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# Mathematical Physics Laboratory Seminars Fall 2006 – Winter 2007

### PRE-HAMILTONIAN STRUCTURES FOR INTEGRABLE NONLINEAR SYSTEMS OF PDE

#### Arthemy Kiselev, DMS, Université de Montréal.

Pre-Hamiltonian matrix operators in total derivatives are considered; they are defined by the property that their images are subalgebras of the Lie algebra of evolutionary vector fields. This construction is close to the Lie algebroids over infinite jet spaces, and we study the corresponding algebraic and geometric properties. We assign a class of these operators and the brackets induced in their pre-images to integrable KdV-type hierarchies of symmetry flows on hyperbolic Euler-Lagrange Liouville-type systems (e.g. the 2D Toda lattices associated with semisimple Lie algebras following arXiv:math-ph/0703082; joint with J. van de Leur.

### SYMBOLIC COMPUTATION OF TRAVELLING WAVE SOLUTIONS OF NONLINEAR PARTIAL DIFFERENTIAL AND DIFFERENTIAL-DIFFERENCE EQUATIONS

#### Willy Hereman, Colorado School of Mines, U.S.A.

The talk addresses hyperbolic and elliptic function methods to symbolically compute travelling wave solutions of a large classes of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs). Typical examples of PDEs include the Korteweg-de Vries, Boussinesq, and Kuramoto-Sivashinski equations. The exact solutions model travelling waves in fluid dynamics, plasmas, electrical circuits, optical fibers, bio-genetics, etc. Nonlinear DDEs play an important role in numerical simulations of nonlinear PDEs, queing and network problems, and discretizations in solid state and quantum physics. Examples of DDEs that can be solved with the tanh-method include the Toda, Volterra, and Ablowitz-Ladik lattices. The methods have been implemented in MATHEMATICA. A demonstration of the software packages to automatically compute travelling wave solutions of nonlinear PDEs and DDEs will be given.

### A « THEORY OF EVERYTHING » CONFRONTS THE REAL WORLD

#### Lieu : McGill University, Ernest Rutherford Physics Building R.E. Bell Conference Room (103)

A description of the « standard model » of particle physics, with an emphasis on its symmetry groups, will be given. The notion of symmetry will then be generalized to supersymmetry, that is, symmetry between quantum states of different spin. The five supersymmetric strings theories will be introduced. It will be shown how any one of these is potentially a « theory of everything ». The five superstring theories will then be unified into a single theory: M-theory. A discussion of how our world and the standard model of particle physics emerges from M-theory will be given. In the process, the notion of a « brane » will be introduced. The last part of the lecture will focus on a new theory of early universe cosmology, called « Ekpyrotic » cosmology, which arises naturally in M-theory as the cataclysmic collision of two branes. It will be shown how the hot Big Bang can arise in such a collision and how the observed fluctuation spectrum in the cosmic microwave background (CMB) is accurately predicted.
N.B. This is a special seminar organized jointly with the Department of Physics, McGill University, in connection with the 40th anniversary celebration of the 1967 honours physics/mathematics class. It will be in the style of a colloquium, aimed at a broad audience. All are welcome to attend. It will be preceded by refreshments at 3:00 p.m. Another talk linked with this event will take place in the same location, at 2:00 pm. Title: « My Adventures in Photonic Crystals » Speaker: Eli Yablonovitch (UCLA, Dept. of Electrical Engineering).

### EXTENDED GRAVITY AND DARK ENERGY AS A NON-LINEAR CURVATURE EFFECT (with Salvatore Capozziello)

#### Mauro Francaviglia, Torino, Italy

Dark matter and dark energy can be explained without resorting to exotic fields if one accepts that the geometry of spacetime is governed by suitable generalized (or « extended ») gravitational theories based on Lagrangians that are non-linear in the curvature of a metric and/or a torsionless linear connection, i.e. working either in second order or in first order formalisms. Convenient choices of (nonquadratic) Lagrangians can well fit most of the galactical, extra-galactical, cosmological and solar system requirements imposed by current experimental results, without imposing drastic modifications of Einstein field equations and with FRW Cosmologies preserved as a good approximation of Nature at a first global scale. It can also been shown that non-linear theories can constitute a benchmark for testing the stochastic background of gravitational waves.

