FALL 2005 – WINTER 2006

#### Tuesday, May 30 at 3:30 p.m.

### LIE ALGEBRAS WITH STRONG DEGENERATION

Maribel Tocon, Ottawa University

#### Tuesday, April 25 at 3:30 p.m.

### ALGEBRAS OF UNBOUNDED OPERATORS AND PHYSICAL APPLICATIONS : A SURVEY

Fabio Bagarello, Palermo, Italie

We review some aspects of the algebraic approach to quantum mechanics for systems with infinite degrees of freedom. Our attention is focused on physical applications and to the role of algebras of unbounded operators in this game.

#### Tuesday, April 18 at 3:30 p.m.

### INTEGRABLE AND SUPERINTEGRABLE SYSTEMS WITH SPIN

Ismet Yurdusen, CRM

In this work we study the integrability and superintegrability of the Hamiltonian systems involving particles with spin. More specifically we consider two non-relativistic quantum particles, moving in a plane, one with spin 1/2, the other with spin 0.

#### Tuesday, April 4 at 3:30 p.m.

### TOPOLOGICAL SOLITONS IN INHOMOGENEOUS MEDIA

Wojtek J. Zakrzewski, (Department of Mathematical Sciences, University of Durham, UK)

We give a short introduction to the theory of topological solitons (in 1-3 dimensions) and then discuss their properties when they propagate in inhomogeneous media. In this discussion we concentrate our attention on solitons in (2+1) dimensions and on Sine-Gordon kinks in (1+1) dimensions.

#### Tuesday, March 28 2006 at 3:30 p.m.

### ON DIFFERENTIAL AND INTEGRAL IDENTITIES FOR SPHERICAL HARMONICS AND FOURIER TRANSFORMS WITH APPLICATIONS TO SYMMETRY REDUCTION OF QUANTUM SYSTEMS

Aleksander Strasburger, (Warsaw University of Agriculture)

A number of results concerning spherical harmonics may be stated in the form of an identity concerning all polynomials of a given class. Examples include a famous canonical decomposition of homogeneous polynomials into spherical harmonics of decreasing orders, Bochner-Hecke identities for the Fourier transform of spherically symmetric functions and many more. In the talk a unified approach to such identities is presented, based on group representations theoretical principles, together with some new results and proofs. We also discuss the problem of symmetry reduction of quantum mechanical systems in the perspective offered by the above approach.

#### Tuesday, March 21 2006 at 3:30 p.m.

### CONTRACTION OF LIE ALGEBRAS INVARIANTS. DEPENDANCE PROBLEMS.

Rutwig Campoamor-Stursberg (UCM, Madrid)

The objective of this talk is to present problems from generalized Inönü-Wigner theory of contractions together with the invariants obtained by the coadjoint representation. We will present criterions for the analysis Casimir invariants dependance/independance and their generalization with respect to contractions, motivated by the Gel’fand method applied to nonhomogeneous groups.

#### Tuesday, March 14 at 3:30 p.m.

### RANK-K SOLUTIONS OF HYPERBOLIC SYSTEMS OF FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS IN MANY DIMENSIONS

Michel Grundland (UQTR, CRM)

In this talk we employ a “direct method” in order to obtain rank-k solutions of any hyperbolic system of first order quasilinear differential equations in many dimensions. We discuss in detail the necessary and sufficient conditions for existence of these type of solutions written in terms of Riemann invariants. The most important characteristic of this approach is the introduction of specific first order side conditions consistent with the original system of PDEs, leading to a generalization of the Riemann invariant method of solving multi-dimensional systems of PDEs. We have demonstrated the usefulness of our approach through several examples of hydrodynamic type systems; new classes of solutions have been obtained in a closed form.

