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# Mathematical Physics Laboratory Seminars Fall 2003 – Winter 2004

### RECURRENCE COEFFICIENTS FOR THE GENERALIZED CHARLIER SEMI-CLASSICAL ORTHOGONAL POLYNOMIALS

#### Norbert Hounkonnou, IMSP, Universite de Abomey-Calavi, Benin

Generalized Charlier polynomials are introduced as semi-classical orthogonal polynomials of class 1 with one parameter, satisfying a system of non-linear recurrence relations. Instead of the previous known results, we give, in a rigorous way, a pleasant simplification of these relations. Then, we provide an algorithm to compute the recurrence coefficients and derive their asymptotics.

### AFFINE LIE ALGEBRAS, THEIR WEYL GROUPS, GRIDS AND COLOURED TABLEAUX

#### R. King, University of Southampton

Each affine Lie algebra has associated with it an affine Weyl group generated by reflections. It is shown that the action of any Weyl group element on any weight vector may be codified by means of a two dimensional periodic grid. This action produces coloured tableaux. A subset is identified of those particular Weyl group elements that have a special role in determining characters of irreducible representations of the affine Lie algebra. The relevant contributions to the character formula are shown to be defined rather simply by coloured tableaux consisting of certain cores supplemented with sequences of boundary strips of a particular length.

### PARTIALLY INVARIANT SOLUTIONS OF MODELS OBTAINED FROM THE NAMBU-GOTO ACTION

#### Alexander Hariton, CRM and Department of Mathematics, Université de Montréal

The concept of partially invariant solutions is discussed in the framework of the group analysis of models derived from the Nambu-Goto action. In particular, we consider the non-relativistic Chaplygin gas and the relativistic Born-Infeld theory for a scalar field. Using a systematic approach based on subgroup classification methods, several classes of partially invariant solutions with defect structure $\delta =1$ are constructed.

### THE STOCHASTIC LIMIT IN THE ANALYSIS OF THE OPEN BCS MODEL

#### Fabio Bagarello, Universita di Palermo, Palermo, Italy

We show how the perturbative procedure known as stochastic limit may be useful in the analysis of the open BCS model first introduced by Buffet and Martin as a spin system interacting with a fermionic reservoir. In particular we show how the same values of the critical temperature and of the order parameters can be found with a significantly simpler approach. We also discuss some possibile generalization of this model and the consequences on the critical temperature.

### TWO-PARAMETRIC DIFFERENCE DEFORMATIONS OF DARBOUX-POSCHL-TELLER MODEL

#### Vladimir Matveev, Universite de Bourgogne

We start by describing the classical case of integrability of the Sturm-Liouville equation found independently by Darboux in 1982 and 50 years later by Poschl and Teller. We describe the isospectral deformations of the DPT model corresponding to the KdV hierarchy. Next, we consider two difference linear integrable models . Both models represent (non isospectral) deformations of the classical Darboux-Poschl-Teller equation . One of these deformations corresponds to a special case of the L-operator of the functional-difference KdV equation and another one to a special case of the L-operator of the functional-difference Toda equation. Both models are labeled by two integers and a continuous deformation parameter h ( the difference step). When one of the integers is equal to zero, the first of those models reduces to the case studied recently by Ruijsenaars, and by van Diejen and Kirillov . In a more special form of the lattice, when x=n, with n integer and h=1, the 1-parametric reduction of the same model was also studied by Zhedanov and Spiridonov. The second model also generalizes in a natural sense another 1-parametric lattice model proposed by Zhedanov and Spiridonov : again their case is recovered by setting one of the integers equal to zero, the difference step equal to one and x=n . We construct the general solution for both models following the original strategy of Darboux for the DPT case. When the difference step tends to zero we recover the known results for the DPT model by a simple limiting procedure. The presented results summarize recent work by P. Gaillard and the speaker.

