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# Mathematical Physics Seminar: January 2002 to February 2003

Date                                     Le mardi 15 janvier 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Pavel Winternitz (CRM et DMS)
Titre / Title:                             « Systèmes intégrables et superintégrables
dans les espaces avec courbures »

Résumé/Abstract:

La conférence est consacrée aux systèmes superintégrables classiques et quantiques dans un espace Riemannien ou pseudo-Riemannien de dimension deux. Les systèmes considérés admettent  trois  intégrales de mouvement quadratiques dans les impulsions (incluant le Hamiltonien). Les espaces considérés incluent les espaces à courbure constante, mais ne sont pas limités à ce cas.

Date                                     Le mardi 22 janvier 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Dmitri Korotkin, Concordia University and CRM
Titre / Title:                             « Some integrable systems on Hurwitz spaces »

Résumé/Abstract:

We introduce a new class of integrable systems, naturally associated to spaces of rational maps. The critical points of the maps play the role of « times ». Our systems provide a natural generalization to the Ernst equation. For the scalar case we generalize our systems to Hurwitz spaces of arbitary genus; the systems obtained in this way can be naturally called the generalized Euler-Darboux equations. We show that any solution of these equations defines a Darboux-Egoroff metric via its tau-function; a subclass of these metrics corresponds to some known classes of Frobenius manifolds.

Date                                     Le mardi 29 janvier 2002
Heure/Time:                           15h30
Conférencier / Lecturer:    S. Gravel (CRM et Département de physique, UdeM)
Titre / Title:                             « Systèmes superintégrables avec symétries du troisième ordre »
Résumé/Abstract:                .

On discutera de systèmes classiques et quantiques, dans un espace plat à deux dimensions, qui possèdent à la fois une intégrale du premier et du troisième ordre dans les impulsions. Il appert que certains systèmes quantiques non triviaux existent qui se ramènent, dans la limite classique, au cas  trivial du hamiltonien sans interactions.

Date                                     Le mardi 5 février 2002
Heure/Time:                           15h30
Titre / Title:                            « Higher order supersymmetric periodic potentials«
Résumé/Abstract:

The talk will start with a quick revision of the basic idea behind « supersymmetric quantum mechanics »: Darboux tranformation or factorization method, also known as intertwining operator technique. In this talk, the stress will be placed on second order intertwining operators. Next, it will be shown how the technique can be applied to periodic Hamiltonians which have a band-structure spectrum. The method will be applied to the family of Lam\’e potentials, showing new supersymmetric potentials with the same band structure as the original one. Finally, some comments will be made on the generation of new exactly solvable potentials related with the Scarf potential, which, in addition to be periodic, has singularities (a fact which makes impossible to apply directly the theorems known from the theory of periodic differential equations).

Date                                     Le mardi 12 février 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Mikhail Babich, Concordia University and Steklov Mathematical Institute, St.Petersburg, Russia
Titre / Title:                            « Higher order supersymmetric periodic potentials«
Résumé/Abstract:

Schlesinger system with 4 points, its symmetries, connections with Painleve VI system and the algebraic surface theory » We consider a geometrical interpretation of the Schlesinger system and demonstrate its connection with the theory of algebraic surfaces. Other points of view on the same subject, the theory of isomonodromic deformations of linear differential equations, and the Painleve VI system, draw us to algebraic surfaces too. All these surfaces are birational equivalents of each other, and all these systems are equivalent, but we demonstrate how symmetries that are evident for one representation become very non-trivial for another one.

Date                                     Le mardi 19 février 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Frederic Lesage, CRM
Titre / Title:                            « Théorie des champs intégrable et dualité«
Résumé/Abstract:

Les dualités entre couplages faibles et forts dans les théories des champs ont généré beaucoup d’attention au cours des dernières années. Ce séminaire montre un exemple d’une telle dualité dans une théorie intégrable avec bord. Ici l’intégrabilité permet d’exhiber la dualité de facon explicite et l’identification d’une théorie infra-rouge bien définie.

Date                                     Le jeudi 21 février 2002
Heure/Time:                           15h30
Conférencier / Lecturer:    Jean-Louis Verger-Gaugry, Département de Mathématiques Institut Fourier,
Université de Grenoble, France

Titre / Title:                           « Un théorème de compacité pour l’ensemble des ensembles
uniformément discrets de Rn et ses sous-ensembles :
réseaux, ensembles modèles, ensembles de Delaunay, clusters
. »