#### Tuesday, March 7 at 3:30 p.m.

### INEQUIVALENT QUANTIZATIONS OF THE THREE-PARTICLE CALOGERO MODEL

Laszlo Feher (KFKI RMKI Budapest et Univ. Szeged)

We study the inequivalent quantizations of the 3-body Calogero model with harmonic and inverse square pair potential that result by allowing all possible self-adjoint local boundary conditions for the angular and radial Hamiltonians in solving the Schrodinger equation by separation of variables. The inverse square coupling constant is taken to be 2v(v-1) with 1/2 v 3/2 and then the angular Hamiltonian is shown to admit a 2-parameter family of inequivalent quantizations compatible with the dihedral D(6) symmetry of its potential term. We describe the qualitative features of the angular eigenvalues and classify the eigenstates under the D(6) symmetry in all cases, and also analyze all self-adjoint versions of the radial Hamiltonian that arise for specific angular eigenvalues. The particles can pass through each other in the generic case of the boundary condition, and in some cases the energy is not bounded from below, even for a repulsive inverse square potential. There are 4 explicitly solvable cases, one reproduces Calogero’s solution and another one smoothly tends to the harmonic oscillator as the inverse square coupling vanishes. The talk is based mainly on joint work with T. Fulop and I. Tsutsui.( Nucl. Phys. B 715 (2005) 713-757).

#### Tuesday, February 28 at 3:30 p.m.

### ISOMODROMIC EQUATIONS, BIORTHOGONAL POLYNOMIALS, INTEGRAL REPRESENTATIONS AND DUALITY

John Harnad (Concordia, CRM)

For paired sequences of polynomials that are biorthogonal with respect to measures determined by polynomial potentials, integral representations of the fundamental systems satisfying the recursion and differential equations will be derived. These will be shown to be bilinearly paired with integral representations of the fundamental systems associated to the dual sequence of Fourier-Laplace transforms of the polynomials. They satisify differential systems which determine the spectral curve and deformation equations that determine the partition function of coupled random 2-matrix models. The integral representations lead to a Riemann-Hilbert characterization determining the biorthogonal sequences, which may be used as the starting point in large N asymptotic analysis. These integral representations will be related to another one, recently proposed by Kujlaars and Mclaughlin, based on multi-orthogonal polynomials, in which the jump matrices are nonconstant functions.

#### Tuesday, February 21 2006 at 3:30 p.m.

### WHAT IS A DYNAMICAL R-MATRIX

Axel Winterhalder, Universidade Estadual do Maranhao, Brazil

The concept of an R-Matrix has its origins in the theory of integrable systems and found an adequate mathematical interpret$ recent results about an extension of this concept, the so called dynamical R-matrix discussing it in the context of one of the most traditional classes of integrable models in mechanics – the Calogero models.

#### Tuesday, February 14 2006 at 3:30 p.m.

### TWO DIMENSIONAL TOPOLOGICAL GRAVITY AND VOLUMES OF MODULI SPACES OF CURVES

Peter Zograf, Steklov Institute, St-Petersbourg, Russie

The famous Witten conjecture, originally proven by Kontsevich, states that the free energy of two dimensional topological gravity is a solution to the KdV hierarchy. On the other hand, moduli spaces of curves carry a natural metric (called Weil-Petersson metric), and their Weil-Petersson volumes play a role in many applications both in mathematics and theoretical physics. This talk will explain how to express the volumes in terms of the free energy, how to explicitly compute them and how they behave asymptotically for large genus and the number of marked points.

#### Tuesday, February 7 2006 at 3:30 p.m.

### MAXWELL-BLOCH EQUATIONS, MAGNETIC SPHERICAL PENDULUM AND SOLITONS

Pavle Saksida, University of Ljubljana

Maxwell-Bloch equations are a well-known integrable system of PDE’s which describes the resonant interaction between light and an optically active medium. I will show that the MBE can be represented as a continuous chain of interacting charged spherical pendula which move in the fields of magnetic monopoles. This description yields a rich family of solitonic solutions of the MBE, and also a new Hamiltonian structure for this system.