### POISSON-LIE T-PLURALITY AND 3-DIMENSIONAL CONFORMALLY INVARIANT SIGMA MODELS

#### Libor Snobl, CRM

Starting from the classification of 6-dimensional Drinfeld doubles and their decomposition into Manin triples we construct 3-dimensional Poisson-Lie T-dual, or more precisely Poisson-Lie T-plural models. Of special interest are those that are conformally invariant on the 1-loop level, i.e. satisfy vanishing beta function equations. Examples of such models will be presented.

### A NEW CLASS OF PAINLEVE TRANSCENDENTS: MONODROMY-SOLVABLE SOLUTIONS

#### Yousuke Ohyama, Osaka University, Osaka, Japan

The Painleve equations can be considered as determining isomonodromic deformations of linear differential equations. In general, the monodromy data are transcendental functions of the coefficients of the linear equations. But in some cases, we can calculate the monodromy data exactly. In this case, the correponding Painleve function is called « monodromy- solvable ». We will show that some known special solutions, algebraic or of the Riccati type, are monodromy-solvable. Moreover we will see examples of new monodromy-solvable solutions which are not classical solutions in the sense of Umemura.

### RIEMANN-HILBERT PROBLEM APPROACH TO A « PHYSICAL » SOLUTION OF THE PAINLEVE FIRST EQUATION

#### Andrei Kapaev (Research Center of Innovation and Developments, St. Petersburg,)

Using the Riemann-Hilbert approach, the $\Psi$-function corresponding to the solution $y\sim\sqrt{-x/6}$ as $|x|\to\infty$ of the first Painleve equation, $y »=6y^2+x$, is explicitly constructed. In this way, the exponentially small jump in the dominant solution is described precisely, and the coefficient asymptotics in its power-like expansion are found.

### CATEGORICAL RELATIVITY

#### Zbigniew Oziewicz

Categorical relativity is based on a new concept of velocity that is a morphism in a groupoid category of observers. The addition of velocities-morphisms is associative. Albert Einstein in 1905 made the reciprocity assumption that the inverse of relativistic addition of velocities must be the same as for the Galilean absolute time. The Einstein reciprocity axiom, that the mutual speeds of two frames differ by sign solely, contradicts the relativity of time, implies that such velocities must be preferred-laboratory-dependent, and that the addition of such velocities must be non associative. The non associativity of the Einstein addition of velocities was discovered by Abraham Ungar in 1989.

### SELF-DUAL METRICS AND THE PAINLEVE EQUATIONS

#### S. Okumura, Osaka University, Japan

We classify the SU(2)-invariant anti-self-dual metrics with signature (+,+,-,-), which are NOT diagonalizable. Such metrics are determined by solutions of the Painleve equations of the types VI, V, III, or II. We will explain the geometric meaning of the metrics specified by each type of Painlevé function. This is a generalization of the work of Hitchin and Dancer about diagonalizable metrics.

### CONSTRAINED REDUCTIONS OF 2D DISPERSIONLESS TODA HIERARCHY, HAMILTONIAN STRUCTURE AND INTERFACE DYNAMICS

#### Oksana Yermolaeva , CRM

Finite-dimensional reductions of the 2D dispersionless Toda hierarchy, constrained by the »string equation » are studied. These include solutions determined by polynomial, rational or logarithmic functions, which are of interest in relation to the »Laplacian growth » problem governing interface dynamics. The consistency of such reductions is proved, and the Hamiltonian structure of the reduced dynamics is derived. The Poisson structure of the rationally reduced dispersionless Toda hierarchy is also derived.

### REALIZATIONS OF LOW-DIMENSIONAL LIE ALGEBRAS

#### Roman Popovych, Institute of Mathematics, Kiev

Using a technique based on the notion of a megaideal, we construct a complete set of inequivalent realizations of Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject.