### SYMMETRIES OF THE 6j SYMBOLS AND PAINLEVÉ VI

#### Philip Boalch, ENS, Paris

We will relate the surprising Regge symmetry of the Racah-Wigner 6j symbols to the surprising Okamoto symmetry of the Painleve VI differential equation. This then presents the opportunity to give a conceptual derivation of the Regge symmetry, as the representation theoretic analogue of the author’s previous derivation of the Okamoto symmetry.

### ON THE LAX PAIRS OF THE SIXTH PAINLEVÉ EQUATION

#### Robert Conte, CEA, Saclay

The dependence of the sixth equation of Painlevé on its four parameters is holomorphic, therefore one expects all its Lax pairs to display such a dependence. This is indeed the case of the second order scalar « Lax » pair of Fuchs, but the second order matrix Lax pair of Jimbo and Miwa presents a meromorphic dependence on (and a holomorphic dependence on the three other ). We analyze the reason for this feature and make suggestions to suppress it.

### POLYNOMES DE MACDONALD ET GEOMETRIE DES VARIETES DE REPRESENTATION DES SURFACES DE RIEMANN

#### Emmanuel Letellier, CRM

Nous expliquerons comment on peut utiliser la theorie des fonctions symetriques (fonctions de Hall-Littlewood et de Macdonald) pour etudier la topologie des varietes de representations des surfaces de Riemann et des varietes de connections correspondantes. Nous ferons aussi le lien avec les algebres de dimension infinie (Kac-Moody).

### METRICS AND CLUSTER PARAMETRISATION FOR THE ANALYSIS OF GENE EXPRESSION DATA. CASE STUDY : E.COLI.

#### Maia Angelova, Northumbria, England

Clustering models are widely used in the analysis of high-throughput gene expression data from microarray experiments [1]. The genes with similar expression patterns can be clustered together. Clustering techniques are helpful to understand gene function, gene regulation, cellular processes and sub-types of cells. However clustering results are very sensitive to the choice of distance or similarity metrics and the parametrisation of the clusters. The choice of similarity measure determines the output of the clustering algorithm and the interpretation of the results. The estimation of the number of clusters is necessary prior to the analysis in a number of clustering approaches such as K-means, EM, hierarchical clustering and self organised maps. It determines the results of the clustering algorithm. The choice of the optimal number of clusters is not straightforward and is a result of a complex optimisation procedure (see for example [1-4]). We have investigated the impact of metrics and cluster parameterisation in the analysis of gene expression for two clustering models, K-means and EM, and propose a method of optimisation of cluster parameters based on homogeneity, cluster stability and separateness. The concept of entropy is explored to solve the optimisation problem. We have used our method to analyse the gene expression data of E.coli bacteria. The data are from DNA microarrays experiments designed to study the effects of knocking out the methionine repressor gene on the E.coli transcriptome [5]. We have used distance measures (Euclidean, Manhattan, Minkowski) and similarity measures (Pearson, cosine, Jaccard, dice) in the two clustering models. Our results indicate that, whilst the choice of metrics and cluster parameters does influence the clustering results, some key highly expressed genes are nearly always clustered together. The gene expression results can be used to construct gene regulation network for E.coli. Now we are investigating gene regulation for the simplest systems, such as the lac operon and ë-phage using models based on differential equations.
1. Stekel D. (2003). Microarray Bioinformatics, Cambridge University Press, Cambridge UK. 2. Jiang D., Tang C. and Zhang A. (2004). Cluster analysis for gene expression data: a survey, IEEE Trans. Knowledge Data Engineering 16 1370-1386. 3. Familili A.F., Liu G, Liu Z. (2004). Evaluation and optimization of clustering in gene expression data analysis, Bioinformatics 20, 1535-1545. 4. Bolshakova N., Azuaje F. and Cunningham P. (2005). An integrated tool for microarray data clustering and cluster validity assessment, Bioinformatics 21, 2546-2547. 5. Marincs F., Manfield I., Stead J., McDowall K. and Stockley P. (2006). Transcript analysis reveals an extended regulon and the importance of protein cooperativity for the Escherichia coli methionine repressor, Biochemical Journal 396, 227-234.