#### Tuesday, January 31 2006 at 3:30 p.m.

### FINITE DIFFERENCE SCHEMES WITH CONTINUOUS SYMETRY GROUPS AND THEIR NUMERICAL APPLICATIONS

Anne Bourlioux, Pavel Winternitz, Université de Montréal

The talk will consist of two parts. In the “group theory” part, we will show how we can discretize an ordinary differential equation and obtain an invariant differences scheme with respect to the same Lie transformation groups. In the “numerical” part, we will consider many nonlinear differential equations of order 2 and 3 and show that the invariant schemes can provide higher precision than standard schemes without significantly increasing the complexity of computations.

#### Tuesday, January 24 2006 at 3:30 p.m.

### HAMILTON-JACOBI THEORY OF ORTHOGONAL SEPARATION OF VARIABLES IN THE FRAMEWORK OF CARTAN’S GEOMETRY

Roman Smirnov, Dalhousie University, Halifax, N.S.

The main purpose of this talk is to show that E. Cartan’s geometry, stemming from the celebrated Erlangen Program of F. Klein, forms a natural foundation for Hamilton-Jacobi theory. In particular, I will discuss how the method of moving frames, which is central to Cartan’s approach to geometry, can be effectively used to solve the problem of the determination of orthogonally separable coordinates in the Hamilton-Jacobi equation, as well as other problems of Hamiltonian mechanics.

#### Tuesday, January 17 2006 at 3:30 p.m.

### POISSON-LIE T-PLURALITY OF SIGMA MODELS

Libor Snobl, CRM

Starting from rather trivial examples, I shall briefly discuss the notion of Poisson-Lie T-dual, or more precisely Poisson-Lie T-plural models. Then I will describe the construction of 3-dimensional conformally invariant Poisson-Lie T-plural models and explain what kind of difficulties may arise. Finally, I will present some examples.

#### Tuesday, Dec 6 2005 at 3:30 p.m.

### QUANTUM TEICHMULLER AND THURSTON THEORIES

Leonid Chekhov, Steklov Institute, Moscow, CRM and Concordia

Based on papers with V.Fock and R.Penner, we propose the way to quantize Teichmuller and Thurston theories for Riemann surfaces with holes (punctures). These surfaces admit, in the Poincare uniformization, a graph description due to Penner and Fock. The corresponding parameters are the coordinates on the Teichmuller space and the mapping class group (modular) transformations can be explicitly constructed. Introducing the Poincare structure compatible with the Goldman Poisson brackets for geodesic functions that follow from 2+1-dimensional Chern-Simons theory, we are able to quantize the structure thus producing the quantum mapping class group transformations and quantum geodesic functions. The arising algebras, in some particular cases, are related to Nelson-Regge algebras of geodesics, or to algebras of Stokes parameters in isomonodromic deformations. In the second part of the talk, we consider the Thurston theory of measured geodesic laminations and show that, under the proper definition pertaining to the so-called tropical limit of mapping class group transformations, we can define the (classical and quantum) limits of the geodesic functions for arbitrary measured lamination and prove the existence of these limiting geodesic lengths, or of the corresponding operators describing quantum geodesic lengths.

#### Tuesday, Nov 27 2005 at 3:30 p.m.

### REDUCTIVE PERTURBATION TECHNIQUE FOR PARTIAL DIFFERENCE EQUATIONS

Decio Levi, Universita Roma Tre

We introduce the tools necessary to perform the Reductive Perturbation Technique on multiple lattices and apply them to the well known lattice potential Korteweg de Vries equation (lKdV). On doing so we show that the far field behaviour of the lKdV is governed by a completely discrete local Nonlinear Schroedinger equation.