### DEVELOPPEMENT N GRAND POUR LES MODELES DE MATRICES ET GEOMETRIE

#### Bertrand Eynard, CEN, Saclay, France

Les developpements a N grand de l’energie libre des modeles de matrices sont des series generatrice comptant les surfaces polygonales de genre fixe. L’energie libre des modeles de matrices est egalement une fonction tau. A l’ordre dominant en N, l’energie libre est reliee a une courbe algebrique. L’ordre suivant a d’abbord ete calcule si la courbe algebrique est de genre 1 OU 2, et des progres recents ont permis de le calculer pour des courbes de tous genres.

### REAL DOUBLES OF HURWITZ FROBENIUS MANIFOLDS

#### V.Shramchenko, Concordia

New family of flat potential (Darboux-Egoroff) metrics on the Hurwitz spaces and corresponding Frobenius structures are found. We consider a Hurwitz space as a real manifold. Therefore the number of coordinates is twice as big as the number of coordinates used in the construction of Frobenius manifolds on Hurwitz spaces found by B.Dubrovin more than 10 years ago. The branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed Frobenius manifolds. We introduce a new family of Darboux-Egoroff metrics in terms of the Schiffer and Bergmann kernels, find corresponding flat coordinates and prepotentials and G-functions of associated Frobenius manifolds.

### CHARACTER EXPANSIONS OF MATRIX INTEGRALS

#### John Harnad, Concordia et CRM

Character expansions play an important role in determining the large N asymptotics of matrix integrals, spectral densities, etc. A simple case is the 2-matrix partition function, and variants, such as the normal matrix one, which admit a Shur function expansion that generally is divergent, except for special choices of deformation parameters in the measures. For other values, the integral may still be interpreted as a Borel sum of the divergent series. Various methods for deriving the Schur function expansion – including a simple application of the Cauchy-Lttlewood identity, and free field fermionic constructions. (Based on joint work with A. Orlov)

### SPECTRAL STABILITY OF LOCAL DEFORMATIONS OF AN ELASTIC ROD: HAMILTONIAN FORMALISM

#### S. Lafortune, University of Arizona

Hamiltonian methods are used to obtain a necessary and sufficient condition for the spectral stability of pulse solutions to two coupled nonlinear Klein-Gordon equations. These equations describe the near-threshold dynamics of an elastic rod with circular cross-section. The present work completes and extends a recent analysis of the authors’ (Physica D 182, 103-124 (2003)), in which a sufficient condition for the instability of particular cases of these pulses was found by means of Evans function techniques. This work was done in collaboration with J. Lega from the University of Arizona.

### SUBSTITUTION POINT SETS FOR PURE POINT DIFFRACTION

#### Jeongyup Lee, University of Alberta, Edmonton

In the mid 1980’s, there was the first discovery of a real metallic alloy (quasicrystal), which showed pure bright peaks in its physical diffraction pattern with forbidden rotational symmetries in crystals. This discovery called for a mathematical interpretation of this unknown atomic structure. Interpreting atoms as points, we are interested in understanding the geometric structure of discrete point sets which have the diffraction spectrum consisting only of pure bright peaks (pure point diffraction spectrum). Substitution is an easy way to generate point sets and tilings with non-periodic structure. We connect substitution point sets and substitution tilings in such a way as to respect the substitution. We will discuss when a pure point diffraction spectrum is equivalent to a pure point dynamical spectrum. Then we will give equivalent criteria for discrete point sets to have pure point dynamical spectrum in substitution. In the case that the point sets are on a lattice, we have an easy way to check if they have a pure point dynamical spectrum. This checking method is parallel to Dekking’s coincidence criterion. This says that point sets coming from an equal-length substitution sequence satisfy coincidence if and only if they have a pure point dynamical spectrum. Our method is more general in the sense that it works for an arbitrary dimension n.

### PARTITION FUNCTION OF MATRIX MODELS AND ISOMONODROMIC TAU FUNCTIONS

#### Marco Bertola, Concordia, Montreal

It is the aim of this talk to elucidate the precise relation between the partition function of Hermitean matrix models and the Isomonodromic tau function of an associated linear system. The (recent) result of ours is the proof that the two objects are in fact (almost) the same. The result can be formulated in a more general setting of normal matrices with spectra on (homology classes of) contours in the complex plane.