### DETERMINANTS OF LAPLACIANS

#### Andrew McIntyre, CRM, CONCORDIA

I will discuss some recent results on the regularized determinant of various Laplacian operators, defined on the moduli space of 2 dimensional surfaces of constant curvature. In particular, I will touch on some connections to theta functions, zeta functions, and scattering theory for hyperbolic 3-manifolds.

### INVARIANTS OF ALGEBRAIC CURVES, COMBINATORICS OF MAPS AND INTEGRABLE SYSTEMS

#### B. Eynard, SPHT Saclay

For any algebraic curve P(x,y)=0, we construct an infinite sequence of invariants F_g(P), which have many interesting properties, under symplectic transformations of P, under modular transformations, and under deformations of P, and under singular limits. When P is the large N limit of a matrix model’s spectral curve, the matrix integral is Z=\exp{\sum_{g=0}^\infty N^{2-2g} F_g(P)}. This works for 1-matrix model, 2-matrix model, marix model with external field (in particular Kontsevitch’s integral), and scaling limits of matrix models. In those cases, the F_g are the generating functions for counting maps of given genus, or intersection numbers.

### GENERALIZED FISHER-HARTWIG ASYMPTOTICS FOR HANKEL DETERMINANTS. THE RIEMANN-HILBERT APPROACH.

#### Alexander Its, IUPUI

We will discuss some new results concerning the Fisher-Hartwig type asymptotics for Hankel determinants whose symbols have jumps and\or root singularities. The method, based on the asymptotic solution of the relevant Riemann-Hilbert problems, will be outlined. The talk is based on the joint works with I. Krasovsky and on some recent works of I. Krasovsky (the root singularities).

### PARTIAL *-ALGEBRAS : AN OVERVIEW AND SOME EXAMPLES

#### Place : Concordia University, Library Building, 1400 Blvd. de Maisonneuve O, Room LB-921-04 (Conference room)

We review the main steps in the development of partial *-algebras. First we discuss the algebraic structure stemming from the partial multiplication. Then we study in some detail the locally convex partial *-algebras, in particular, the Banach partial *-algebras, and we describe a number of concrete examples. Next we consider the partial *-algebras of closable operators in Hilbert spaces (partial O*-algebras), with a special emphasis on their automorphisms. Finally we sketch the representation theory of abstract partial *-algebras and give some instances of possible physical applications.

### A HOLOMORPHIC REPRESENTATION OF THE JACOBI/SCHROEDINGER ALGEBRA VIA COHERENT STATES

#### Place : Concordia University, Library Building, 1400 Blvd. de Maisonneuve O, Room LB-921-04 (Conference room)

The coherent states (CS), invented by Erwin Schroedinger in the first days of Quantum Mechanics, are a powerful bridge between Classical and Quantum Mechanics. Here I shall present some applications of the group theoretic generalization of Perlomov’s CS in representations of a class of groups on homogeneous spaces.

### SYSTEMS OF TWO SECOND ORDER ODEs INVARIANT WITH RESPECT TO THE LOW-DIMENSIONAL LIE ALGEBRAS

#### Maryna Nesterenko, National Academy of Sciences of Ukraine

Regular and singular systems of two second-order ordinary differential equations invariant with respect to real three or four-dimensional real Lie algebras are exhaustively described. Necessary and sufficient condition for systems of two ODEs to be presented in normal form is presented.

### LARGE REPRESENTATION TYPE OF JORDAN ALGEBRAS

#### Iryna Kashuba, Universidade de Sao Paulo

This is a joint work with S.Ovsienko and I. Shestakov. The talk is devoted to the problem of the clasification of indecomposable Jordan bimodules over finite dimensional algebras.

We assume the base field k to be algebraically closed and of characteristics 2,3. Recall, that for a Jordan algebra J the category J-bimod of k-finite dimensional J-bimodules is equivalent to the category U-mod of (left) finitely dimensional modules over an associative algebra U = U(J), which is called the universal multiplication envelopeof J. If J has finite dimension the algebra U is finite dimensional as well. It allows us to apply to the category J-bimod all the machinery developed in the representation theory of finite dimensional algebras. In particular, in accordance with the representation type of the algebra U one can define Jordan algebras of the finite, tame and wild representation types. As in the case of associative algebra the distinction of the objects of finite and tame representation type is an interesting problem, especially because in these cases we can obtain a complete classification of finite dimensional bimodules over J.

