Tuesday, October 14 2008 at 11:00 a.m.

### LIE ANTIALGEBRAS ASYMPTOTIC ANALYSIS OF THE SEMICLASSICAL SINE-GORDON EQUATION

#### Robert Buckingham, CRM

The small dispersion limit of the completely-integrable sine-Gordon equation models magnetic flux propagation in long Josephson junctions. We present two new families of initial data for which the scattering data can be computed. Furthermore, it is possible to compute the exact solutions via inverse scattering for arbitrarily small dispersion values. Plots of the solutions reveal regions of pure librational and rotational motion, as well as regions of multi-phase waves separated by nonlinear caustics. We use the Riemann-Hilbert method to study the leading asymptotic solution in the zero dispersion limit and explain the transitions between certain regions. This is joint work with Peter Miller.

#### Tuesday, October 21 2008 at 11:00 a.m.

### STABILITY OF THE SPIKED MODEL IN THE SMALL PERTURBATION REGIME

#### Dong Wang

In this talk we consider a special case of the Wishart ensemble, the spiked model, and concentrate on the even more special rank 1 case. We try to analyze the phenomenon that the distribution of the largest eigenvalue is the same as the null Wishart (Laguerre) ensemble, when the perturbation is small. We discuss all the 3 ensembles: complex, quaternionic and real, from easy to difficult.

#### Tuesday, November 4th 2008 at 11:00 a.m.

### THE GROUND STATE OF THE QUARTIC OSCILLATOR : FROM THE ANHARMONIC OSCILATOR TO THE DOUBLE WELL

#### A. Turbiner, UNAM, Mexico and CRM

#### Tuesday, november 11 2008 at 11:00 a.m.

### Exceptionnally, the seminar will be held at Concordia University.

Concordia University, Library Building, 1400 Boul de Maisonneuve Ouest, room 921.04

### SUSY, INTERTWINING OPERATORS AND RELATED VECTOR COHERENT STATES

#### Fabio Bagarello, Universita di Palermo

I discuss a strategy which produces, stating from a fixed hamiltonian, a second operator with the same eigenvalues. The eigenvectors are related as well, essentially as in ordinary SUSY quantum mechanics. I will also show some relations between these partner hamiltonians and a family of vector coherent states.

#### Tuesday, november 18 2008 at 11:00 a.m.

### COUNTING COLORED 3D YOUNG DIAGRAMS WITH VERTEX OPERATORS

#### Benjamin Young, CRM

I will show how to compute some multivariate generating functions for 3D Young diagrams (otherwiseknown as « plane partitions »). Each box in a 3D Young diagram gets assigned a « color » according to a certain pattern; the variables keep track of how many boxes of each color there are. My generating functions also turn out to be orbifold Donaldson-Thomas partition functions for C^3/G, where G is a finite abelian subgroup of SO(3).

#### Thursday, november 20 2008 at 11:00 a.m.

### INTEGRABILITY OF PARTIAL DIFFERENCE EQUATIONS BY THE GENERALIZED SYMMETRY METHOD

#### Decio Levi, Università Roma Tre

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Using the existence of a generalized symmetry as an integrability condition, we derive a few integrability conditions suitable for testing and classifying equations of that class. Those conditions are used to test for integrability discretizations of some well-known hyperbolic equations.

#### Tuesday, november 25 2008 at 11:00 a.m.

### UNIVERSALITY CLASSES IN RANDOM MATRICES

#### Marco Bertola, Concordia and CRM.

In this talk of introductory nature, I will explain what universality is in random matrices and then focus on certain universality results that we studied recently on the « microscopic » and « mesotropic » colonization of a spectral band.

This is joint work with S.Y. Lee and M. Y. Mo.