Résumé/Abstract

On montrera que l’ensemble UD des ensembles uniformément discrets de Rn de constante r > 0 peut être muni d’une métrique (d) pour lequel il est compact. La classe des ensembles de Delaunay de Rn de constantes (r, R) est soit vide soit un sous-espace compact de (UD,d). Ce théorème de compacité implique en particulier le théorème de sélection de Mahler pour les réseaux, ingrédient essentiel en Géométrie des Nombres par exemple pour l’existence de réseaux extrêmes. La démonstration montre qu’il s’agit d’une topologie uniforme et utilise des outils élémentaires qui généralisent l’approche de Chabauty pour les réseaux. Par définition, un cristal est un élément de UD dont la tranformée de Fourier de l’autocorrélation est une mesure discrète. Une question ouverte est de savoir si la sous-classe des cristaux est ouverte ou fermée dans cette topologie et de connaître sa structure. Mêmes questions avec la sous-classe des ensembles modèles constructibles à l’aide de nombres algébriques et la sous-classe des beta-entiers (ensembles de Meyer) où beta est un nombre de Perron.

Date                                     Le mardi 26 février 2002
Heure/Time:                           15h30
Conférencier / Lecturer:    Dmitry Jakobson, Université McGill
Titre / Title:                            « Spectra of elements in the group ring of SU(2)«
Résumé/Abstract:

Let g_1, g_2, …, g_n be n rotations in SO(3). Consider the operator T=T(g_1,…,g_n) acting on functions as follows: Tf(x)=f(g_1(x))+f(g_1^{-1}(x))+…+f(g_n(x))+f(g_n^{-1}(x)). We discuss the spectrum of this operator acting on the space of spherical harmonics of weight k. In particular, we consider the existence of spectral gaps, limiting eigenvalue distribution (as k->infinity), the rate of convergence to that distribution, and eigenvalue spacings. This is a joint work with Peter Sarnak and Alex Gamburd.

Date                                     Le mardi 12 mars 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Charles P. Boyer, University of New Mexico
Titre / Title:                            « Sasakian-Einstein Geometry »
Résumé/Abstract:

The geometry of Sasakian-Einstein manifolds has recently become of interest in the so-called AdS/CFT correspondence in Physics. In particular, 5 dimensional Sasakian-Einstein manifolds are related to the low energy limit on D3 branes, and 7 dimensional Sasakian-Einstein manifolds are related to the low energy limit on M2 branes. I describe recent work in collaboration with K. Galicki, and M. Nakamaye which proves the existence of Sasakian-Einstein metrics on many simply connected compact 5-manifolds. Results in dimension 7 are also mentioned, but I will concentrate the lecture on dimension 5.

Date                                     Le mardi 19 mars 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     J. Harnad (CRM, Univ. Concordia)
Titre / Title:                            « Matrix models, integrable systems, duality, and all that … »
Résumé/Abstract:

This talk will review some known results and introduce some new ones concerning the connection between matrix model integrals and integrable systems. The emphasis will be on results recently obtained (with M. Bertola and B. Eynard) concerning a class of 2-matrix models. The topics will include: 1) A discussion of why matrix model partition functions are KP tau functions and, in some cases, also isomonodromic tau functions 2) An introduction to new results concerning « spectral duality » of pairs of isomonodromic deformation families of covariant deribative operaators on the Riemann sphere with punctures, in relation to the biorthogonal polynomials associated to 2-matrix model integrals. 3) (If time permits) Schur function expansion of 2-matrix partition functions.

Date                                     Le mardi 26 mars 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Maria Cristina Ciocci, University of Gent (Belgium)
Titre / Title:                            « KAM for reversible systems »
Résumé/Abstract:

The main issue of the KAM theory is the persistence of quasi-periodic invariant tori in integrable systems for small perturbations. Here we are interested in the occurrence of quasi-periodicity in integrable _reversible_ systems. Integrable’ refers to a toroidal symmetry, so that the invariant tori in the integrable approximation are of Floquet type. By a scaling device, this perturbation problem is translated to the case where integrable’ is replaced by linear and integrable’. The cases of resonance are notoriously hard in the KAM, but our method can handle at least 1:1 resonances. We also study excitation of normal modes, which leads to reversible quasi- periodic Hopf bifurcation.

Date                                     Le mardi 2 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:    Jan Zich, Technical University, Prague and CRM
Titre / Title:                            « Voronoi and Delone tiling of quasicrystals »
Résumé/Abstract:

A new general method is presented which allows one to find all distinct Voronoi and Delone tiles in any quasicrystal from a large family. That includes the tiles which may be present with arbitrarily low density $>0$. At all stages the method requires only consideration of a (possibly large) finite number of cases. We study the Voronoi and Delone tilings given by two dimensional point sets, quasicrystals’, arising from the $A_4$-root lattice space by means of the standard projection to a 2-dimensional plane with the irrationality $\tau=(1+\sqrt{5})/ 2$. For the method in general, we require that the acceptance window be bounded with non-empty interior. Specific results are provided for the circular acceptance windows of any radius $0<\infty$. Within one quasicrystal the tiles are distinguished by their shape, size, and orientation.

