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# Mathematical Physics Laboratory Seminars Fall 2004 – Winter 2005

### A SUPER-INTEGRABLE DISCRETIZATION OF THE CALOGERO MODEL

#### Hideaki Ujino, CRM

A time-discretization that preserves the super-integrability of the classical Calogero model is obtained by application of the integrable time-discretization of the harmonic oscillator to the projection method for the classical Calogero model with continuous time. In particular, the difference equations of motion, which provide an explicit scheme for time-integration, are explicitly presented for the two-body case. Numerical demonstrations exhibit that the scheme conserves all the(=3) conserved quantities of the (two-body) Calogero model with a precision of the machine epsilon times the number of iterations.

### FOUR GENERALIZATIONS OF THE COSINE TRANSFORM IN TWO DIMENSIONS AND THEIR DISCRETIZATION ON FINITE ABELIAN SUBGROUPS OF COMPACT SEMISIMPLE LIE GROUPS OF RANK TWO

#### Art Zaratsyan, CRM

We develop and decribe four variants of 2-dimensional generalizations of the cosine transform. Each variant is based on a compact semisimple Lie group of rank 2. The cosines are generalized as the corresponding orbit functions of the Lie group. An orbit function is the contribution to an irreducible character from one orbit of the Weyl group. An explicit description is provided for Fourier expansions into series of orbit functions of the four compact semisimple Lie groups of rank two. The main goal is a study and comparison of the four versions of such expansions when the orbit functions are replaced by their values on the grid given by an Abelian subgroup of elements of finite order in the Lie group.

### RANK ONE CONDITIONS, HIROTA BILINEAR DIFFERENCE EQUATIONS AND THE KP GENERATORS

#### Michael Gekhtman, Notre Dame University and CRM

The Hirota Bilinear Difference Equation is a 3-term relation that encodes the KP hierarchy and its reductions. I will discuss a simple geometric approach to the derivation of this equation based on natural maps between Grassmannians that leads to alternative choices of generators of the KP flows. If time permits, I will also discuss the role that 3-term relations play in the Poisson geometry of finite-dimensional Grassmannians.

### CLUSTER ALGEBRAS AND POISSON GEOMETRY OF REAL GRASSMANNIANS

#### Michael Gekhtman, Notre Dame University et CRM

I will use the Poisson structure on Grassmannians viewed as Poisson homogeneous spaces as an example that leads to the notion of cluster algebras recently introduced by Fomin and Zelevinsky. As an application the number of connected components in so-called refined Schubert cells in a real Grassmannian will be computed.

### ON A RAMANUJAN RELATION AND SYMMETRIC MONOPOLE

#### Victor Enolskii, Concordia University

We describe explicitly a family of trigonal curves satisfying to the constraints defining BPS-monopole. The family contains the known curve corresponding to the tetrahedral 3-monopole as well as a number of new ones. The moduli of the curves appeared to be algebraic numbers due to a Ramanujan hypergeometric identity.

### A FAMOUS INTEGRAL OVER THE UNITARY GROUP AND SOME APPLICATIONS TO RANDOM MATRICES

#### Marco Bertola, Concordia University

In this talk I’ll present a famous integral over the group of unitary matrices the Harish-Chandra, Itzikson, Zuber integral. I’ll review some interesting ways of calculating it that connect with Hamiltonian geometry and/or Character Theory and point out some extremely important applications to Random Matrix Theory.

### THE SEMICLASSICAL FOCUSING NONLINEAR SCHRODINGER EQUATION

#### S.Venakides, Duke University

The work presented is joint with Alex Tovbis (UCF) and Xin Zhou (Duke). The NLS equation describes solitonic transmission in fiber optic communication and is generically encountered in propagation through nonlinear media. One of its most important aspects is its modulational instability: regular wavetrains are unstable to modulation and break up to more complicated structures. The inverse scattering approach linearizes the NLS initial value problem by associating it with a linear 2 by 2 first order ODE eigenvalue problem, the Zakharov Shabhat (ZS) equation. The scattering information of ZS constitutes the input to a Riemann-Hilbert problem (RHP) that is to nonlinear integrable systems what the Fourier integral represetation is to the solution of a linear PDE. Solving it provides the solution to NLS together with the dependence of the eigenfunctions of ZS in the spectral parameter for any point in space time). The time evolution of the input is as simple as the evolution of the Fourier transform in a linear PDE. We will outline the asymptotic methods that lead to the following results: 1) Proof of existence and basic properties of the first breaking curve in space-time above which a phase teransition occurs and show that for pure radiation no further breaks occur. 2) Post-break structure of the solution. 3) Rigorous error estimate. 4) Rigorous asymptotics for the large time behavior of the system in the pure radiation case.