#### Tuesday, November 22 2005 at 3:30 p.m.

### SOLVABLE N-BODY MODELS WITH NEAR-NEIGHBORS INTERACTIONS AND EXCHANGE OPERATOR FORMALISM

Alberto Enciso Carrasco, Universidad Complutense Madrid

In this talk we shall discuss three N-body spin models in one dimension where the particles only interact with their nearest neighbors. These models can be regarded as cyclic generalizations of Calogero–Sutherland (CS) Hamiltonians of type, where any two particles interact via a two-body potential . The study of solvable nearest-neighbors (NN) N-body problems goes back to Jain and Khare (1999), who considered them in connection with the theory of random matrices. Two years later, Deguchi and Ghosh managed to calculate some eigenfunctions for spin- NN models. Here we shall present a systematic approach to polynomial subspaces invariant under NN Hamiltonians which hinges on a modification of the exchange operators for CS models and provides at least one algebraic eigenfunction for each energy. Possible applications to spin chains will be briefly mentioned.

#### Tuesday, Nov 8 2005 at 3:30pm

### BASES MONOMIALES EN THÉORIE DES CHAMPS CONFORMES

Pierre Mathieu, Universite Laval, Quebec

La solution d’un problème en théorie des champs conforme passe, dans une large mesure, par la théorie des représentations irréductibles de l’algèbre de Virasoro. Cette approche mène, entre autre, à des expressions pour les caractères sous forme de sommes alternées. Ces expressions sont dites “bosoniques”. Au milieu des années 90, une autre facon de décrire ces caractères est découverte. Ces nouvelles expressions, manifestement positives, s’écrivent en terme multisommes. Ces formules sont dites “fermioniques”. Elles correspondent è une description de l’espace des états en terme d’un remplissage par des quasi-particules, remplissage sujet è des restrictions qui prennent la forme d’un principe de Pauli généralisé. Des exemples de telles bases de quasi-particules seront presentés.

#### Tuesday, Oct. 25 2005 at 3:30 p.m.

### A NEW PARADIGM OF CHAOS IN DYNAMICAL SYSTEMS ?

David Gomez-Ullate, Politècnica de Catalunya, Barcelon

We will present through a toy example given by three coupled first order ODEs a mechanism of transition from simple periodic orbits to motions of increasing complexity. The mechanism is based on the presence of many separatrices (eventually a dense set) and can be understood by regarding the evolution in real time as a curve on a Riemann surface, the one associated to the solution (as a function of some suitably defined “complex time”) of the equations of motion. The presence of a high amount of branching produces complicated orbits. We will also discuss in which sense this mechanism also features sensitive dependence on the initial data, albeit with no exponential divergence of nearby trajectories. In some cases we will show how the periods of the orbits are closely connected to the partial quotients of the development in continued fraction of the coupling constants, hence providing another source of irregular, unpredicatable motions. The explanation will be complemented by the projection of computer simulations of the different motions obtained through a numerical integration of the equations. All these concepts will be explained for the toy model in question, although the mechanism is believed to be present in many systems governed by differential equations. This is joint work with Francesco Calogero (Roma La Sapienza), Paolo Santini (Roma La Sapienza) and Matteo Sommacal (SISSA Trieste).

**Tuesday, Oct. 18 2005, 3:30 p.m.**

### SPIN CALOGERO MODELS OBTAINED FROM DYNAMICAL r-MATRICES AND GEODESIC MOTION

Bela Gabor Pusztai, CRM et Concordia

We study classical integrable systems based on the Alekseev-Meinrenken dynamical r-matrices corresponding to automorphisms of self-dual Lie algebras, G. We prove that these r-matrices are uniquely characterized by a non-degeneracy property and apply a construction due to Li and Xu to associate spin Calogero type models with them. The equation of motion of any model of this type is found to be a projection of the natural geodesic equation on a Lie group G with Lie algebra G, and its phase space is interpreted as a Hamiltonian reduction of an open submanifold of the cotangent bundle T^*G, using the symmetry arising from the adjoint action of G twisted by the underlying automorphism. This shows the integrability of the resulting systems and gives an algorithm to solve them. As illustrative examples we present new models built on the involutive diagram automorphisms of the real split and compact simple Lie algebras, and also explain that many further examples fit in the dynamical r-matrix framework. This is a joint work with Laszlo Feher.