#### Tuesday, december 2 2008 at 11:00 a.m.

### PATH INTEGRATION, PERTURBATION THEORY AND COMPLEX ACTIONS

#### Manu Paranjape, Université de Montréal

We consider Euclidean functional integrals involving actions which are not exclusively real. This situation arises, for example, when there are t-odd terms in the the Minkowski action. Writing the action in terms of only real fields (which is always possible), such terms appear as explicitly imaginary terms in the Euclidean action. The usual quanization procedure which involves finding the critical points of the action and then quantizing the spectrum of fluctuations about these critical points fails. In the case of complex actions, there do not exist, in general, any critical points of the action on the space of real fields, the critical points are in general complex. The proper definition of the function integral then requires the analytic continuation of the functional integration into the space of complex fields so as to pass through the complex critical points according to the method of steepest descent. We show in two simple examples where this procedure can be carried out explicitly that the procedure of finding the critical points of the real part of the action and quantizing the corresponding fluctuations, treating the (exponential of the) complex part of the action as a bounded integrable function fails, at least perturbatively.

#### Tuesday, december 9 2008 at : Part I : 11:00 a.m. – Part II : 12:30 p.m.

### QUANTUM SUBGROUPS OF LIE GROUPS AND MODULAR INVARIANCE IN CONFORMAL FIELD THEORIES

#### Robert Coquereaux (Directeur de recherche, CNRS. CPT, Luminy-Marseille)

For quantum groups at roots of unity, one can construct a monoidal category of representations that admits, for special values of the chosen root, module-categories, ie additive categories on which the previous one acts. In the case of quantum SU2, those « quantum subgroups » are classified by the usual ADE Dynkin diagrams. This classification is equivalent to another problem solved long ago in the case of SU2 by theoretical physicists, in the context of conformal field theories with boundaries, namely the classification of modular-invariant sesquilinear forms, for the Hurwitz – Verlinde representations of SL(2,Z). Each such quantum subgroup is associated with a weak Hopf algebra of a special kind (an Ocneanu quantum groupoid) that admits two, usually distinct, representations theories whose multiplicative structures can be encoded by graphs: the fusion graph and the graph of quantum symmetries. The purpose of the seminar is to provide a general introduction to the above ideas and to describe what happens when SU2 is replaced by more general Lie groups. This leads in particular to higher analogues of Coxeter-Dynkin diagrams (that will be presented for SU3 and SU4) and to higher graphs of quantum symmetries.

#### Tuesday, January 13 2009 at 3:30 p.m.

### QUANTUM GRAVITY IN THREE DIMENSIONS

#### Alex Maloney, Physics Dept., McGill

Quantum gravity in a world with three dimensions (that is, with two spatial dimensions and one time dimension) provides a fascinating theoretical laboratory where we can test precisely our ideas about quantum general relativity. Remarkably, for a certain « chiral » version of the theory we can compute the quantum partition function of the theory exactly. In this case many objects which appear quite mysterious, such as black holes, can be understood precisely at the quantum level.

#### Tuesday, January 20 2009 at 3:30 p.m.

### CONVOLUTION SYMMETRIES OF INTEGRABLE HIERARCHIES, MATRIX MODELS AND TAU-FUNCTIONS

#### John Harnad, Concordia and CRM

Generalized convolution symmetries of integrable hierarchies of KP-Toda and 2KP-Toda type have the effect of multiplying the Fourier coefficients of the Baker-Akhiezer function by a specified sequence of constants. The induced action on the associated fermionic Fock space is diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The coefficients in the single and double Schur function expansions of the associated tau- functions, which are the Pluecker coordinates of a decomposable element, are multiplied by the corresponding diagonal factors. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type which are also KP-Toda or 2KP-Toda tau-functions. More general multiple integral representations of tau functions are similarly obtained, as well as finite determinantal expressions for them.

#### Tuesday, January 27 at 3:30p.m. 15h30

### INVARIANT SOLUTIONS OF THE SUPERSYMMETRIC SINE-GORDON EQUATION

#### Alexander Hariton, CRM

A comprehensive symmetry analysis of two different forms of the supersymmetric sine-Gordon equation is carried out. We begin by studying a system of partial differential equations corresponding to the coefficients of the various powers of the fermionic independent variables. Next, we consider the super-sine-Gordon equation expressed in terms of a bosonic superfield. In each case, a Lie (super) algebra of symmetries is determined and a classification of all of its one-dimensional subalgebras is performed. The method of symmetry reduction is systematically applied in order to derive invariant solutions of the supersymmetric model.