### MATRIX MODELS: FROM WDVV TO EFFECTIVE FIELD THEORY

#### Leonid Chekhov, Steklov Institute, Moscow and Perimeter Institute

Special, multisupport, solutions of matrix models, as was shown by Dijkgraaf and Vafa, describe vacuum solutuions of N=2 SUSY theories. We start with the demonstration that these solutions become Krichever–Whitham systems in the limit of large matrix size (planar limit). We prove that these solutions satisfy the associativity (WDVV) equations in the leading order of genus expansion (planar diagrams). This property, which turned out to be quite general (it was observed in the one-matrix and two-matrix models, as well as in the normal matrix model) is intrinsic for topological theories. Concentrating on one-matrix model, we derive the next-to-leading order correction (torus correction) and on its base discuss possibility to construct the effective field theory (the ground ring in the topological theory lexic).

### TWISTED WAKIMOTO REALIZATIONS OF AFFINE LIE ALGEBRAS

#### Bela Gabor Pusztai, CRM and University of Szeged (Hungary)

In a vertex algebraic framework we present an explicit description of the twisted Wakimoto realizations of the affine Lie algebras in correspondence with an arbitrary finite order automorphism and a compatible integral gradation of a complex simple Lie algebra. This yields generalized free field realizations of the twisted and untwisted affine Lie algebras in any gradation. This is a joint work with Laszlo Feher.

### FOURIER TRANSFORMS ON LIE GROUPS AND APPLICATIONS FOR IMAGING AND DATA PROCESSING

#### Armen Atoyan, CRM

We are working on development of a new method for Fourier-like transforms of discrete functions {G_i} sampled on uniform grids of rectangular and triangular/hexagonal symmetries, based on the orbit functions of compact Lie groups. Discrete orbit-function transform (OFT) allows a natural continuous extension (CE) which interpolates well the sampled function {G_i} between the grid points, and has valuable properties that make OFT very different from the standard DFT (discrete Fourier transform). The following are those properties of CEOFT that will be discussed and exemplified. (a) Similarily to DFT, also OFT is an exact discrete transform which returns the values of the original function on the points of the grid. But unlike the continuous extension of the standard DFT, the CEOFT does not converge to the original continuous function G(x) with the increasing number N of grid point. (b) For sufficiently large N, the first and second derivatives of CEDCT approximate the derivatives of G. (c) For CE DOT the « localization principle » known for canonical continuous Fourier transforms holds as well. Examples of application of CEOFT for imaging in case of 2D rectangular and hexagonal grids, which are realized in cases of SU(2)xSU(2) and SU(3) groups, respectively, will also be given.

### APPLICATION DE L’EXTENSION CONTINUE D’UNE TRANSFORMÉE DE TYPE FOURIER DANS L’ANALYSE DES DONNÉES RADAR

#### Mickael Germain

Les transformées de type Fourier sur les groupes de Lie semi-simples et compactes, (également notées TGD pour les transformées en groupe discrètes) apparaissent comme de nouveaux outils plus efficaces qu’une simple transformée de Fourier discrète (TFD) dans certaines applications de traitement d’images ou de données. L’objectif de notre projet est d’appliquer les TGD dans un contexte de détection et de reconnaissance de cibles sur les images Radar. Nous proposons d’utiliser l’extension continue sur la TGD du groupe SU(2)xSU(2), également connue sous le nom de la transformée en cosinus discret (TCD). Les propriétés de l’extension continue de la TCD (ECTCD) telles que la convergence, la localisation et la différentiabilité peuvent être utiles à la détection et la reconnaissance d’objets. Les applications en imagerie Radar sont réalisées en partenariat avec Lockheed Martin Canada, et intégrées dans leur logiciel d’expertise dans l’analyse des données de télédétection. Des tests ont été effectués sur les données du capteur Radar polarimétrique Convair-580. Les résultats mettent en valeur une meilleure classification des caractéristiques Radar avec l’utilisation de l’ECTCD. Dans ce contexte, la fusion des données impliquant de telles caractéristiques devrait apporter une amélioration dans l’identification et la reconnaissance de cibles.

### INTEGRATING ODE’S WITH SYMMETRY REVISITED

#### Mark Fels, Utah State University

In this talk I will look at the problem of integrating scalar ordinary differential equations with symmetry. This problem when written in terms of exterior differential systems (EDS) admits a simple geometric interpretation. The EDS approach will be compared with the reduction of order technique known to Lie. The emphasis will be on examples, including systems of ordinary differential equations.

