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# Working Seminar on Integrable Systems, Random Matrices, and Random Processes

## Fall 2008 – Winter 2009

#### Time/Date/Heure: Tuesday, September 16; mardi le 16 septembre 2008, at / à 3:30 p.m. / 15h30 Title/titre: « Dimer models and Donaldson Thomas Invariants » Speaker/Conférencier: Benjamin Young (McGill and CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Donaldson-Thomas theory is a « curve-counting » theory: it gives a virtual count of subschemes of a Calabi-Yau threefold X. If X should happen to be a toric threefold, then the computation reduces to the combinatorics of three-dimensional Young diagrams placed at the toric fixed points of X.
A similar phenomenon happens when computing « non-commutative » Donaldson-Thomas invariants associated to the path algebra of certain quivers; there, the problem reduces to the dimer model on the square lattice.
In this first working seminar, I’ll review the combinatorial tools one can use to deal with these objects: the dimer model, height functions, vertex operators, etc.; I will also to address their relationship with Donaldson-Thomas theory. (to be continued)

#### Time/Date/Heure: Tuesday, October 21; mardi le 21 octobre 2008, at / à 12:30 p.m. / 12h30 Title/titre: « Vertex operators for combinatorics » Speaker/Conférencier: Benjamin Young (McGill and CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

I will give an overview of vertex operators and the infinite wedge formalsim, with an application to enumeration of 3D Young diagrams or pyramid partitions. The textbook is the collected appendices of A. Okounkov’s papers.

#### Time/Date/Heure: Tuesday, October 25; mardi le 25 octobre 2008, at / à 12:30 p.m. / 12h30 Title/titre: « PNG droplet and airy process: part 2 » Speaker/Conférencier: Dong Wang (CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

This is an introduction to the paper « Scale invariance of the PNG droplet and the Airy Process » by M. Praehofer and H. Spohn.

#### Time/Date/Heure : Tuesday, December 9; mardi le 9 octobre 2008 at / à : Part I : 11:00 a.m. / 11h – Part II : 12:30 a.m. / 12h30 Title/titre : « Quantum subgroups of Lie groups and modular invariance in conformal field theories » Speaker/Conférencier: Robert Coquereaux (Directeur de recherche, CNRS. CPT, Luminy-Marseille) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

« 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. »

#### Time/Date/Heure : Friday, January 9; vendredi le 9 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Hermitian matrix model with external source » Speaker/Conférencier: Seung-Yeop Lee, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

« I will introduce my on-going project with Robert Buckingham and Virgil Pierce about the hermitian matrix model with external source. I will describe the phenomena of « jumping outliers. »

#### Time/Date/Heure : Friday, January 16; vendredi le 16 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Tensor networks in graph combinatorics » Speaker/Conférencier: Peter Zograf Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

A tensor network is an assignment of a tensor to each vertex of a graph together with a one-to-one correspondence between tensor indices and half-edges incident to the vertex. We will show how tensor networks can be applied to some of the well-known combinatorial problems like the dimer problem, edge coloring, the Hamiltonian cycle problem, etc.

#### Time/Date/Heure : Friday, January 23; vendredi le 23 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « KdV equation and computational geometry of moduli spaces of curves » Speaker/Conférencier: Peter Zograf Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

We present a fast algorithm for computing intersection numbers on moduli spaces of complex algebraic curves in terms of the coefficients of special solutions of the KdV equation. As an application, we derive conjectural large genus asymptotics of these numbers (in particular, Weil-Petersson volumes).

#### Time/Date/Heure : Friday, January 29; vendredi le 29 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Convolution symmetries of integrable hierarchies, matrix models and tau-functions (Part 3) » Speaker/Conférencier: John Harnad Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Les symétries convolutives généralisées des hiérarchies intégrables du type KP-Toda et 2KP-Toda ont l’effet de multiplier les coefficients de Fourier de la fonction de Baker-Akhiezer par une suite de constantes donnée. L’action induite sur l’espace de Fock fermionique associé est diagonale dans la base orthonormale standard déterminée par les sites occupés et étiquetés par les partitions. Les coefficients dans les développements simples et doubles en fonctions de Schur de la fonction-tau associée, qui sont les coordonnées de Pluecker d’un élément décomposable, sont multipliés par des facteurs diagonaux correspondants. Appliquant de telles transformations aux intégrales de matrices, nous obtenons des nouveaux modèles de matrices du type extérieurement couplé qui sont également des fonctions-tau KP-Toda ou 2KP-Toda. Des représentations intégrales multiples plus générales des fonctions-tau sont également obtenues, aussi bien que des expressions déterminantales finies pour elles.

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 (e.g. determinental correlation functions). (The language of the presentation will determined at the time of the lecture.)

Abstract

#### Time/Date/Heure : Friday, February 13; vendredi le 13 février 2009 at / à 2:00 p.m. / 14h Title/titre : « On (iso)monodromic deformations » Speaker/Conférencier: Marco Bertola, Concordia Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Given a system of first order ODEs in the complex plane with rational coefficients one can (univocally) associate other data, which I will call « Birkhoff data ». They consist in the « isomonodromic times » and the « generalized monodromy data », namely, the monodromy representation of the ODE together with the datum of the Stokes’ matrices.
Solving this forward problem amounts to solving the ODE. The inverse problem consists in reconstructing the ODE (or its solution) starting from the Birkhoff data: this is an inherently harder problem, which involves the solution of a Riemann–Hilbert problem (i.e. a singular-integral equation).
It is known that for generic Birkhoff data, the inverse problem admits a (unique) solution; however there are special data, called the « Malgrange Theta divisor » where this inversion problem fails.
The invertibility of the problem hinges on the invertibility of a suitable Toeplitz operator and hence the Malgrange divisor could be thought of as the locus of data where the « determinant » of the operator vanishes (with the nasty detail that the operator in question does not have a notion of determinant).
In a parallel approach, in the eighties, Jimbo-Miwa-Ueno gave a computable definition of a closed differential of the isomonodromic times which involved purely algebraic manipulations of the coefficients of the ODE. The (locally defined) function with this differential has the property that it vanishes precisely and only on the Malgrange divisor, as proved by Malgrange for the case of Fuchsian systems and Palmer for systems with irregular singularities.
The goal of this seminar is to show how to transform JMU’s differential into an object more closely related to the Riemann–Hilbert problem and hence extend in a natural way its definition so as to allow us to compute the derivatives of the tau function with respect to the {\em monodromy} data (Stokes etc.)
I will try to give a no-nonsense approach with details on the construction and properties of the Birkhoff data.
Some applications to random matrices and Painleve’ equations will be outlined.
The seminar is based on ongoing research.