### POLYNOMIAL WAVELET-TYPE EXPANSIONS ON THE SPHERE

#### A.A. Hemmat, Vali-Asr University of Rafsanjan, Iran et McMaster University, Hamilton

We present a polynomial wavelet-type system on S^d such that any continuous function can be expanded with respect to these « wavelets. » The order of the growth of the degrees of the the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a « dual wavelet system. » The » dual wavelets system » is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulae for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.

### CONTINUOUS AND DISCRETE HOMOTOPY OPERATORS WITH APPLICATIONS IN INTEGRABILITY TESTING

#### Willy Hereman, Department of Mathematical and Computer Sciences Colorado School of Mines, Golden, CO

The continuous and discrete homotopy operators will be presented together with their application in integrability testing of nonlinear PDEs and semi-discrete lattices. The continuous homotopy operator is a little known, yet powerful tool for integration by parts on the jet space. The discrete analogue allows one to invert the total difference operator. As an application, we will show how homotopy operators can be used to compute fluxes corresponding to nonlinear PDEs and differential-difference equations. We implemented the homotopy operators in Mathematica. A demonstration of the symbolic code will be given.

### A LAPLACE OPERATOR AND HARMONICS ON QUANTUM VECTOR SPACE

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

A Laplace operator on the quantum vector space (on which the quantum unitary group acts) is introduced and studied. Harmonic polynomials on this space are defined: they are solutions of the corresponding Laplace equation. It is proved that (similar to the classical case) a restriction of these polynomials onto the quantum sphere uniquely determines these polynomials. Zonal and asociated spherical functions are constructed. A q-analogue of separation of variables is given.

### DYNAMICAL r-MATRICES AND CALOGERO-MOSER MODELS

#### Bela Gabor Pusztai, CRM

In a joint work with L. Feher we further developed the construction of dynamical r-matrices building mainly on the seminal paper of Etingof and Varchenko. We managed to find generalizations of Felder’s dynamical r-matrices. An interesting problem is to find applications of the generalizations of Felder’s r-matrices in integrable systems. In this respect, it appears promising to seek for generalized Calogero-Moser systems, since certain spin Calogero-Moser systems are known to be closely related to Felder’s r-matrices.

### LAPLACE TRANSFORMATIONS AND SPECTRAL THEORY OF PERIODIC TWO-DIMENSIONAL SEMI-DISCRETE AND DISCRETE SCHROEDINGER OPERATORS

#### Alexei V. Penskoi, CRM

We introduce Laplace transformations of 2D semi-discrete Schroedinger operators and show their relation to a semi-discrete 2D Toda lattice. We develop an algebro-geometric spectral theory of 2D semi-discrete Schroedinger operators and solve the direct spectral problem for 2D discrete Schroedinger operators. (The inverse problem for discrete operators was already solved by Krichever). Using the spectral theory we investigate spectral properties of Laplace transformations of these operators. (Joint work with Alexei A. Oblomkov, MIT)

### SOLITONS AND PARTICLE SYSTEMS VIA ALGEBRAIC SURFACES

#### Thomas A. Nevins, University of Michigan

A puzzling discovery in the theory of integrable systems is that the motion of poles of meromorphic solutions of systems of nonlinear PDE (such as the KP hierarchy) is often governed by many-body systems (such as the Calogero-Moser system). I will present a geometric explanation of this phenomenon (joint work with D. Ben-Zvi) using (noncommutative) algebraic geometry. More precisely, we realize the KP hierarchy as a collection of flows on « the Hilbert scheme of points on a noncommutative surface, » and a geometric Fourier transform then identifies this Hilbert scheme with the phase space of the CM system.

