors. As an illustration we will employ the
KillingTensor package to present a comprehensive analysis of the Calogero-Moser
potential and other superintegrable potentials defined in Euclidean space.
Petr Somberg (Karlovy (Charles) University)
The Uniqueness of the Joseph Ideal for the Classical Groups
Same abstract as of 7/17 poster session.
Petr Somberg (Karlovy (Charles) University)
The Uniqueness of the Joseph Ideal for the Classical Groups
The Joseph ideal is a unique ideal in the universal enveloping algebra of
a simple Lie algebra attached to the minimal coadjoint orbit. For the classical
groups, its uniqueness - in a sense of the non-commutative graded deformation
theory - is equivalent to the existence of tensors with special properties.
The existence of these tensors is usually concluded abstractly via algebraic
geometry, but we present explicit formulae. This allows a rather direct
computation of a special value of the parameter in the family of ideals used
to determine the Joseph ideal.
Petr Somberg (Karlovy (Charles) University)
The Uniqueness of the Joseph Ideal for the Classical Groups
The Joseph ideal is a unique ideal in the universal enveloping algebra of
a simple Lie algebra attached to the minimal coadjoint orbit. For the classical
groups, its uniqueness - in a sense of the non-commutative graded deformation
theory - is equivalent to the existence of tensors with special properties.
The existence of these tensors is usually concluded abstractly via algebraic
geometry, but we present explicit formulae. This allows a rather direct
computation of a special value of the parameter in the family of ideals used
to determine the Joseph ideal.
Vladimir Soucek (Karlovy (Charles) University)
Analogues of the Dolbeault complex and the separation of variables
The Dirac equation is an analogue of the Cauchy-Riemann equations
in higher dimensions. An analogues of the del-bar operator in the theory
of several complex variables in higher dimensions is the Dirac operator D
in several vector variables. It is possible to construct a resolution
starting with the operator D, which is an analogue of the Dolbeault complex.
A suitable tool for study of the properties of the complex is the separation
of variables for spinor valued fields in several vector variables and the
corresponding Howe dual pair.
I will first report on some recent work on generalising
Shapiro-Lopatinski condition to overdetermined problems. The technical
difficulty in this extension is that the parametrices are no longer
pseudodifferential operators, but Boutet de Monvel operators. Then I
discuss some numerical work related to these issues, and present one
possibility to treat overdetermined problems numerically. In this
approach there is no need to worry about inf-sup condition: for example
one can stably compute the solution of the Stokes problem with P1/P1
formulation.
Mikhail Vasiliev (P. N. Lebedev Physics Institute)
I will discuss nonlinear equations of motion of higher spin
gauge fields. The driving idea is to study most symmetric field theories,
assuming that whatever theory of fundamental interactions is it should be
very symmetric. The formulation is based on the unfolded dynamics formalism
which is an overdetermined multidimensional covariant extension of the
one-dimensional Hamiltonian dynamics. General properties of the unfolded
dynamics formulation will be discussed in some detail with the emphasize
on symmetries and coordinate independence.
Alfredo Villanueva (University of Iowa)
A Method to Find Symmetries of the Yamabe Operator
We present our research through two examples; first for 1-forms on
curved and non curved spaces, and secondly for a trace-free symmetric
2-tensor on non curved spaces. We use an overdetermined system as a
starting point, from here representation theory and generalized
gradients are used to analyze the bundles where the covariant
derivatives land. We obtain formulas where higher derivatives are
written in terms of a finite number of independent jets. Then if Y is
the Yamabe Operator, D and P are differential operator with unknown
coefficients, we set YD-PY = 0, and use our formulas to find the right
coefficients for D and P.
These lectures will cover the following topics
1. Definition and basic properties.
2. Lie symmetries and higher order symmetries.
3. Quadratic superintegrability and the separation of variables in spaces
of constant and variable curvature.
4. Superintegrability and exact solvability.
5. Superintegrability without separation of variables. Third order
integrals of motion. Velocity dependent forces.
6. Integrable and superintegrable systems involving particles with spin.
Thomas Wolf (Brock University)
Classification of 3-dimensional scalar discrete integrable equations
A new field of discrete differential geometry is presently emerging on
the border between differential and discrete geometry.Whereas classical differential geometry investigates smooth geometric
shapes (such as surfaces), and discrete geometry studies geometric
shapes with finite number of elements (such as polyhedra), the
discrete differential geometry aims at the development of discrete
equivalents of notions and methods of smooth surface theory.Current interest in this field derives not only from its importance in
pure mathematics but also from its relevance for other fields like
computer graphics. Recent progress in discrete differential geometry,
reported in a review by A.Bobenko and Yu. Suris (see www.arxiv.org,
math.DG/0504358) has lead, somewhat unexpectedly, to a better
understanding of some fundamental structures lying in the basis of the
classical differential geometry and of the theory of integrable
systems.In particular it was discovered that classical transformations of
remarkable classes of smooth surfaces (Baecklund transfromations,
Ribaucour transformations etc.) after discretization of the
respective classes of surfaces become just their extension with an
"extra discrete dimension" in an absolutely symmetric way.The requirement of consistency of the original difference systems with
the operation of adding such "extra discrete dimension" gives an BIG
overdetermined system of equations for the coefficients of the
original difference equation describing the discrete system in
question. This reqirement was considered in the review by A.Bobenko
and Yu. Suris as the fundamental property giving a criterion of
"discrete integrability".In this talk we describe our recent results on complete classification
of a class of 3-dimensional scalar discrete integrable equations.
Keizo Yamaguchi (Hokkaido University)
Geometry of linear differential systems - towards "contact geometry of
second order"
Starting from the geometric construction of jet spaces,
defining the symbol algebra of these canonical (contact) systems, the
goal of these lectures is to formulate submanifolds of 2-jet spaces as
PD-manifolds (R,D1,D2), i.e. D1 and D2 are a pair of subbundles of the
tangent bundle of R. This will also serve as a preparation to
symmetries of p.d.e. and to parabolic geometry associated with various
p.d.e.s, which will be discussed in later weeks.
Jin Yue (Dalhousie University)
The 1856 Lemma of Cayley Revisited, II. Fundamental Invariants
We continue the study of vector spaces of Killing tensors
defined in the Minkowski plane from the viewpoint of the invariant theory
initiated in an earlier paper. This work is based on the inductive version
of the
moving frames method developed by Irina Kogan. Thus we develop an
algorithm to compute complete sets of fundamental invariants of the
isometry group action in the vector spaces of Killing tensors of arbitrary
valence defined in the Minkowski plane. This is joint work with Roman
Smirnov.
My talk is devoted to the equivalence problem of non-holonomic vector
distributions and it is based on the joint work with Boris Doubrov. The
problem was originated by E. Cartan in the beginning of twenty century,
who treated the first nontrivial case of the fields of planes in a
five-dimensional ambient space with his method of equivalence. In our talk
we would like to describe a new rather effective approach to this problem,
which we call the symplectification procedure. The starting point of this
procedure is to lift the distribution to a special submanifold of the
cotangent bundle, foliated by the characteristic curves. The invariants of
the distributions can be obtained from the study of the dynamics of this
lifting along the characteristic curves. The case of rank two
distributions (fields of planes) will be discussed in more detail. In this
case we succeeded to construct the canonical frame and to find the most
symmetric models for the arbitrary dimension of the ambient manifold,
generalizing the mentioned work of Cartan. The new effects in the case of
distributions of rank greater than two will be discussed as well.