### QUARTIC MANY-BODY ANHARMONIC OSCILLATORS

#### A.Turbiner, UNAM, Mexique

Anharmonic effects are very important in nature. Even small anharmonic terms lead to a conceptual difference from unperturbed problems. We introduce two quantum quartic anharmonic many-body oscillators. One of them is an anharmonic generalization of the celebrated Calogero model ($A_n$ rational model) which carries the same symmetry as the original Calogero model. Another model is the three-body Wolfes model ($G_2$ rational model) with additional quartic anharmonic interaction, which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by a variational method.

### SOME COMPUTATIONS INVOLVING THETA-FUNCTIONS

#### Chris Eilbeck, Heriot-Watt University, UK

I will discuss two topics involving the efficient evaluations of theta functions connected with algebraic curves. Both are of interest on both practical and theoretical grounds. One is the use of the Richelot transformation to evaluate genus two hyperelliptic integrals, a generalization of the Algebraic-Geometric Mean of Gauss. The other is the study of reducible period matrices, when the algebraic curve is a cover of one or more of lower genus. In this case the higher genus theta function can be written as a sum of products of lower genus (often g=1) theta functions.

### MULTISCALE ANALYSIS OF DISCRETE NONLINEAR EVOLUTION EQUATIONS: THE REDUCTION OF THE DNLS

#### D. Levi, Universita’ degli Studi Roma Tre and Sezione INFN, Roma Tre

We consider multiple lattices and functions defined on them. We introduce slowly varying conditions and define a multiscale analysis on the lattice. We apply the obtained results to the case of the multiscale expansion of the differential-difference Nonlinear Schroedinger equation.

### A SIMPLE WAY OF TURNING A HAMILTONIAN SYSTEM INTO A BI-HAMILTONIAN ONE

#### Artur Sergyeyev, Silesian University, Opava, Czech Republic

Given a Poisson structure (or, equivalently, a Hamiltonian operator) P, we show that its Lie derivative L_X(P) along a vector field X defines another Poisson structure, which is automatically compatible with P, if and only if [L_X^2(P),P]=0, where [,] is the Schouten bracket. We further prove that if a Poisson structure P on a finite-dimensional manifold M has dim ker P < 2 and P is of locally constant rank, then all Poisson structures compatible with P are locally of the form L_X(P), where X is a local vector field such that L_X^2(P)=L_Y(P) for some other local vector field Y. This leads to a remarkably simple construction of bi-Hamiltonian dynamical systems. We also present a generalization of these results to the infinite-dimensional case. In particular, we provide a new description for pencils of compatible local Hamiltonian operators of Dubrovin–Novikov type and associated bi-Hamiltonian systems of hydrodynamic type.

### MONOPOLES AND AN IDENTITY OF RAMANUJAN

#### Harry Braden, Edinburgh University, UK

Magnetic monopoles, or the topological soliton solutions of Yang-Mills-Higgs gauge theories in three space dimensions, have been objects of fascination for over a quarter of a century. BPS monopoles in particular have been the focus of much research. Many striking results are now known, yet, disappointingly, explicit solutions are rather few. We bring techniques from the study of finite dimensional integrable systems to bear upon the construction. The transcendental constraints of Hitchin may be replaced by (also transcendental) constraints on the period matrix. For a class of curves we show how to these may be reduced to a number theoretic problem. A recently proven result of Ramanujan enables us to solve these and construct the corresponding monopoles.

### SEMICLASSICAL ORTHOGONAL POLYNOMIALS, MATRIX MODELS, ISOMONODROMIC TAU FUNCTIONS, AND VIRASORO GENERATORS

#### John Harnad, Concordia University and CRM

Orthogonal polynomials with respect to arbitrary semiclassical measures supported on contours in the complex plane, together with « second type » solutions of the associated recursion relations satisfy compatible systems of deformation equations obtained from varying the measures. These preserve the generalized monodromy of an associated sequence of rank-2 rational covariant derivative operators. The corresponding isomonodromic tau functions are shown to coincide with the partition functions for matrix models consisting of unitarily diagonalizable matrices with spectra supported on these contours. The coefficients of the associated hyperelliptic spectral curves are uniquely determined by the application of certain Virasoro generators to the logarithm of the partition function (or, more generally, to the gap probablities). The isomonodromic deformation parameters and the logarithmic derivatives of the partition function with respect to these are given by simple residue formulae evaluated on the spectral curve. These derivatives are interpretable as nonatonomous Hamiltonians generating the deformations via the classical R-matrix Poisson structure.