#### Tuesday, February 3 2009 at 3:30p.m.

### SUPERINTEGRABILITY WITH SECOND AND THIRD ORDER INTEGRALS OF MOTION, CUBIC ALGEBRAS AND SUPERSYMMETRY

#### Ian Marquette, CRM et département de physique

We consider a superintegrable Hamiltonian system in a two-dimensional space that allows one quadratic and one cubic integral of motion. We construct the most general cubic Poisson algebra generated by these integrals for the classical case. For the quantum case, we construct the most general cubic algebra and we present specific realizations in term of parafermionic algebras. We use them to calculate the energy spectrum. All classical and quantum superintegrable potentials separable in Cartesian coordinates with a third order integral are known. Some of these superintegrable potentials involve PainlevÃ© transcendents and are very interesting. We will also discuss a relation between these potentials and supersymmetric quantum mechanics.

#### Tuesday, February 10 2009 at 3:30p.m.

### CONVOLUTION SYMMETRY FLOWS AND INTEGRABLE HIERARCHIES

#### John Harnad, CRM et Concordia

Flows consisting of generalized convolution symmetries are shown to give rise to an alternative fermionic operator representation of tau-functions for KP-Toda and 2KP-Toda hierarchies. Applications include: generating functions for random plane partitions; computation of overlap amplitudes between electronic coherent states and shifted boundary states; partition functions and correlators for SSYM.

#### Tuesday, February 17, 2009 at 3:30p.m.

### BERGMAN TAU-FUNCTION AND DEGENERATING RAMIFIED COVERINGS

#### Alexey Kokotov, Concordia University

We discuss the spectral theory interpretation of the Bergman tau- function on the moduli space of ramified coverings of the Riemann sphere. We also study the asymptotic behavior of the tau-function when the underlying Riemann surface degenerates to a nodal curve with two irreducible components. It turns out that the tau-function demonstrates the « factorization property » in complete analogy with asymptotic behavior of Faltings’ delta-invariant near the boundary of the moduli space of Riemann surfaces.

#### Tuesday, February 24, 2009 at 3:30p.m.

### SUPERINTEGRABLE SYSTEMS AND THEIR HIDDEN SYMMETRIES

#### Willard Miller Jr., School of Mathematics, University of Minnesota

A classical (or quantum) superintegrable system is an n-D system with potential that admits 2n-1 functionally independent constants of the motion, polynomial in the momenta, the maximum possible. If the constants are quadratic the system is second order. Such systems have remarkable properties, including a quadratic algebra of symmetries whose irreps yield spectral information about the SchrÃ¶dinger operator and deep connections with special functions.

#### Tuesday, March 3 2009 at 4:00 p.m.

#### Salle : Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214

### THE ASYMMETRIC SIMPLE EXCLUSION PROCESS : INTEGRABLE STRUCTURE AND LIMIT THEOREMS

#### Prof. Craig A. Tracy, UC Davis, Chaire Aisenstadt 2008-2009.

Since its introduction by FRANK SPITZER nearly forty years ago, the asymmetric simple exclusion process (ASEP) has become the « default stochastic model for transport phenomena. » Some have called the ASEP the « Ising model for nonequilibrium physics. » In ASEP on the integer lattice Z particles move according to two rules: (1) A particle at x waits an exponential time with parameter one (independently of all the other particles), and then it chooses y with probability p(x,y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x and restarts the clock. † The adjective « simple » refers to the fact that allowed jumps are one step to the right, p(x,x+1)=p, or one step to the left, p(x,x-1)=1-p=q. The asymmetric condition means p ≠ q so that there is a net drift to either the right or the left.

The first lecture will discuss the integrable structure of ASEP on Z. † We show how ideas from Bethe Ansatz can be applied in a novel way to ASEP. The second lecture continues with a discussion of some basic limit theorems which prove KPZ Universality for ASEP.

This work is joint work with Harold Widom.

Coffee will be served at 3:30 pm in Salon Maurice-L’Abbé – Room 6245.