We introduce two new notions for Jordan algebras : a diagram and a tensor algebra of a module, which prove to be very useful. In particular, we classify half-unital (or one-sided) representation type for Jordan matrix algebras with radical square zero. The results obtained are similar to the corresponding classical results for associative algebras.

### CLIFFORD ALGERA DERIVATIONS OF TAU FUNCTIONS FOR TWO DIMENSIONAL INTEGRABLE MODELS WITH POSITIVE AND NEGATIVE FLOWS

#### Johan Van de Leur, Utrecht University

We use a Clifford algebra to define multi-component tau functions as expectation values of certain multi-component Fermi operators that satisfy simple bilinear commutation relations. The tau functions contain both positive and negative flows and satisfy the 2n-component KP hierarchy. The hierarchy equations can be formulated in terms of pseudo-differential equations for n × n matrix wave functions derived in terms of tau functions. These equations are cast in form of Sato-Wilson relations. A reduction process leads to the AKNS, two-component Camassa-Holm and Cecotti-Vafa models and the formalism provides simple formulas for their solutions. This talk is based on joined work with Henrik Aratyn from the University of Illinois at Chicago.

### FERMIONIC CONSTRUCTION OF TAU FUNCTIONS AND RANDOM PROCESSES

#### A. Yu. Orlov, CRM and Oceanology Institute, Moscow

Via the fermionic construction of tau functions we shall show that discrete time versions of asymmetric exclusion processes are related to discrete analogs of certain matrix ensembles. (Joint work with J. Harnad)

### HALF-DIFFERENTIALS ON HYPERELLIPTIC SURFACES AND THEIR RELATIONS TO THE STRONG ASYMPTOTICS OF ORTHOGONAL POLYNOMIALS

#### Marco Bertola, CRM et Université Concordia

Orthogonal polynomials and generalizations thereof where the (pseudo)–measure is complex analytic and supported on paths in the plane can be studied in the limit of large degrees and the accumulation set of their zeroes characterized. Interestingly it is convenient to describe their asymptotics in terms of half-differentials on certain hyperelliptic curves and the construction is in a certain sense natural. Furthermore, using techniques of conformal glueing, it is possible to construct classes of pseudo orthogonal polynomials whose zeroes accumulate on certain trees (or forests) of Jordan arcs in the plane.

### TEICHMÜLLER THEORY OF BORDERED SURFACES

#### Leonid Checkov, Steklov Mathematical Institute

We propose the graph description of Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) of the Penner-Fock type. Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We can then describe both classical and quantum theories having the proper number of Thruston variables (graph-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. We enlarge the mapping class group allowing transformations that effectively push marked points from one boundary component to another and/or interchange marked points on one component. We describe the classical and quantum braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and discuss the possible relation to integrable systems of Frobenius type.

### EXTENDED AFFINE LIE ALGEBRAS

#### Erhard Neher, Ottawa University

Extended affine Lie algebras are generalizations of affine Lie algebras, which are in general not Kac-Moody algebras. Unlike arbitrary Kac-Moody algebras, extended affine Lie algebras have a nice realization, using toroidal Lie algebras and their generalization as a replacement of loop algebras. In this talk I will describe the general structure theory of extended affine Lie algebras.

### QUANTUM VERTEX ALGEBRAS

#### Iana Anguelova, CRM and Concordia University

Vertex algebras are characterized by a collection of fields which form in a sense a module-algebra (over the Hopf algebra generated by the infinitesimal translation) with singular commutative multiplication. If we replace the commutativity by braided commutativity, and also introduce a braiding in the action of the infinitesimal translation we obtain (one specific type of) quantum vertex algebras. We then discuss the connection between these quantum vertex algebras and the deformed chiral algebras defined by E. Frenkel and N. Reshetikhin. This is joint work with Maarten Bergvelt (UIUC).

### INVARIANT FOLIATIONS AND NILPOTENT LIE ALGEBRAIC INVARIANTS FOR THE POISSON BRACKETS OF HYDRODYNAMIC TYPE

#### Oleg Bogoyavlenskij, Queen’s University

An invariant foliation with an induced non-degenerate metric <v,w>of constant curvature K is discovered for any degenerate Poisson bracket of hydrodynamic type on a manifold with (2,0)-tensor of rank m < n. An invariant dynamical system V on is introduced that is tangent to the leaves of the foliation . The dynamical system V is applied for constructing the scalar and tensor invariants of the Poisson bracket. Invariant (n-m)-dimensional nilpotent Lie algebras are found that are embedded into the cotangent spaces .

