Mathematical Physics Laboratory

Université de Montréal

Pavillon André Aisenstadt, 2920, ch. de la Tour

Salle/Room 4336

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

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)

Title/titre: "Dimer models and Donaldson Thomas Invariants (Part II)"

Speaker/Conférencier: Benjamin Young (McGill and CRM)

Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

Title/titre: "Dimer models and Donaldson Thomas Invariants (Part III)"

Speaker/Conférencier: Benjamin Young (McGill and CRM)

Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

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

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

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

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

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

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

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

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.)

Title/titre : "Integrable Lagrangians and modular forms"

Speaker/Conférencier: Alexander Odesski, Brock University

Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336

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

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

Working seminar on Integrable Systems, Random Matrices, Random Processes: Spring 2008

For further information: *bertola@CRM.UMontreal.CA*
or *anguelov@CRM.UMontreal.CA*