#### Tuesday, Oct. 11 2005, 15h30

### DETERMINANT OF LAPLACIANS FOR RIEMANN SURFACES AND KLEINIAN GROUPS

Andrew McIntyre, CRM et Concordia University, Montreal

The regularized determinant of the Laplacian on a constant curvature surface forms a function on the moduli space M_g of Riemann surfaces of genus g. It is a partition function for a conformal field theory. Up to “anomaly” factors, it is the modulus square of the determinant of the dbar operator, which is a holomorphic function on M_g and a tau function for an integrable hierarchy. I will discuss how this latter function generalizes the Dedekind eta function, from genus 1 to higher genus. If time permits, I will sketch extensions to quasifuchsian groups and to other bundles (necessary for applications to theta functions), and connections with work of Kokotov and Korotkin. Some of this work is joint with Leon Takhtahjan and Lee-Peng Teo.

#### Tuesday, Oct. 4, 2005 at 3:30 p.m.

### PERIODICITY OF THE DETERMINANT LINE BUNDLE

Frederic Rochon, State University of New York, Stony Brook

The determinant line bundle associated to a family of elliptic operators was introduced by Quillen. In this talk, we will discuss how Bott periodicity arises in this context. To do so, we will describe the determinant bundle in terms of principal bundles. The periodicity result will be obtained using properties of the harmonic oscillator in quantum mechanics. If time permits, we will discuss how this periodicity result can be used to obtain a pseudifferential generalization of a theorem of Dai and Freed about the trivialization of certain determinant line bundles by the eta invariant. The subject of this talk is joint work with Richard Melrose.

#### Tuesday, Sept. 27, 2005 at 3:30 p.m.

### SURFACES ASSOCIATED WITH GRASSMANNIAN SIGMA MODELS ON MINKOWSKI SPACE

Libor Snobl, CRM

We construct and investigate surfaces in su(N) algebras associated with the Grassmannian sigma models on Minkowski space. The structural equations of such surfaces are studied using moving frames adapted to the surfaces. The first and second fundamental forms of these surfaces as well as the relations between them as expressed in the Gauss-Weingarten and Gauss-Codazzi-Ricci equations are found. The scalar curvature and the mean curvature vector expressed in terms of a solution of Grassmanian sigma model are obtained. We present the relation between CP^1 sigma models and the sine-Gordon equation. In collaboration with A.M. Grundland.

#### Tuesday, Sept. 20, 2005 at 3:30 p.m.

### RESONANT ISOMONODROMIC PROBLEM

Man Yue Mo, CRM

Despite a lot of work was carried out on the non-resonant isomonodromic problem, little is known about the resonant case. In recent years, resonant isomonodromic problems arise in the studies of biorthogonal polynomials for the multi-matrix models. In a joint work with M. Bertola, we have generalized the well-known results of non-resonant isomonodromic problems to resonant ones. In our studies, A^{\alpha}_r is allowed to have any Jordan block form while a condition is imposed onto A^{\alpha}_{r+1}. We have constructed the zero curvature representation and generalized the notion of the tau-function.

#### Tuesday, Sept. 13, 2005 at 3:30 p.m.

### KILLING TENSORS AS IRREDUCIBLE REPRESENTATIONS OF THE GENERAL LINEAR GROUP

Robert Milson, Dalhousie University

We show that the vector space of fixed valence Killing tensors on a space of constant curvature is naturally isomorphic to a certain irreducible representation of the general linear group. The isomorphism is equivariant in the sense that the natural action of the isometry group corresponds to the restriction of the linear action to the appropriate subgroup. As an application, we deduce the Delong-Takeuchi-Thompson formula on the dimension of the vector space of Killing tensors from the classical Weyl dimension formula.