Mathematical Physics Laboratory

Centre de recherches math´┐Żatiques

Atelier sur les matrices aléatoires, déformations
isomonodromiques et problèmes de Riemann-Hilbert
Working Seminar on Random Matrices, Isomonodromic Deformations and Riemann-Hilbert Problems

hiver/Winter 2006

Concordia University
Library Building, 1400 de Maisonneuve Blvd. West
LB 921-04 (Ninth floor meeting room)

Time/Date/Heure: Thursday/jeudi, Jan. 19, le 19 jan. at/à 09:30-10:30
Title/titre: "The inverse spectral problem for arbitrary rational Lax matrices"
Speaker/Conférencier: Marco Bertola (Concordia and CRM)

We provide an extension of the old Baker-Akhiezer approach in order to give explicit formulae for the entries of an arbitrary rational Lax matrix knowing certain spectral data.

Time/Date/Heure: Thursday/jeudi, Jan. 26, le 26 jan. at/à 09:30-10:30
Title/titre: "Determinants of Laplacians, moduli spaces of Riemann surfaces and Abelian differential"
Speaker/Conférencier: Alexey Kokotov (Concordia)

We generalize the well-known Ray-Singer formula for the determinant of the Laplacian on elliptic surfaces to the case of compact Riemann surfaces of higher genus. To illustrate the potential applications of this result we study the determinant of the Laplacian as a Morse function on the moduli space of Riemann surfaces of genus two. Its stationary points are identified as surfaces with large groups of automorphisms. As a corollary we calculate the orbifold Euler characteristics of the moduli space and its symmetric strata.

Time/Date/Heure: Thursday/jeudi, Feb. le 2 fév. at/à 09:30-10:30
Title/titre: "Remarks on complex geometry of 3-monopole"
Speaker/Conférencier: Victor Enolski (CRM, Concordia)

We develop the Ercolani-Sinha construction of SU(2) monopoles and make this effective for (a five parameter family of centred) charge 3 monopoles. In particular we show how to solve the transcendental constraints arising on the spectral curve. For a class of symmetric curves the transcendental constraints become a number theoretic problem and a recently proven identity of Ramanujan provides a solution. The Ercolani-Sinha construction provides a gauge-transform of the Nahm data. The full text of the article the article is given in ArXiv: math-ph/0601040

Time/Date/Heure: Thursday/jeudi, Feb. 9, le 9 fév. at/à 09:30-10:30
Title/titre: "Effective integration of the Nonlinear Vector Schrödinger Equation"
Speaker/Conférencier: Victor Enolski (CRM, Concordia)

A comprehensive algebro-geometric integration of the two component Nonlinear Vector Schrödinger equation (Manakov system) is developed. The associated spectral variety is a trigonal Riemann surface, which is described explicitly and the solutions of the equations are given in terms of theta-functions of the surface. The final formulae are effective in the sense that all entries, like transcendental constants in exponentials, winding vectors etc. are expressed in terms of prime forms on the curve and well algorithmized operations on them. This makes the result available for direct calculations in applied problems implementing the Manakov system. The simplest solutions in terms of Jacobian theta-functions are given as particular case of general formulae and discussed in detail.
(Based on collaborative work with A.Its and J.Elgin.)

Time/Date/Heure: Thursday/jeudi, Feb. 16, le 16 fév. at/à 09:40-11:00
Title/titre: "Isomonodromic equations satisfied by biorthogonal polynomials, integral representations and duality"
Speaker/Conférencier: John Harnad (CRM, Concordia)

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.
(Based on collaborative work with M. Bertola and A.Its.)

Time/Date/Heure: Thursday/jeudi, March 2, le 2 mars at/à 09:40-11:00
Title/titre: What is a Dynamical R-Matrix?
Speaker/Conférencier: Prof Axel Winterhalder (Universidade Estadual do Maranhao, Brazil)

R-matrices have their origins in the theory of integrable models. They found an adequate mathematical interpretation in the theory of Hopf algebras and quantum groups where they play a pivotal role in the development of this area. In our talk we present recent results about an extension of the concept of an ordinary R-matrix, 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.

Time/Date/Heure: Thursday/jeudi, March 9, le 9 mars at/à 09:40-11:00
Title/titre: The inverse asympto-monodromic method: asymptotics of orthogonal polynomials via Strebel, Stokes, Deift--Zhou and Its (Part I.)
Speaker/Conférencier: Marco Bertola
(Concordia and CRM)

This (informal) talk/series of talks has three parts. In a previous talk we applied the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials. Using purely geometrical arguments we showed heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve, thus reproducing from "first principles" formulae appearing in the works of Deift et al. on the subject. In the second part we show that given a (possibly nodal) hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part. In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeros may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.