### SUBGROUPS OF LIE GROUPS AND THE SEPARATION OF VARIABLES

#### Zora Thomova , SUNY, Utica, NY

Separable coordinate systems for Hamilton-Jacobi and Schrodinger equations are introduced in complex and real four-dimensional spaces. We consider maximal Abelian subgroups of E(n,C) and E(p,q) to generate coordinate systems with a maximal number of ignorable variables. The results are presented also graphically to illustrate the connections between subgroups and coordinates. The talk is based on a recent article with E.G. Kalnins and P.Winternitz.

### MAXWELL-BLOCH EQUATIONS AND CHAINS OF NEUMANN SYSTEMS

#### Pavle Saksida , University of Ljubljana, Slovenia

The Maxwell-Bloch equations are an integrable system of PDE’s which, roughly speaking, serve as a model for a laser. We will consider a certain discretization of this system with respect to the space variable, leaving the time continuous. We obtain a system of ODE’s which can be thought of as a chain of globally interacting Neumann systems. I shall describe a method which yields integrals of motion of this Neumann chain. This method is a modification of the zero-curvature condition. The existence of the integrals stems from the fact that a certain curvature takes values in a suitable subalgebra of the Lie algebra of the structure group, but this curvature is in general non-zero. I shall also mention a new Hamiltonian structure of the Maxwell-Bloch equations.

### ON THE CLASSIFICATION OF SOLVABLE LIE ALGEBRAS WITH GIVEN NILRADICALS

#### Libor Snobl, CRM

An approach to the classification of solvable Lie algebras starting from the knowledge of nilpotent algebras (i.e. their nilradicals) will be presented. This approach was developed in the classification of low dimensional Lie algebras by Mubarakzyanov, Turkowski and others. It can be used also in the classification of solvable Lie algebras of arbitrary dimension, as performed by Winternitz, Rubin, Ndogmo and Tremblay in the case of Heisenberg, Abelian and triangular nilradicals. The general procedure will be applied to the case of nilradicals with maximal degree of nilpotency. The classification of solvable Lie algebras possessing a nilradical with an (n-1) dimensional Abelian ideal will be presented. The generalized Casimir invariants of such solvable Lie algebras will be discussed.

### QUANTIZATION GENERATED BY GEODESICS AND MAGNETIC FIELD

#### Tom Osborne, University of Manitoba

A quantization of the cotangent bundle is called perfect if
– it is exact, rather than expressed via asymptotic series;
– it has an operator representation in a Hilbert space (corresponding to Schrodinger quantum mechanics);
– it is given explicitly and purely in symplectic terms.
A perfect quantization over a curved manifold M for a system of charged particles moving in an external electromagnetic field is constructed. An exact, coordinate invariant integral formula for the star product is found. The resulting symbol calculus is manifestly gauge invariant. The manner in which the Riemannian connection Gamma on M intertwines with the magnetic field F via the quantization process to produce a symplectic phase space connection is described.

### PROLONGATION STRUCTURE AND INTEGRABILITY OF THE COUPLED KdV-mKdV SYSTEM

#### Ismet Yurdusen, Middle East Technical University, Ankara et CRM

Recently, Kersten and Krasil’shchik constructed the recursion operator for a coupled KdV-mKdV system, which arises as the classical part of one of superextensions of the KdV equation. In this work, we study the integrability of this system using the Painleve test. Then, we use the Dodd-Fordy algorithm for the Wahlquist-Estabrook prolongation technique in order to obtain the Lax pair. We find a 3 x 3 matrix spectral problem for the Kersten and Krasil’shchik system. We also show that the Lax pair obtained is a true Lax pair since the spectral parameter cannot be removed by a gauge transformation, as can be proven by a gauge-invariant technique.

### ETATS LOCALISES DE CHAMPS BOSONIQUES

#### Stephan De Bièvre, Université de Lille, CRM et McGill

Je presenterai la generalisation d’un theoreme de Knight, qui affirme que les etats localises d’un champ de Klein-Gordon ne peuvent pas etre construits avec un nombre fini de particules. Je montrerai qu’un theoreme analogue est valable pour tout champ bosonique libre ou un enonce de ce type a un sens. Un lien avec le theoreme de Reeh-Schlieder sera aussi etabli.