#### Thursday, March 5 2009 at 4:00 p.m.

#### Salle : Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214

### THE ASYMMETRIC SIMPLE EXCLUSION PROCESS : INTEGRABLE STRUCTURE AND LIMIT THEOREMS

#### Prof. Craig A. Tracy, UC Davis, Chaire Aisenstadt 2008-2009.

Since its introduction by FRANK SPITZER nearly forty years ago, the asymmetric simple exclusion process (ASEP) has become the « default stochastic model for transport phenomena. » Some have called the ASEP the « Ising model for nonequilibrium physics. » In ASEP on the integer lattice Z particles move according to two rules: (1) A particle at x waits an exponential time with parameter one (independently of all the other particles), and then it chooses y with probability p(x,y); (2) If y is vacant at that time it moves to y, while if y is occupied it remains at x and restarts the clock. † The adjective « simple » refers to the fact that allowed jumps are one step to the right, p(x,x+1)=p, or one step to the left, p(x,x-1)=1-p=q. The asymmetric condition means p ≠ q so that there is a net drift to either the right or the left.

The first lecture will discuss the integrable structure of ASEP on Z. † We show how ideas from Bethe Ansatz can be applied in a novel way to ASEP. The second lecture continues with a discussion of some basic limit theorems which prove KPZ Universality for ASEP.

This work is joint work with Harold Widom.

Coffee will be served at 3:30 pm in Salon Maurice-L’Abbé – Room 6245.

#### Friday, March 6 2009 at 4:00 p.m.

#### Suitable for a general audience.

#### Room : Pavillon André-Aisenstadt, 2920, ch. de la Tour, Salle / Room 6214

### INTEGRABLE MODELS IN STATISTICAL PHYSICS AND ASSOCIATED UNIVERSALITY THEOREMS AND CONJECTURES

#### Prof. Craig A. Tracy, UC Davis, Chaire Aisenstadt 2008-2009.

This lecture, designed for a general audience, will survey « exactly solvable » models in statistical physics. The three main examples will be the 2D Ising model, Random Matrix Models, and the Asymmetric Simple Exclusion Process. The underlying theme is the connection with integrable differential equations of Painlevé type.

Coffee will be served at 3:30 pm and a reception will follow the lecture in Salon Maurice-L’Abbé – Room 6245.

#### Tuesday, March 10 2009 at 3:30 p.m.

### 4D YOUND DIAGRAMS, QUASICRYSTALS, DIMERS AND FERMIONS

#### Benjamin Young, CRM and McGill University

This is an introduction to a long-standing, difficult problem: describe the set of four-dimensional Young diagrams (solid partitions). I will describe the history of the problem, as well as some work in progress which makes use of the other subjects mentioned in the title.

#### Tuesday, March 17 2009 at 3:30p.m.

### A NEW CONTINUOUS FAMILY OF TWO-DIMENSIONAL EXACTLY-SOLVABLE AND (SUPER) INTEGRABLE SCHROEDINGER EQUATIONS

#### Alexander Turbiner, IHES, Bures-sur-Yvette et Universidad Nacional Autonóma de México, México.

#### Tuesday, March 24 2009 at 3:30p.m.

### DETERMINANTS OF LAPLACIANS ON ELLIPTIC CURVES

#### Yuliya Klochko, Concordia

We study the determinant of the Laplacian as a functional on the space of pairs (X, m), where X is a compact Riemann surface of genus one and m is a conformal flat conical metric on X. We give an explicit expression for this functional. This formula generalizes the well-known Ray-Singer result for a flat smooth torus. We also give a new proof of Troyanov’s theorem stating the existence of a conformal flat conical metric on a compact Riemann surface of arbitrary genus with a prescribed divisor of conical points.

#### Tuesday, March 31 2009 at 3:30 p.m.

### SPECTRUM OF RANDOM HERMITIAN MATRICES WITH A SMALL-RANK EXTERNAL SOURCE

#### Robert Buckhingham

The random Hermitian matrix model with source arises in the study of complex Hamiltonians with an external field. We consider the case when the source has small rank (a finite or sublinear number of nonzero eigenvalues). We establish universality of certain spectral densities for a large family of potentials in the supercritical, subcritical, and critical regimes. This is joint work with Marco Bertola, Seung-Yeop Lee, and Virgil Pierce.