### EXTENDED CHIRAL ALGEBRAS, GENERALISED COMMUTATION RELATIONS, AND FERMIONIC CHARACTERS

#### David Ridout, CRM, Université Laval

There has recently been much renewed interest in conformal field theories whose chiral algebras are defined by so-called generalised commutation relations. Often, examples of these algebras can be constructed from better known theories by extending the chiral algebra by the modes of a simple current. For example, the Virasoro algebra of the minimal models M(p’,p) can be so extended when p > p’ > 2, as can the affine Lie algebras of the SU(N) Wess-Zumino-Witten models.
Such extended algebras have not received as much attention in the literature because they are not Lie algebras: The generalised commutation relations which define the algebra structure involve an infinite number of terms. I will discuss some aspects of the representation theory of these extended algebras, in particular the reducibility of the « Verma modules » and bases generalising the Poincare-Birkhoff-Witt construction. Amazingly, the physical representations of the extended chiral algebra turn out to be faithful. This has numerous interesting ramifications, including combinatorial derivations of so-called fermionic character formulae.

### SOLVABLE CRITICAL DENSE POLYMERS

#### Jorgen Rasmussen, University of Melbourne

We discuss a model for critical dense polymers described in terms of the planar Temperley-Lieb algebra. The model involves a set of commuting transfer matrices satisfying an inversion identity which we solve exactly for finite sizes. The finite-size corrections and the associated selection rules are determined, allowing us to extract the bulk and boundary free energies as well as information on the conformal properties. The latter give rise to a CFT with c=-2 and spectrum corresponding to an infinitely extended Kac table. The physical combinatoris is also discussed.

### FUZZY HYPERBOLOIDS

#### Place : Concordia University, Library Building, 1400 Blvd. de Maisonneuve O., room LB-921-04 (Meeting room)

We first recall the main features of the (Madore) fuzzy sphere. A construction of the 2d and 4d fuzzy de Sitter hyperboloids is carried out by using a (vector) coherent state quantization. We get a natural discretization of the dS « time » axis based on the spectrum of Casimir operators of the respective maximal compact subgroups SO(2) and SO(4) of the de Sitter groups SO(1,2) and SO(1,4). The continuous limit at infinite spins is examined.

### SOME REDUCTION ASPECTS OF THE MANY-BODY PROBLEM

#### Place : Concordia University, Library Building, 1400 Blvd. de Maisonneuve O., room LB-921-04 (Meeting room)

We consider a system of N identical bosons which interact by pair potentials and obey the semirelativistic Salpeter equation. The Hamiltonian H = sqrt(-Delta + m^2) + V for a single particle is a non-local operator, but is otherwise well behaved mathematically. For example, the discrete spectrum can be characterized variationally, and the corresponding many-body problem is well defined. The non-individuality of identical particles greatly simplifies the analysis of this many-body problem because the necessary permutation symmetry of the allowed states is a powerful constrain on the motion of the particles : if you know what any two are doing, you know almost all.

### COMPUTATION OF INVARIANTS OF LIE ALGEBRAS BY MEANS OF MOVING FRAMES

#### Vyacheslav Boyko, Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., Kyiv-4, 01601 Ukraine

A new purely algebraic algorithm is presented for computation of invariants (generalized Casimir operators) of Lie algebras. It uses the Cartan’s method of moving frames and the knowledge of the group of inner automorphisms of each Lie algebra. The algorithm is applied, in particular, to computation of invariants of real low-dimensional Lie algebras and invariants of solvable Lie algebras of general dimension n<= ∞ restricted only by a required structure of the nilradical. A number of examples are calculated to illustrate its effectiveness and to make a comparison with the same cases in the literature.