### GAP SOLITON INTERACTION IN A ONE-DIMENSIONAL PHOTONIC CRYSTAL

#### Igor V. Melnikov, University of Toronto

We consider the stability of Bragg solitons in a resonant photonic crystal composed of alternating layers of dielectric and two-level atoms. The perturbation of general form motivates a new analysis of the bi-directional Maxwell-Bloch equations. Such forms of effective steering of the slow soliton as reflection, trapping, and inelastic interaction can be approached by means of excitation of weak linear modes of the field inside the crystal or by incoherent pump of its small length.

### AN EXTENDED VARIATIONAL PRINCIPLE FOR THE SHERRINGTON-KIRKPATRICK MODEL

#### Shannon Lee Starr (CRM)

There has been a lot of recent progress in understanding the Sherrington-Kirkpatrick, mean field, spin glass model. Notably are Guerra and Toninelli’s proof of the existence of the thermodynamic limit for all temperatures, and Guerra’s proof that Parisi’s ansatz gives a one-sided bound for the true free energy. I will review these results and describe the extension of these ideas by Aizenman, Sims and myself, which results in a variational principle for the free energy density in the thermodynamic limit.

### SUR LES SOLUTIONS CONDITIONNELLEMENT INVARIANTES DES EQUATIONS DE LA MAGNETOHYDRODYNAMIQUE

#### Philippe Picard, CRM et Département de physique, Université de Montréal

Dans cet exposé, je présenterai une nouvelle version de la méthode des symétries conditionnelles afin d’obtenir des ondes multiples qui s’expriment en termes des invariants de Riemann. Pour cela, on construit une distribution abelienne de champs vectoriels qui sont des symétries du système d’EDP étudié et qui sont sujet a certaines contraintes différentielles du premier ordre. Ainsi, je demontrerai la pertinence de nouvelle approche en présentant les ondes simples et doubles qui sont décrites par le systeme des équations de la magnétohydrodynamique. Egalement, j’en ferai une comparaison avec la méthode générale des caracteristiques appliquée au système MHD. è

### TAU FUNCTION OF ALGEBRAIC AND ANALYTIC CURVES, FREE ENERGY OF RANDOM MATRIX MODELS AND DISPERSIONLESS TODA HIERARCHY : PART 2

#### Marco Bertola, Concordia University, Montreal

In this talk we present recent development on the explicit representation of the free energy of the two-matrix model in the so called planar limit. Such functional naturally lives on the moduli space of algebraic curves with certain additional structure and has links to classical structures in complex geometry (Bergmann kernel and prime forms).

We also illustrate the connections of this function with the 2-dimensional electrostatic energy of a density of charge on a domain in the plane, which is called « the tau function of an analytic curve », the curve here being the boundary of the domain. This second setting (which is a proper subset of the previous one) is the object of active research because of its connections to certain problems in oil extraction. Time permitting we link both setting to the theory of the dispersionless Toda hierarchy.

### SOME RIGOROUS RESULTS ON THE SHERRINGTON KIRKPATRICK SPIN GLASS MODEL

#### Shannon Starr, CRM/McGill

Abstract

The goal of this minicourse will be to outline some recent progress in the analysis of the Sherrington Kirkpatrick mean field, spin glass model. This is primarily work of

1. Guerra and Toninelli,

2. Guerra himself,

3. Aizenman, Sims and Starr

In the first two weeks we will introduce the problem. We will describe the physical motivation of the SK model, some initial failed attempts by physicists to analyze the model, and Parisi’s amazing, but highly heuristic exact solution by the replica symmetry breaking. We will also compare to the deterministic analogue, which is the Curie-Weiss, mean field spin system. The CW model really is exactly solvable as can be proved using a large deviation principle, and is a typical first example in a basic statistical mechanics class. The SK model is harder.