Time/Date/Heure: Thursday/jeudi, March 16 le 16 mars at/à 09:40-11:00
Title/titre: The inverse asympto-monodromic method, part II: Boutroux admissible curves and Stokes--Kirchoff differentials
Speaker/Conférencier: Marco Bertola
(Concordia and CRM)

We derive the properties that are necessary for the Deift--Zhou--Its steepest descent method in the general setting of pseudo-semiclassical orthogonal polynomials. We introduce the Stokes--Kirchoff differentials as a tool to implement the Stokes parameters. They form a (nonlinear) manifold of dimension g in the space of certain meromorphic differentials.

Time/Date/Heure: Thursday/jeudi, March 23 le 23 mars at/à 09:40-11:00
Title/titre: The inverse asympto-monodromic method, part III: Boutroux admissible curves and Stokes--Kirchoff differentials
Speaker/Conférencier: Man Yue Mo (CRM)

Using the properties of admissible Boutroux curves we define the orthogonal polynomials whose asymptotic data are given by the chosen curve and an arbitrarily chosen Kirchoff--Stokes differential. Near a turning point with three branchcuts a new parametrix (still of Airy type) is introduced.

Time/Date/Heure: Thursday/jeudi, March 30 le 30 mars at/à 11:00-12:00 (N.B. Time change exceptionally for this date)
Title/titre: The inverse asympto-monodromic method, part IV: reconstruction of arbitrary admissible Boutroux curves from a forest of decorated trivalent loop-free trees using Strebel weldings
Speaker/Conférencier: Marco Bertola (Concordia and CRM)

We show that for an arbitrary forest of decorated trivalent loop-free trees (to be defined in the lecture) we can reconstruct an admissible Boutroux curve. This provides several surprising explicit examples in the theory of asymptotics of orthogonal polynomials where their zeroes become dense on the given forest of Jordan arcs.

Time/Date/Heure: Thursday/jeudi, April 13 le 13 avril at/à 10:30-12:00 (N.B. Time change)
Title/titre: Determinant of Laplacians and Schottky groups
Speaker/Conférencier: Andrew McIntyre (Concordia and CRM)

My intention is to explain some work on determinant of Laplacian and Schottky groups in more detail than before. In particular:

1. Schiffer and Bergman kernels; Zograf's product formula
2. Generalization of Riemann normalization of abelian differentials to higher order differentials
3. Variational formula for det laplacian on higher order differentials in "fermionic" form
Item 2 is self-contained and may be of independent interest.

Time/Date/Heure: Thursday/jeudi, April 20 le 20 avril at/à 10:30-12:00 (N.B. Time change.)
Title/titre: On the connection of free fermions and integrable hierarchies: the case for quantum dynamics
Speaker/Conférencier: Paul Wiegmann, James Frank Institute, Enrico Fermi Institute, Univ. of Chicago

The connection between fermionic systems and integrable equations goes back to the seminal work of Date, Jimbo, Kashiwara and Miwa in 1983. There the tau-function of the KP hierarchy was represented as a matrix element of a coherent state evolving under the action of a current algebra. The generators of the current algebra are $J_k=sum_n c^dag_{n+k}c_n$ where $c_k$ are fermionic modes. Despite numerous applications of this theory, physical problems require the study of evolution of fermionic systems in real time, i.e, under the action of Hermitian Hamiltonians $H_k=sum_n n^k c^dag_{n}c_n$ rather than non-Hermitian current operators $J_k$. The first three Hamiltonians are simply chemical potential, momentum, energy. Remarkably this evolution in real time is also connected to integrable hierarchies, but in a subtle manner. In this talk I describe this connection and a few physical applications based on recent, yet unpublished works of A. Abanov, E. Bettelheim and myself.

Time/Date/Heure: Thursday/jeudi, April 27 le 27 avril, at/à 10:30-12:00 (N.B. Final seminar in this semester)
Title/titre: Schur functions expansions of algebro-geometric tau functions
Speakers/Conférenciers: Viktor Enolskii (Kiev, CRM and Concordia) and John Harnad (Concordia and CRM)

This will be a two-part talk:

Part I. (J.H.) The Segal-Wilson Grassmannian, determinantal line bundles and Schur function expansions in general

Part II. (V.E.) The Schur function expansion for tau functions of genus g=1 (Weierstrass), 2 (Baker) and, for trigonal curves of g=3 (Enolskii et al)

Working seminar on random matrices, isomonodromic deformations and iemann-Hilbert Problems: Fall 2005

For further information: bertola@CRM.UMontreal.CA