#### Tuesday, April 7 2009 at 3:30p.m.

### INVOLUTIVE DISTRIBUTIONS OF OPERATOR-VALUED EVOLUTIONARY VECTOR FIELDS

#### Arthemy Kiselev, Department of Mathematics, Universiteit Utrecht

#### Tuesday, April 14 2009 at 3:30p.m.

### BERGMAN TAU-FUNCTIoN ON SPACES OF RATIONAL FUNCTIONS AND RELATIONS BETWEEN BOUNDARY DIVISORS OF THIS SPACE

#### Dmitri Korotkin, CRM

Bergman tau-function on Hurwitz spaces (spaces of meromorphic functions on Riemann surfacs) is a section of an appropriate line bundle on a Hurwitz space; this function appears as isomonodromic Jimbo-Miwa tau-function of a class of Riemann-Hilbert problems related to Hurwitz spaces, it also plays an important role in the theory of hermitian matrix models and Frobenius manifolds. In this talk we use the Bergman tau-function to construct a true function on the space of rational functions, which is non-singular and non-vanishing outside the boundary of this space. The boundary consists of several components, called « caustics », « Maxwell stratum » and actual « boundary ». We compute the degree of our function on the different boundary components, which provides a relation between corresponding divisors in the Picard group. These relations turn out to coincide with relations derived some time ago by Lando and Zvonkine.

#### Tuesday, April 21 2009 at 3:30 p.m.

### HAMILTONIANS SEPARABLE IN POLAR COORDINATES WITH A THIRD ORDER INTEGRAL OF MOTION

#### Frédérick Tremblay, CRM

Following the classification achieved in cartesian coordinates by Gravel and Winternitz in 2004, we propose a classification of Hamiltonians that admit separation of variables in polar coordinates and allow the existence of a third order integral of motion in a two-dimensional Euclidean space. Once again, the existence of a third order constants of motion in quantum mechanics implies that some potentials are expressed in term of an elliptic function or Painlevé transcendent. In the classical limit, those potentials are reduced to the free potential or rational function of trigonometric functions.

#### Tuesday, April 28 2009 at 3:30p.m.

### FACTORIZATION OF SECOND-ORDER PARTIAL DIFFERENTIAL OPERATORS, PSEUDOANALYTIC FUNCTIONS AND APPLICATIONS

#### Vladislav V. Kravchenko, Dept of Mathematics, CINVESTAV del IPN, Mexique

Given a particular solution of a linear second-order ordinary differential equation, this equation can be written in a factorized form leading to its general solution. In the case of a linear second-order partial differential equation (pde) the knowledge of a particular solution in general does not offer much information on a general solution of the equation. The aim of this talk is to present a new technique which makes it possible under quite general conditions to obtain explicitly infinite and even complete systems of solutions of linear second-order pde’s when a particular solution of the equation is available. The technique is based on a factorization of the pde and application of pseudoanalytic function theory.

#### Tuesday, May 5 2009 at 3:30 p.m.

### DISCRETIZATION OF TORI OF COMPACT SIMPLE LIE GROUPS

#### Jiri Hrivnak, CRM

For a given compact simply connected simple Lie group G and a positive integer M, we describe a certain finite set of lattice points F_M. The set F_M is a subset of the Lie algebra of the maximal torus of G. The group G and corresponding affine Weyl group induce the symmetry of the grid F_M, the number M specifies the density of F_M. We review a construction of the set F_M and present counting formulas for the numbers of its points |F_M|. The relation between the numbers |F_M| and numbers of elements of finite order in G is discussed. We recall the definition of C- and S-functions and study their properties on F_M. We describe the maximal sets of pairwise orthogonal C- and S- functions. These finite sets allow us to calculate Fourier like discrete expansions of arbitrary discrete functions. Application of these discrete transforms to interpolation is presented.