In the second portion of class, we will describe the work of Guerra and Toninelli to prove the existence of a thermodynamic limit at all temperatures; the work of Guerra to prove that the Parisi-ansatz free energy is a rigorous lower bound; and the work of Aizenman, Sims and myself to obtain an extended variational principle for the free energy density in the thermodynamic limit. These works all use one trick, which is « quadratic interpolation ».

We note that the problem of proving Parisi’s ansatz is not solved by these techniques. Only certain parts of the problem.

In the last part of class we will describe Ruelle’s random probability cascades, and show how they can be used to derive Parisi’s ansatz as a variational bound, starting from the extended variational principle (thus rederiving 2). A less detailed and more heuristic version of this second derivation is included in the nice book of Mezard, Parisi and Viraso, « Spin Glass Theory and Beyond ». But the mathematically rigorous version does not seem to be well known. It remains a mystery (to me) to understand how Parisi came up with his first derivation. I believe there must be a good answer and it probably leads to new math!

### TAU FUNCTION OF ALGEBRAIC AND ANALYTIC CURVES, FREE ENERGY OF RANDOM MATRIX MODELS AND DISPERSIONLESS TODA HIERARCHY : PART 1

#### Marco Bertola, Concordia University, Montreal

In this talk we present recent development on the explicit representation of the free energy of the two-matrix model in the so called planar limit. Such functional naturally lives on the moduli space of algebraic curves with certain additional structure and has links to classical structures in complex geometry (Bergmann kernel and prime forms).

We also illustrate the connections of this function with the 2-dimensional electrostatic energy of a density of charge on a domain in the plane, which is called « the tau function of an analytic curve », the curve here being the boundary of the domain. This second setting (which is a proper subset of the previous one) is the object of active research because of its connections to certain problems in oil extraction. Time permitting we link both setting to the theory of the dispersionless Toda hierarchy.

### DEFORMATION QUANTIZATION AND OPERATOR CALCULI

#### M. Englis, Mathematical Institute, Prague

We use methods of harmonic analysis on symmetric spaces to deduce some elementary results about bilinear differential operators, and then apply these, in combination with a new generalization of a classical theorem of Borel, to obtain a result from deformation quantization.

### DISCRETE DIFFERENTIAL GEOMETRY: SURFACES MADE FROM CIRCLES

#### Prof. Alexander Bobenko (Technische Universitat, Berlin)

Discrete differential geometry aims at the development of discrete equivalents of the geometric notions and methods of differential geometry. The latter appears then as a limit of refinements of the discretization. We discuss special classes of discrete surfaces (such as minimal, constant curvature, isothermic etc.) and show how they can be constructed from circle patterns using variational principles and methods from the theory of integrable systems.

Department of Mathematics and Statistics, Concordia University and CRM, Univ. de Montreal

(514) 343-2491 or (514) 848-3242

### MULTIMODEL REGIONAL ENSEMBLE FORECASTING SYSTEM. BASIC CONCEPTS AND PRELIMINARY RESULTS

#### Roman Zelazny, Institute of Plasma Physics, Warsaw, Poland

Responsable: Pavel Winternitz (wintern@crm.umontreal.ca, (514) 343-7271)

### LIE SYMMETRIES, LAGRANGIANS, CONSERVATION LAWS AND SOLUTIONS OF DIFFERENCE EQUATIONS

#### P.Winternitz, CRM et DMS, Universite de Montreal

Abstract: The main part of the lecture will be devoted to Lie point symmetries of difference schemes involving one dependent and one independent variable. The symmetries will act on the equation and on the lattice. We will show that if the lattice is suitably adapted, difference schemes will have essentially the same symmetries as differential equations obtained as their continuous limits. We will show how symmetries can be used to classify difference schemes, to decrease the order of the scheme, and to obtain exact solutions. Variational symmetries are particularly useful in this context. Consequences for numerical analysis will be discussed, as well as generalizations to multivariable difference schemes. Finally we will show that if we wish to obtain interesting symmetry results for difference equations on fixed lattices, we must go beyond the concept of point symmetries.