• Local symmetries and conservation laws of spin s>0 fields Stephen Anco (sanco@brocku.ca) Abstract A complete and explicit
classification of all locally constructed (higher order) conserved currents
and (generalized) symmetries is obtained for massless linear symmetric
spinor fields of any spin s>0 in four dimensional flat spacetime. These
results generalize a recent classication in the spin s=1 case of all conserved
currents and symmetries locally constructed from the electromagnetic spinor
field. The present classification of currents yields spin s>0 analogs
of the well-known electromagnetic stress-energy tensor and Lipkin's zilch
tensor, as well as a spin s>0 analog of a novel chiral tensor that was
found in the spin s=1 case. In addition, the symmetry classification yields
spin s>0 analogs of the second order symmetries discovered by Fushchich
and Nikitin for electromagnetic fields. |
• Natural Quintessence and the Braneworld Cliff Burgess (cliff@physics.mcgill.ca
) Abstract Current cosmological
measurements tell us that the universe contains 5% baryons, 25% dark matter
and 70% dark energy. The nature of the dark matter and the dark energy
is completely unknown, apart from the fact that they satisfy different
equations of states. This talk briefly summarizes the difficulties which
are encountered in explaining the dark energy. The focus is on naturalness
issues, which I argue provide very strong restrictions on the kinds of
cosmologies which can be considered. |
• Congruence subgroups of the modular group Chris Cummins (cummins@alcor.concordia.ca) Abstract In recent years modular
invariance has played an increasing important role in many areas of mathematics
and physics. In this talk I shall discuss the computation and classification
of congruence subgroups of the modular group of small genus. This is joint
work with Sebastian Pauli. |
• Asymptotic SU(2) Wigner functions from the weight diagram Hubert de Guise
(hdeguise@mail.lakeheadu.ca) Abstract By comparing portions
of the SU(2) weight diagram with physical systems, I will show how one
can infer the essential features of the asymptotic form of the SU(2) Wigner
functions. If time permits, the same method will be used to deduce some
features of the SU(3) Wigner functions. |
• Physical applications of a metric formulation of Galilean invariance Marc de Montigny
(montigny@phys.ualberta.ca)
Abstract We describe a five-dimensional metric formulation of Galilean covariance and illustrate it with various examples in field theory. As an first illustration, we recover the two Galilean limits of electromagnetism first examined by Le Bellac and L\'evy-Leblond. Then we describe the field theoretical formulation of some fluids and superfluids models : Navier-Stokes equation, Takahashi Lagrangian for irrotational fluids and Thellung-Ziman Lagrangian for Helium II. Our last examples treat non-relativistic Bhabha equations for spin 0 and 1 particles, and Dirac equation for spin 1/2. |
• Dedekind eta and Jacobi theta function identities Terry Gannon (tgannon@math.ualberta.ca) Abstract In Gannon-Lam, geometrical lattice equivalences were used to produce identities involving the Jacobi theta functions. Here we strengthen and extend the method, and find identities also involving the Dedekind eta function (and more generally any Dirichlet twists of the theta functions). We find over 100 new quadratic eta function identities and conjecture we've found them all. |
• Simple examples of Berezin-Toeplitz Quantization: finite sets and unit interval unit interval Jean-Pierre Gazeau
(gazeau@ccr.jussieu.fr)
Abstract We present a quantization scheme of an arbitrary measure space based on overcomplete families of states and generalizing the Klauder and the Berezin approaches. The procedure is illustrated by elementary examples in which the measure space is a N-element set and the unit interval. |
• Weierstrass representation for surfaces in multidimensional spaces Michel Grundland
(grundland@CRM.UMontreal.CA)
Abstract We introduce a Weierstrass-type system of equations corresponding to CP^N harmonic maps, which generalize the systems previously constructed for CP^1 and CP^2. We use a set of conserved quantities for CP^N model in order to suggest a certain geometric interpretation of the generalized Weierstrass system describing surfaces immersed in multidimensional spaces. |
• Bob Sharp: teacher, researcher, colleague, friend John Harnad (harnad@crm.umontreal.ca) Abstract Just some brief personal
recollections. |
• Boson realizations of the semi-simple Lie algebras Miloslav Havlicek
(dekan@fjfi.cvut.cz) Abstract A short review of the construction of the boson realizations of the semi-simple Lie algebras is given. The method is illustrated on examples. An application for the construction of matrix differential realizations is mentioned and their properties are studied. It is shown how alternatively the concept of a Lie field can be used. |
• Stretched Littlewood-Richardson and Kostka coefficients Ronald King (R.C.King@maths.soton.ac.uk) Abstract Littlewood-Richardson
and Kostka coefficients both arise in character theory and also in the
study of symmetric functions; the former as tensor product multiplicities
and as coefficients in the expansion of products of Schur functions, and
the latter as weight multiplicities and as transition matrix elements
between Schur functions and the monomial symmetric functions. In each
case these coefficients are specified by means of partitions. If the parts
of the partitions are all multiplied by some fixed positive integer number
$N$, then the values of the coefficients will change as a function of
the stretching parameter $N$. It has been first conjectured and then proved
that a stretched coefficient is non-zero if and only if the corresponding
unstretched coefficient is non-zero. Furthermore a stretched coefficient
is $1$ if and only if the corresponding unstretched coefficient is $1$.