Date                                     Le jeudi 4 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Michel Racine, Université d’Ottawa
Titre / Title:                            « Superalgèbres simples »
Résumé/Abstract:

Après avoir défini la notion de superalgèbre pour les variétés d’algèbres usuelles, c’est-à-dire, commutative, associative, associative à involution, de Jordan et de Lie, on essaie de résumer la structure des superalgèbres simples de dimension finie.

Date                                     Le jeudi 9 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     G.Pogosyan UNAM, Mexico and JINR, Dubna, Russia
Titre / Title:                            « Superintegrable potentials in N-dimensional Euclidean space »
Résumé/Abstract:

We obtain all separable coordinates for the two superintegrable potentials generalizing Kepler-Coulomb and oscillator systems in N-dimensional Euclidaen space. The algebra of second order symmetries of the resulting Schroedinger equation is given.

Date                                     Le jeudi 16 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Marco Bertola, CRM
Titre / Title:                            « Fundamental systems of solutions for infinite recurrence relations arising in two-matrix models; applications to the Riemann-Hilbert problem »
Résumé/Abstract:

In the context of the biorthogonal polynomial approach to two–matrix models, we show how to construct explicitly a fundamental system of solutions to the differential and multiplicative recurrence relations (which will be recalled). This system can be used to analize the Stokes phenomenon for a certain system of ODEs with polynomial coefficients which arises naturally in this context.

Date                                     Le mardi 23 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Jacek Szmigielski (Univ. de Saskatchewan)
Titre / Title:                            « Discrete classical strings, Weyl functions, and integrable systems »
Résumé/Abstract:

I will describe a method of explicit integration of three Hamiltonian systems: interacting peakons, finite Toda, and Calogero-Francoise systems. The method employs the classical string with discrete masses. In this setup the three systems can be viewed as isospectral deformations of certain boundary value problems and the solution can be obtained by solving the inverse spectral problem. The main technical tool is the Weyl function which encodes all the information about the boundary value problem and for which the inverse spectral problem is intrinsically connected to the continued fraction expansion of the Weyl function and the associated classical moment problem. This is based on joint work with R.Beals and D. Sattinger.

Date                                     ( Exceptionnellement) Le jeudi 25 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:    Miroslav Englis Mathematics Institute, Academy of Sciences, Prague
Titre / Title:                            « A review of (Berezin and other) quantization methods »
Résumé/Abstract:

The talk will present a survey (somewhat biased in favour of results obtained by the author) of recent progress, new ideas and open problems in several approaches to quantization form a mathematician’s viewpoint. In particular, we will discuss some current developments in nonlinear, Berezin, Berezin-Toeplitz and prime quantization, as well as mention some related questions from the theory of Fourier integral operators, several complex variables, group theory and complex geometry.

Date                                     Le mardi 30 avril 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Bertrand Eynard, Saclay (France) et CRM
Titre / Title:                            « An Ansatz for the large n asymptotics of bi-orthogonal polynomials. The genus zero case. »
Résumé/Abstract:

The large n spectral statistics of two coupled random matrices is related to the large n asymptotics of some bi-orthogonal polynomials in some appropriate scaling regime. Up to now, the large n asymptotics of bi-orthogonal polynomials are not known. The Riemann Hilbert method will probably give the answer (work in progress with M. Bertola and J. Harnad). The Riemann-Hilbert method needs a starting point, and in this purpose, we propose an Ansatz, motivated by a non-rigorous matricial saddle-point method. The topic of this seminar is to present the Ansatz and explain its origin. For simplicity, only the genus zero case will be presented.

Date                                     Le mardi 7 mai 2002
Heure/Time:                          14h00 (exceptionellement)
Conférencier / Lecturer:     T. A. Osborn ( Department of Physics and Astronomy, Univ. de Manitoba)
Titre / Title:                            « Magnetic Curvature of Quantum Phase Space »
Résumé/Abstract:

A gauge invariant Weyl quantization in a closed integral form is developed over a linear phase space endowed with a Faraday 2-form. The resulting structure is a quantum phase space wherein the classical commutative product of functions is replaced by a noncommutative (star) product. This star product is determined by a symplectic area phase. It is shown how a reflection symmetry associated with the quantization process defines a connection and how the asymptotic expansions of the star product displays this curvature structure. Semiclassical representations of the of the Weyl symbol corresponding to the Schrodinger evolution operator are obtained.

