CRM : Mathematical Physics Laboratory seminars

Centre de recherches 

Mathematical Physics Laboratory seminars

Fall 2010 - Winter 2011

CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

September 14   September 21   September 28   October 5   October 12   October 19   October 26   November 2   November 23  

Tuesday, September 14 2010, 15:30

Ward identities and integrable differential equations for quantum correlation functions

Benjamin Doyon, Department of Mathematics, King's College London

Some time ago, Fonseca and Zamolodchikov proposed a method for deriving the known differential equations for correlation functions in the thermally perturbed Ising model. The main idea of the method is to consider extra symmetries emerging after ``doubling' the model, and the associated Ward identities. Although the derivation used particularities of the Ising conformal field theory, similar-looking methods in the theory of classical integrability are known to generally produce bilinear differential equations for tau functions. In this talk, I will explain, in the context of QFT, how the main idea may be applied to correlation functions of twist fields in arbitrary quadratic density matrices of free massive (1+1-dimensional) fermionic models. To show how it works, I will develop the case of the U(1)-twist fields in the massive Dirac theory. This work was done in collaboration with my present PhD student James Silk (who did most of the calculations).

Tuesday, September 21 2010, 15:30

Structure of Levi extensions of nilpotent algebras

Libor Snobl, Department of Physics, Czech Technical University in Prague

We consider Lie algebras which can be decomposed into a nontrivial semidirect sum $mathfrak{g}= mathfrak{p} dotplus mathfrak{r}$ of a semisimple subalgebra $mathfrak{p}$ and the radical $mathfrak{r}$. We assume that the nilradical of $mathfrak{r}$ is given and investigate restrictions on possible Levi factors $mathfrak{p}$ coming from the structure of characteristic ideals of the nilradical. We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9. In particular we explain, mostly by a simple dimensional analysis, why the majority of 5- and 6-dimensional nilpotent algebras do not possess any Levi extension.

Tuesday, September 28 2010, 15:30

Differential Geometry, Lie Algebras and Lie Groups with Maple

Ian Anderson, Dept. of mathematics and statistics, Utah State University

In this talk I will describe a new suite of Maple packages for symbolic computations in differential geometry, Lie groups and Lie algebras. The talk will begin with an overview of the functionality of this software. I will then give a brief demonstration of the basic features of the software and conclude by presenting some recent applications to the integration of ordinary differential equations and to the equivalence problem for space-times and solutions to the Einstein equations.

Tuesday, October 5 2010, 15:30

The Excitation Spectrum for Weakly Interacting Bosons

Robert Seiringer, Dept. of mathematics, McGill

We investigate the low energy excitation spectrum of a Bose gas with weak, long range repulsive interactions. In particular, we prove that the Bogoliubov spectrum of elementary excitations with linear dispersion relation for small momentum becomes exact in the mean-field limit.

Tuesday, October 12 2010, 15:30

Families of Superintegrable Systems


In this talk, I will discuss several new results stemming from the recent discovery of a family of exactly-solvable deformations of the simple harmonic oscillator by Turbiner, Tremblay and Winternitz (J. Phys. A, 2009, 42, 242001 ) (denoted TTW). The conjecture of superintegrability lead to several new methods of analysis of superintegrable systems which were successfully applied to the TTW system. I will also describe a transformation of integrable systems, known as coupling constant metamorphosis, and its application to the TTW system to obtain a new family of superintegrable deformations, this time of the Kepler-Coulomb potential.

Tuesday, October 19 2010, 15:30

Two-variable Wilson polynomials and the generic superintegrable system on the 3-sphere

Willard Miller, University of Minnesota

We show that the symmetry operators for the quantum superintegrable system on the 3-sphere with generic 4-parameter potential form a closed quadratic algebra with 6 linearly independent generators that closes at order 6 (as differential operators). Further there is a algebraic relation at order 8 expressing the fact that there are only 5 algebraically independent generators. We work out the details of modeling physically relevant irreducible representations of the quadratic algebra in terms of divided difference operators in two variables The representation spaces turn out to be those of two variable Wilson and Racah polynomials with arbitrary parameters. The quadratic algebra structure breaks the degeneracy of the spaces of these polynomials. In an earlier paper we found a similar characterization of one variable Wilson and Racah polynomials in terms of irreducible representations of the quadratic algebra for the quantum superintegrable system on the 2-sphere with generic 3-parameter potential. This indicates a general relationship between 2nd order superintegrable systems in n dimensions and multivariable orthogonal polynomials. Joint work with Ernie Kalnins and Sarah Post.

Tuesday, October 26 2010, 15:30

A Missing System of Classical Orthogonal Polynomials

Alexei Zhedanov, Donetsk Institute for Physics and Technology

We study a new class of "classical" orthogonal polynomials which satisfy (apart from 3-term recurrence relation) an eigenvalue problem with Dunkl-type differential operator. These polynomials thus satisfy the same bispectrality property as classical orthogonal polynomials. Explicit expression for these polynomials as well as their recurrence coefficients and the weight function are obtained. We also show that these polynomials provide a nontrivial realization of the Askey-Wilson algebra for q=-1.

Tuesday, November 2 2010, 15:30

The groundstate of the XXZ model and the quantum Knizhnik-Zamolodchikov equation

Tiago Fonseca, LPTHE, Paris 6 et maintenant CRM

In 2000, Razumov and Stroganov conjectured an intriguing relation between two statistical integrable models, the XXZ spin chain and the 6-Vertex model. Naturally, we started computing the XXZ groundstate (using the quantum Knizhnik-Zamolodchikov equation). This led us to the proof of some old results and to new conjectures.

Tuesday, November 23 2010, 15:30

The Riemann-Hilbert approach to the transition from the Pearcey to the Airy process

Mattia Cafasso, CRM

I will consider the gap probability for the Airy and the Pearcey processes. Using a Riemann-Hilbert approach I will show that the Pearcey process, in a large gap/large time limit, factorizes in two independent Airy processes. This talk is based on a joint work with Marco Bertola.

For further information: SAINT@CRM.UMontreal.CA