University of Minnesota
http://www.umn.edu/

## Symmetries and Overdetermined Systems of Partial Differential Equations

### July 17-August 4, 2006

Event ID: #1883, Published: October 31, 2006 10:56:57

Stephen Anco (Brock University)

Group-invariant soliton equations and bi-Hamiltonian geometric curve flows in Riemannian symmetric spaces and Lie groups

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.

Douglas N. Arnold (University of Minnesota Twin Cities) http://www.ima.umn.edu/~arnold/

Finite element exterior calculus and its applications

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.

Peter A. Clarkson (University of Kent at Canterbury) http://www.kent.ac.uk/ims/personal/pac3/

Special polynomials associated with rational solutions of the Painleve equations and applications to soliton equations

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.

Michael Cowling (University of New South Wales)

The rigidity problem for Carnot groups

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 without moving frames

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).

Symmetries of the Square of the Laplacian

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.

Final Discussion Session

Diagram

The Work of Thomas P. Branson
(Michael Eastwood, moderator)

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.

Mark Fels (Utah State University)

Exterior differential systems with symmetry

Given a symmetry group of an exterior differential system, we'll investigate ways to use the group to simplify finding integral manifolds.

Eugene Ferapontov (Loughborough University)

Integrable equations of the dispersionless Hirota type and hypersurfaces of the Lagrangian Grassmanian

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.

Evelyne Hubert (Institut National de Recherche en Informatique Automatique (INRIA)) http://www.inria.fr/cafe/Evelyne.Hubert

Rational and Algebraic Invariants of a Group Action

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.

Peter Hydon (University of Surrey)

Discrete symmetries and Lie algebra automorphisms

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.

Jens Jonasson (Linköping University) http://www.mai.liu.se/~jejon/

Multiplication of solutions for systems of partial differential equations

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

No abstract.

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.

Jonathan Kress (University of New South Wales)

Quadratic algebras and superintegrable systems

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.

Jonathan Kress (University of New South Wales)

Final Discussion Session

Chalk board photos

Joseph Landsberg (Texas A & M University) http://www.math.tamu.edu/~jml/

Projective differential geometry with moving frames

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.

Thomas Leistner (University of Adelaide)

Ambient connections realising conformal Tractor holonomy

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)

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.

Debra Lewis (University of Minnesota Twin Cities) http://math.ucsc.edu/~lewis/

Geometric integration and control

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)

A simple criterion for involutivity

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.

Vladimir S. Matveev (Katholieke Universiteit Leuven) http://home.mathematik.uni-freiburg.de/matveev/

New integrable system on the sphere

Joint work with Holger Dullin.

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.

Vladimir S. Matveev (Katholieke Universiteit Leuven) http://home.mathematik.uni-freiburg.de/matveev/

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.

Vladimir S. Matveev (Katholieke Universiteit Leuven) http://home.mathematik.uni-freiburg.de/matveev/

Superintegrable systems and the solution of a S. Lie problem

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)

Willard Miller Jr. (University of Minnesota Twin Cities) http://www.ima.umn.edu/~miller/

Final Discussion Group
Willard Miller Jr., moderator

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.

Willard Miller Jr. (University of Minnesota Twin Cities) http://www.ima.umn.edu/~miller/

Second order superintegrable systems in two and three dimensions. (Solving a system in multiple ways)

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.

Willard Miller Jr. (University of Minnesota Twin Cities) http://www.ima.umn.edu/~miller/

Variable separation and second order superintegrability

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.

Pawel Nurowski (University of Warsaw) http://www.fuw.edu.pl/~nurowski

Differential equations and conformal structures

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.

Peter J. Olver (University of Minnesota Twin Cities) http://www.math.umn.edu/~olver

Introduction to Moving Frames and Pseudo-Groups

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.

Bent Orsted (Aarhus University)

Geometric analysis in parabolic geometries

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.

Juha Pohjanpelto (Oregon State University)

The Structure of Continuous Pseudogroups

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.

Gregory J. Reid (University of Western Ontario) http://www.orcca.on.ca/~reid/

Algorithmic Symmetry Analysis and Overdetermined Systems of PDE

Topics covered in this talk include

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

Abstract and paper in pdf format only.

Colleen Robles (University of Rochester)

Rigidity of the adjoint varieties

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.

Siddhartha Sahi (Rutgers University)

Equivariant differential operators, classical invariant theory, unitary representations, and Macdonald polynomials

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.

Gerd Schmalz (University of New England) http://mcs.une.edu.au/~gerd

CR-manifolds, differential equations and multicontact structures (tentative)

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

Same abstract as of 7/17 poster session.

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.

Jan Slovak (Masaryk University)

Wünsch's calculus for parabolic geometries

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.

Roman Smirnov (Dalhousie University) http://www.mathstat.dal.ca/~smirnov

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.

Roman Smirnov (Dalhousie University) http://www.mathstat.dal.ca/~smirnov

Geometry "a la Cartan" revisited: Hamilton-Jacobi theory in moving frames

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).

Roman Smirnov (Dalhousie University) http://www.mathstat.dal.ca/~smirnov

The KillingTensor package (presented by Roman Smirnov on behalf of Joshua Horwood)

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.

Jukka Tuomela (University of Joensuu)

Overdetermined elliptic boundary value problems

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)

Higher spin gauge theories and unfolded dynamics

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.

Pavel Winternitz (University of Montreal)

Superintegrable classical and quantum systems

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

Joint work with S. Tsarev and A. Bobenko.

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.

Igor Zelenko (International School for Advanced Studies (SISSA/ISAS)) http://www.sissa.it/~zelenko/zelenkohp_html

Symplectification procedure for the equivalence problem of vector distributions

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.