Here we study the behaviour of these coefficients as a function of $N$
for other values $k$ of the unstretched coefficient. It is conjectured
that this behaviour is always polynomial in $N$. The study makes use of
a hive model of both Littlewood-Richardson and Kostka coefficients, thereby
making a connection with the integer points on and within certain rational
convex polytopes. Many examples of the relevant polynomials will be given,
along with further conjectures regarding their general form and that of
their generating functions. |
• Group actions and closed geodesics for some hyperbolic manifolds Peter Kramer (peter.kramer@uni-tuebingen.de)
Abstract Hyperbolic spaces $H^n \sim SO(n,1,R)/SO(n,R)$ carry a pseudoriemannian metric of constant negative curvature [1]. As universal covering spaces of a closed hyperbolic manifolds $M$ they provide particular models for a cosmos with closed geodesics, compare M Lachieze-Rey and J-P Luminet [2]. Shortest closed geodesics and the autocorrelation may be compared with astrophysical evidence. We extend the continuous group analysis from $M=$ the double torus on $H^2$ given in [3] to $M=$ the hyperbolic dodecahedral space, Weber and Seifert (1932), Best (1971). The first homotopy groups $\pi(M)$ are (normal) subgroups of hyperbolic Coxeter groups. We describe closed geodesics starting at general points, their length and direction on $M$ by actions of the homotopy and related continuous groups on $H^2,H^3$. We analyze the first homology groups, the autocorrelation function and the notion of elementary systems on the manifolds. \vspace{0.2cm} [1] R G Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer, New York 1994 [2] M Lachieze-Rey and J-P Luminet, {\em Cosmic Topology}, Phys. Rep. {\bf 254} (1995) 135-214 [3] P Kramer and M Lorente, {\em The double torus as a 2D cosmos: groups, geometry and closed geodesics}, J Phys {\bf A 35} (2002) 1961-1981 |
• An Ultimate Symmetry in Physics? Harry Lam (Lam@physics.mcgill.ca)
Abstract Over the years, more and more fundamental symmetries of physics have been discovered: from global symmetries of a geometrical nature, to local symmetries that can replace part of our dynamics. Extrapolating from this historical progression, perhaps it is not too ridiculous to expect the dynamical theory of physics will one day be completely replaced by an ultimate symmetry. I will elaborate on this speculation by using examples from existing theories, to argue that even deep symmetries may just be based qualitatively on very simple physical requirements. In this light, perhaps existing and future physical puzzles can lead us to an ultimate symmetry of physics? |
• Formal Characters and Resolutions of Certain Simple Infinite Dimensional $A_n$-Modules Frank Lemire (lemire@uwindsor.ca) Abstract If $V$ is a finite dimensional simple module for a simple Lie algebra Berstein, Gel'fand, Gel'fand provided a formal character formula for $V$ as well as a resolution of $V$ in terms of Verma modules. In this talk we generalize these results to certain simple infinite dimensional $A_n$-modules having bounded weight multiplicities. |
• From representations of G < SU2 to the Monster via semi-affine Coxeter-Dynkin graphs. John McKay (mckayj@CRM.UMontreal.CA) Abstract Bob Sharp took an interest
in generating functions and I shall show how some may be derived using
a q-deformation of the Coxeter-Dynkin diagram. There are also totally
mysterious connections with the Monster and other sporadic finite simple
groups. |
• Fusion rules: generating functions and bases Pierre Mathieu Abstract We review basic results related to the computations of fusion rules in affine Lie algebras that follow from the extension of the Patera-Sharp generating-function approach for tensor products. |
• Transient effects in Wigner distribution phase space for a scattering problem MOSHINSKY MARCOS
(moshi@fisica.unam.mx) Abstract At the Group24 Conference
in July 2002 in Paris several researchers were interested in my analysis
in Wigner distribution space of my work on diffraction in time. It was
suggested that I should extend it when a potential is present and the
simplest case is one dimensional with a delta function at the origin and
the particles coming from the left with given momentum. In this paper
wew carry the analysis of the transient effects of this problem using
the Laplace transform. We discuss the similarities and differences of
our problem in Wigner distribution phase space when compared with the
case of diffraction in time. |
• An Ultimate Symmetry in Physics? Zorka Papadopolos Peter A.B. Pleasants