### ENTANGLEMENT IN THE XY SPIN CHAIN

#### Alexander R. Its, IUPUI, Indianopolis

We consider the ground state of the XY model on an infinite chain at zero temperature. Following Bennett, Bernstein, Popescu, and Schumacher we use entropy of a sub-system as a measure of entanglement. Vidal, Latorre, Rico and Kitaev conjectured that the von Neumann entropy of a large block of neighboring spins approaches a constant as the size of the block increases. We evaluate this limiting entropy as a function of anisotropy and transverse magnetic field. We use novel methods based on integrable Fredholm operators and Riemann-Hilbert problems.

### QUANTIFICATION PAR DÉFORMATION PROJECTIVEMENT INVARIANTE

#### A.M. EL GRADECHI, Université d’Artois

Dans le cadre de la quantification par déformation, nous montrerons l’existence et l’unicité d’un produit * projectivement invariant sur le cotangeant de l’espace projectif.

### ON THE DISCRETE SPECTRA OF SOME BEREZIN-TOEPLITZ OPERATORS

#### S. Twareque Ali, Concordia

Berezin-Toeplitz operators arise in the theory of Berezin quantization as quantum counterparts of classical observables. Such operators also arise in signal analysis, where they are called localization operators. A useful class of such operators, both for quantization theory and for signal analysis, is those which possess completely discrete spectra. There exist specific results in the literature on when this situation prevails. In this talk we shall formulate a number of fairly general criteria which guarantee the existence of discrete spectra. Our results subsume all the earlier results and hold for a significantly wider class of systems.

### SUR LES SURFACES PLONGÉES DANS LES ALGÈBRES su(N+1) ASSOCIÉES AUX MODÈLES CP^N

#### A.M.Grundland, UQTR et CRM

Nous étudions certains aspects géométriques des surfaces orientables à deux dimensions provenant de l’étude des modèles sigma CP^N. Pour cela, nous utilisons une identification de l’espace R^N(N+2) avec l’algèbre de Lie su(N+1) pour construire une formule généralisée de Weierstrass pour l’immersion des surfaces. Les éléments structurels de la surface tels que son repère mobile et ses équations de Gauss-Weingarten et de Gauss-Codazzi-Ricci sont exprimés comme fonctions de la solution du modèle CP^N qui les définissent. De la même facon, les premières et secondes formes fondamentales, ainsi que la courbure gaussienne, le vecteur de courbure moyenne, la fonctionnelle de Willmore, et la charge topologique des surfaces sont exprimées en termes de cette solution. Nous présentons une implémentation detaillée de ces résultats pour les surfaces plongées dans les algèbres de Lie su(2) et su(3).

### DECAY OF SCALAR WAVES IN KERR GEOMETRY

#### Niky Kamran, McGill

We show that the solutions of the scalar wave equation in the Kerr geometry of a rotating black hole in equilibrium, corresponding to Cauchy data compactly supported outside the event horizon tend to zero in $L^{\infty}_{loc}$ as $t$ tends to infinity. This is joint work with Felix Finster, Joel Smoller and Shing-Tung Yau.

### RÉDUCTION DE L’INTÉRACTION DE LA RÉSONANCE DES TROIS ONDES À LA SIXIÈME ÉQUATION GÉNÉRIQUE DE PAINLEVÉ

#### Robert Conte (CEA-Saclay)

Parmi les réductions du système d’intéraction de résonance des trois ondes à des systèmes différentiels à six dimensions, une a spécifiquement été reliée à la sixième équation générique de Painlevé. Nous présentons explicitement ce lien, et nous fournissons une nouvelle pair de Lax pour la transcendante P6 de Painlevé sans aucune restriction sur les exposants de monodromie.

### Tensors on Hilbert Spaces of Generalized Functions and the Problem of Emergence of Classical Space

#### Alexei Krioukov (Wisconsin)

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the space R^n, Hilbert spaces admit various useful realizations as spaces of functions. In the talk this observation will be used to construct a fruitful formalism of local coordinates on Hilbert manifolds. Images of charts on manifolds in the formalism are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations describe then families of functional equations on various spaces of functions. After a brief review of applications of the formalism in linear algebra, theory of generalized functions and differential equations, the significance of the formalism in quantum theory, in particular, in describing the emergence of classical space-time will be discussed.

### Less (precision) is more (information): quantum probability in classical embeddings

#### Paul Busch (Hull)

This is a review of the statistical models approach to quantum mechanics. This approach enables a systematic account of various representations of quantum mechanics, classical or otherwise. It will be shown how these representations highlight the irreducible element of uncertainty and unsharpness characteristic of quantum mechanics. Rather than being merely a limitation to the precision of observation, quantum unsharpness is increasingly found to constitute a resource for enhanced information processing.

#### Responsable: Pavel Winternitz (wintern@CRM.UMontreal.CA)

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