### PERTURBATION DES ENSEMBLES DE MATRICES ALÉATOIRES AVEC SYMÉTRIES UNITAIRES

#### Patrick Desrosiers, Université Laval

La théorie des polynômes orthogonaux est un outil puissant pour l’étude des ensembles de matrices aléatoires dont la densité de probabilité est invariante par rapport aux transformations unitaires. Lorsqu’une source (champ extérieur) est ajoutée à de tels ensembles, la symétrie unitaire est brisée. Il faut alors introduire des polynômes qui satisfont des conditions d’orthogonalité par rapport à plusieurs mesures; i.e., des polynômes (biorthogonaux) multiples. Dans ce séminaire, j’expliquerai comment utiliser les polynômes multiples pour calculer les fonctions de corrélation lorsque les perturbations sont de rang fini. L’effet de cette « brisure douce«  de la symétrie unitaire sera décrit à l’aide de l’analyse asymptotique des polynômes multiples, laquelle fait apparaître des généralisations des fonctions de Bessel et de Airy. J’aborderai aussi la question de la distribution des valeurs propres extrêmes et de leur lien avec la perturbation des équations de Painlevé.

### GEOMETRIC QUANTIZATION OF ALGEBRAIC REDUCTION

#### Place : Concordia University, Library Building, 1400 Blvd. de Maisonneuve O., Room LB-921-04 (Meeting Room)

I shall discuss reduction of symmetries in Hamiltonian mechanics and quantization of the obtained reduced spaces (or their Poisson algebras). In special cases we obtain a generalization of results of Guillemin and Sternberg in commutativity of quantization and reduction valid for non-compact groups and non-compact symplectic manifolds. (Reference : axXiv : math. DG/0609727 v1).

### ON THE CONSTRUCTION OF (SPIN) CALOGERO MODELS BY HAMILTONIAN

#### Bela Gabor Pusztai (Concordia University and CRM)

The reductions of the free geodesics motion on a non-compact simple Lie group $G$ based on the $G_+ \times G_+$ symmetry given by left- and right-multiplications for a maximal compact subgroup $G_+ \subset G$ are investigated. At generic values of the momentum map this leads to (new) spin Calogero type models. At some special values the `spin’ degrees of freedom are absent and we obtain the standard $BC_n$ Sutherland model {\em with three independent coupling constants} from $SU(n+1,n)$ and from $SU(n,n)$. This is a joint work with Laszlo Feher.

### SPECTRA OF OBSERVABLE FOR THE q-OSCILLATOR AT q>1

#### Anatoliy Klimyk, Bogolyubov Institute for Theoretical Physics, Kiev, Ukraine

The position and momentum operators of the Biedenharn-Macfarlane q-oscillator (with the main relation ) are bounded operators when q<1 and their spectra are well-known. We study these operators when q>1. We show that these operators are symmetric but not self-adjoint. They have one-parameter families of self-adjoint extensions. These extensions are derived explicitly. Spectra of these extended operators and and the corresponding eigenfunctions are given explicitly. Spectra of different extensions are discrete and do not intersect. The results show that the creation and annihilation operators and a of the q-oscillator at q>1 cannot determine a physical system without further more precise definition. In order to determine a physical system we have to choose appropriate self-adjoint extensions of the position and momentum operators. A q-analogue of the Fourier transform, corresponding to the q-oscillator with q>1, is derived. Since spectra of the position and momentum operators are discrete, it is a unitary discrete transform, which is given by a unitary matrix.

### POISSON-LIE T-PLURALITY AS CANONICAL TRANSFORMATION

#### Libor Snobl (CRM & Czech Technical University in Prague)

We generalize the prescription realizing classical Poisson-Lie T-duality as canonical transformation to Poisson-Lie T-plurality. The key ingredient is the transformation of left-invariant fields under Poisson-Lie T-plurality. Explicit formulae realizing canonical transformation are presented and the preservation of canonical Poisson brackets and Hamiltonian density is shown.

### TOEPLITZ OPERATORS AND SEGAL BARGMANN ANALYSIS

#### Prof. M. Englis Czech Academy of Sciences, Prague

Toeplitz operators on the Bergman space of the disc behave very nicely under the group of Moebius maps (conformal automorphisms). In this talk we describe a general version of such group-invariant operator calculi, introduced and studied recently by Arazy and Upmeier. In particular, we discuss basic properties like boundedness or Schatten class membership of the resulting operators, and relate them to the localization operators of Gabor and Daubechies. The last item represents a generalization of a recent result of L. Coburn and M.-L. Lo.

For further information: GRUNDLAN@CRM.UMontreal.CA

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