#### Time/Date/Heure : Friday, February 20; vendredi le 20 février 2009 at / à 2:00 p.m. / 14h Title/titre : « Riemann-Hilbert problems associated to Frobenius manifold structures on Hurwitz spaces. » Speaker/Conférencier: Vasilisa Shramchenko Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

There are two (dual to each other) Riemann-Hilbert problems naturally associated to every Frobenius manifold. In the case of Frobenius structures on Hurwitz spaces, the corresponding Riemann-Hilbert problems turn out to be solvable in terms of meromorphic bidifferentials defined on the underlying surface. In this talk, I will present these solutions and discuss their monodromy groups. The monodromy transformations for both solutions are related to the monodromy in the appropriate spaces of contours on the Riemann surface.

#### Time/Date/Heure : Friday, February 27; vendredi le 27 février 2009 at / à 2:00 p.m. / 14h Title/titre : « Riemann–Hilbert problems: Schlesinger and Sato » Speaker/Conférencier: Marco Bertola Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

Given any (sufficiently well-behaved) family of Remann–Hilbert problems where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form $\Omega$ on the deformation space (Malgrange’s differential).
Such a one–form has a pole where the deformation family meets the Malgrange Theta divisor, namely, the set of unsolvable RHP. I will try to review this fact reading through Malgrange and Palmer’s papers.
The differential $\Omega$ fails to be closed in general, but when it does the formula $\tau := {\rm e}^{\int \Omega}$ defines locally a function that vanishes precisely on $\Theta$.
We then introduce the notion of Schlesinger discrete transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Interestingly, even if $\Omega$ is not closed, the difference of $\Omega$ evaluated along solution of the original RHP and the Schlesinger transformed RHP is closed off $\Theta$; in fact, such difference is shown to be the logarithmic differential (on the deformation space) of a function -say- $H$.
I will show how this function $H$, as a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker–Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector.

#### Time/Date/Heure : Friday, March 6; vendredi le 6 mars 2009 at / à 2:00 p.m. / 14h Title/titre : « Asymptotic analysis of random Hermitian matrices with a small-rank external source. » Speaker/Conférencier: Robert Buckingham Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

We will discuss ongoing work on the behavior of the eigenvalues of a large Hermitian matrix drawn from a random ensemble with an external source. We will consider three cases depending on the strength of the source, refered to as supercritical, subcritical, and critical. This is joint work with Seung Yeop Lee and Virgil Pierce and is a continuation of a previous talk by Lee earlier this semester.

#### Time/Date/Heure : Friday, March 20; vendredi le 20 mars 2009 at / à 2:00 p.m. / 14h Title/titre : « A (conjectural) PDE satisfied by the gap probability of the Gaussian matrix model with external source. » Speaker/Conférencier: Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

In this talk I show an attempt to apply the Adler-van Moerbeke method on the Gaussian matrix model with external source.

#### Time/Date/Heure : Friday, March 27; vendredi le 27 mars 2009 at / à 2:00 p.m. / 14h Title/titre :  » r-Airy parametrix in Hermitian matrix model with external source » Speaker/Conférencier: Seung-Yeop Lee, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

In this talk, I consider the hermitian matrix model with external source at the criticality. It is already known that, for Gaussian potential, the kernel is given by r-Airy kernel. We derive this through 3 by 3 Riemann-Hilbert problem hence the result applies to more general potentials. This is work in progress with Marco Bertola, Robert Buckingham and Virgil Pierce.Â

#### Time/Date/Heure : Friday, April 3; vendredi le 3 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Rank 1 real external source model » Speaker/Conférencier: Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

In this talk I will explain how a trick (partially) solves the rank 1 real external source model. The trick resembles the convolution symmetry.

#### Time/Date/Heure : Tuesday, April 7; Mardi le 7 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Rank 1 real external source model (Part II) » Speaker/Conférencier : Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

In this talk I will explain how a trick (partially) solves the rank 1 real external source model. The trick resembles the convolution symmetry.

#### Time/Date/Heure : Friday, April 10; vendredi le 10 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation » Speaker/Conférencier: Alexander G. Abanov, Stony Brook University Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

Calogero-Sutherland model is the model of particles in one-dimension interacting through the inverse square potential. Both classical and quantum versions of the model are known to be integrable. I consider the hydrodynamic description of this model. The hydrodynamic equations derived in the limit of infinitely many particles are also integrable. They present a bidirectional analogue of classical Benjamin-Ono equation which first appeared as an equation for interface waves in deep stratified fluids. I discuss hydrodynamic soliton solutions and dispersive shock waves for this equation.

#### Time/Date/Heure : Friday, April 17; vendredi le 17 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Multiple orthogonal polynomials and nonlinear automorphisms of the Heisenberg-Weyl algebra. » Speaker/Conférencier: Alexei Zhedanov, Donetsk Institute for Physics and Technology Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.

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