Date                                     Le mardi 14 mai 2002
Heure/Time:                           15h30
Conférencier / Lecturer:     Franco Magri (Univ. Milano-Bococca)
Titre / Title:                            « A geometrical characterization of separable systems according to Levi Civita: classical roots and modern perspectives. »
Résumé/Abstract:

The aim of this talk is to give a precise set of sufficient conditions allowing one to state that the spectral curve of the Lax matrix gives actually the separation equations (in the sense of Stackel) of the separable system.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 1er octobre 2002 15 h 30 Pavel Winternitz, CRM et DMS, Université de Montréal Symétries des équations à différences finies, Lagrangiens, intégrales premières et solutions exactes Sophus Lie a demontré qu’une équation différentielle ordinaire peut être integrée en quadratures, si son groupe de symétrie est de dimension 2 (ou plus). Nous allons discuter le cas des équations à différences finies et montrer comment les symétries nous aident à construire des Lagrangiens, des intégrales premières et des solutions exactes.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 8 octobre 2002 15 h 30 Bertrand Eynard, SPHT Saclay (France) et CRM Points critiques des modèles de matrices et la hiérarchie de Painlevé II Au point critique, la densité de valeurs propres s’annule avec un degré 2m. Nous calculons les comportements asymptotiques des polynômes orthogonaux associés (qui permettent de calculer toutes les fonctions de corrélations de valeurs propres) dans cette limite, en fonction de m. Ceci n’était connu que pour m=1, dans le cas pair (spectre symmétrique). Nous trouvons que les asymptotiques sont solution d’un système d’équations différentielles associé à la hiérarchie de Painlevé II.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 15 octobre 2002 15 h 30 Siaka KONE Université de Cocody, Côte d’Ivoire Mixed order systems of ordinary linear differential equations Expansions into eigenfunctions and associated functions of n-th order ordinary scalar differential equations and first order systems of ordinary linear differential equations have been extensively investigated. Here we consider systems of higher order differential equations.In this case,it is not always possible to obtain an associated to first order systems. A mixed order 2×2 system which are equivalent to first order systems is considered.Results on eigenfunction expansion are established.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 22 octobre 2002 15 h 30 Michel Grundland CRM et UQTR Sur les représentations de Weierstrass, applications harmoniques CP^N et surfaces ˆ courbures moyennes constantes Nous donnons une généralisation de la représentation de Weierstrass qui correspond aux application harmoniques CP^N. Cette généralisation nous permet d’étudier les surfaces à courbures moyennes constantes plongées dans des espaces multi-dimensionnels. Nous utilisons cette représentation pour suggérer une interprétation géométrique des applications harmoniques CP^N.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 5 novembre 2002 15 h 30 David Gomez-Ullate (CRM) Quasi-exactly solvable extensions of quantum Calogero-Sutherland models The ideas of quasi-exact solvability in Quantum Mechanics are applied to many-body problems of the Calogero-Sutherland type. In particular, a systematic procedure will be developed to construct many-body problems for interactiong spin or spinless particles, which preserve known finite dimensional subspaces of smooth functions. This allows (at least) a portion of the spectrum to be calculated by algebraic means, since the action of the Hamiltonian in this subspace reduces to a finite dimensional matrix. A complete classification of the models obtainable by this method is performed. The previously known exactly-solvable Hamiltonians appear in the classification (rational, trigonometric and hyperbolic interaction), which also shows that the more general models with elliptic interaction are only quasi-exactly solvable. Some explicit eigenfunctions of an elliptic Calogero-Sutherland model will be presented. The application of some of these models to spin chains with long range interaction will also be discussed.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 3 décembre 2002 15 h 30 Alexei V. Penskoi (CRM) Poisson brackets for the Camassa-Holm equation We will discuss the Camassa-Holm shallow water equation and the geometry of the corresponding Poisson brackets in the case of periodic solutions.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 10 décembre 2002 15 h 30 Pietro-Luciano Buono (CRM) Animal locomotion and symmetric networks of differential equations Animal locomotion comes in many different patterns: bipeds will walk or hop, quadrupeds for instance will walk, trot or pace while hexapods use mostly tripod gaits. It has long been observed that many animal gaits exhibit symmetry properties, in fact spatio-temporal symmetries. Physiologists have provided evidence that locomotion is produced by a network of cells in the central nervous system called the Central Pattern Generator (CPG). In this talk, I will present a model of CPG for animal locomotion consisting of a symmetric network of coupled ordinary differential equations. The properties of this model will be discussed and predictions based on the model will be presented.

 Date Heure – Time Conférencier – Lecturer Titre – Title Résumé – Abstract
 Le mardi 25 février 2003 15 h 30 Thalia D. Jeffres ( Universidad Michoacana, Mexico) Regularity of the Heat Operator on a Cone I will discuss recent work with Paul Loya in which we investigated the mapping properties of the heat operator between appropriately defined spaces of whole and fractional Holder) derivatives. The estimates obtained allow solution of certain semilinear parabolic equations.

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