Abstract Planes orthogonal to 5-, 3- and 2-fold axes of icosahedral phases We are interested in planes in models of icosahedral quasicrystal that are candidates for surfaces in their physical realizations. Icosahedral quasicrystals are derived from 3 basic modules (or from combinations of them), called by Mermin et al. [Phys. Rev. {\bf B35} 5487 (1987)] "primitive", "face-centered" or "body-centered" and obtained by icosahedral projection of the ZZ^6, D_6 or D*_6 six-dimensional lattices, respectively. The densities of the atomic positions in the 2-fold, 3-fold and 5-fold planes are governed by the underlying module and by the window. When the window is a sphere, or a polytope very close to a sphere, the module is the major influence on the densities of these planes, both in the bulk-model and, if there is no reconstruction, on the surface of an alloy. We give the relative densities of these planes for all three modules and we discuss in particular the model for the icosahedral alloys (i-AlPdMn, i-AlCuFe) defined by three windows. |
• Generating functions in group representation theory Jiri Patera (patera@crm.umontreal.ca)
Abstract This topic was for many years undoubtedly the closest to Bob Sharp's heart. A subjective review of Bob's work and an outlook will be presented. |
• Symmetries and Lagrangian time-discretizations of Euler equations Alexei Penskoi (penskoi@crm.umontreal.ca) Abstract In late 80s - early
90s J. Moser and A. P. Veselov considered Lagrangian discrete systems
on Lie groups with additional symmetry conditions imposed on Lagrangians.
They observed that such systems are often integrable time-discretizations
of integrable Euler equations on these Lie groups. In recent papers we
studied Lagrangian discrete systems with additional symmetry requirements
on certain infinite-dimensional Lie groups. We will discuss some interesting
properties of these systems. |
• Quasi-exactly solvable models in quantum optics Miguel A. Rodriguez
(rodrigue@fis.ucm.es) Abstract In this talk, we shall present a discussion on the connections between some models in Quantum Optics, including n-th harmonic generation and photon cascades, and quasi-exactly solvable problems in one dimension. |
• Geometric quantization from a coherent state perspective David J. Rowe (rowe@physics.utoronto.ca)
Abstract The theory of coherent state representations is both a theory of induced representations and of quantization. The different perspectives lead to valuable insights and new ways of unifying and extending the two theories. The coherent state approach, and its vector coherent state generalization, have the merit that their physical content is transparent and highly practical. Indeed, they have been shown in numerous applications to be applicable to a wide range of symmetry problems in physics. In particular, in the language of geometric quantization, they are able to quantize interesting physical systems with intrinsic (gauge) degrees of freedom. |
• Symmetry Math Video Game Used to Identify Genius and Train Creativity Maps onto Knot Theory Gordon Shaw (gshaw@mindinst.org) Abstract Spatial-temporal
(ST) reasoning (making a mental image and thinking ahead in space and
time using symmetry operations, as in chess) is of considerable benefit
to children in understanding math and science. Big Seed, a ST video game,
is challenging, demanding and relevant, even for research mathematicians
and scientists (available for play at this Workshop). The large ST reasoning
capabilities on Big Seed demonstrated by the four groups of young children
far exceeded even the most optimistic expectations in less than 7 hours
of training. A 3 rd grader has been identified as a genius (functionally
defined) in ST performance. We suggest that Big Seed be used for training
and assessing "creativity" (functionally defined) and ST reasoning as
well as discovering genius. The mapping of knot theory onto Big Seed is
presented. |
• Conformal Symmetries of a Model in non-Commutative Space Luc Vinet (vinet@vpa.mcgill.ca) Abstract Conformal Symmetries
of a Model in non-Commutative Space The symmetries of a recently proposed
non relativistic model in two dimensional non-commutative space are examined.
The corresponding constants of motion are shown to form a Poisson-Lie
algebra that conformally extends the Galilei algebra with a two dimensional
central extension. |
• Polytope sums and Lie characters Mark Walton (walton@uleth.ca) Abstract Results on exponential
sums of convex lattice polytopes are applied to the characters of irreducible
representations of simple Lie algebras. The Brion formula is used to write
a useful character formula that manifests weight-multiplicity-degeneracy
beyond that explained by Weyl symmetry. It also allows the derivation
of a character generator relevant to affine fusion, the fusion of Wess-Zumino-Witten
conformal field theories. |