-3 hours lecture

Speaker: Marco Bertola (Concordia University & SISSA/ISAS)
Title: Tau functions for isomonodromic systems and general Riemann-Hilbert problems: Theory and applications
Abstract: In this three hour course I will explain the main theory and properties of the above mentioned tau function. These special functions are in a certain sense a generalization of Riemann Theta functions; they can be also thought of as (regularized) determinants. In certain cases they are genuine (finite or infinite-dimensional) determinants.
As such their vanishing determines the obstruction to the solvability of a certain linear problem.
The tau function was originally defined for isomonodromic systems (e.g. the associated linear systems for the Painleve equations) but then the definition can be extended to more general Riemann-Hilbert problems; this extension does not require a linear ODE. Even in the case of the "isomonodromic" tau function, this extension allows to determine consistently the dependence on the (generalized) monodromy data. I will explain various applications to
1. integrable systems (KdV, Toda, KP)
2. Witten-Kontsevich special tau function and intersection numbers; generating functions.
3. Gap probabilities (Tracy-Widom and generalizations thereof); Fredholm determinants.
4. (Multi)-Orthogonal polynomials and Hankel/Toeplitz determinants. Random Matrices.

Speaker: Davide Guzzetti (SISSA)
Title: Monodromy preserving deformations and Painleve' equations
Abstract: The purpose of this 3-hours short course is to introduce the audience to the theory of monodromy preserving deformations and its application to Painleve' equations. This is a well established topic.
- I will first review the notion of monodromy data of a linear differential system, then I will explain the standard theory of monodromy preserving deformations of linear systems with local holomorphic dependence on parameters, according to Jimbo-Miwa-Ueno's original work.
- The isomonodromic dependence on parameters in encoded into non-linear PDEs, such as the Schlesinger's equations, whose solutions enjoy the Painleve' property. I will give examples related to the classical Painleve' equations. Painleve' transcendents play a central role in pure mathematics and mathematical physics, with applications in a variety of problems, such as number theory, theory of analytic varieties (like Frobenius structures), random matrix theory, orthogonal polynomials, non linear evolutionary PDEs, combinatorial problems, etc. I will focus on the sixth Painleve' equation, reviewing results obtained by the monodromy preserving deformations method, including some recent ones.

Speaker: Dmitry Korotkin (Concordia University)
Title 1: Riemann-Hilbert problem with quasi-permutation monodromies
Abstract 1: An arbitrary Riemann-Hilbert problem with quasi-permutation monodromy matrices is solved in terms of Szeg"o kernel on a branched covering. The isomonodromic tau-function is computed via theta-functions and Bergman tau-function on Hurwitz spaces.

Title 2: Bergman tau-function as higher genus analog of Dedekind's eta-function
Abstract 2: Bergman tau-function is defined on Hurwitz spaces and moduli spaces of holomorphic and quadratic differentials. Study of analytical properties of the Bergman tau-function leads to various (old and new) relations between tautological classes on these moduli spaces.
The lecture is based on joint works with M.Bertola, A.Kokotov and P.Zograf.

Title 3: Symplectic aspects of second order differential equation on a Riemann surface
Abstract 3: We study symplectic properties of monodromy map of second order linear equation with meromorphic potential having only simple zeros  on a Riemann surface. We show that the canonical symplectic structure on the cotangent bundle $T^*{{mathcal M}}_g,n}$ implies under an appropriately defined monodromy map the Goldman Poisson structure on the corresponding character variety.
The talk is based on joint work with M.Bertola and C.Norton.

Speaker: Kanehisa Takasaki (Kinki University)
Title: Integrable hierarchies in melting crystal models and topological vertex
Abstract: The melting crystal models are statistical models of random partitions that stems from the instanton sum of a 5D supersymmetric gauge theory. The topological vertex is a diagrammatic method for constructing the partition functions, or amplitudes, of topological string theory on non-compact toric Calabi-Yau threefolds.  I will review aspects of integrable hierarchies hidden in these combinatorial models of mathematical physics.  Main tools are Schur functions, free fermions, a 2D quantum torus algebra and quantum-dilogarithmic functions. 
The lectures comprise three parts. In the first part, I will introduce the melting crystal models and underlying integrable structures. A technical clue here is a set of algebraic relations, called "shift symmetries'', in the quantum torus algebra.  The relevant integrable structures turn out to be the 1D Toda hierarchy and the Ablowitz-Ladik (or relativistic Toda) hierarchy. 
In the second part, I will briefly explain the method of topological vertex, and show a few cases where open string amplitudes can be computed in a closed form.  Generating functions of these amplitudes turn out to be a tau function of the KP or 2D Toda hierarchy. Shift symmetries again play a crucial role. 
The third part deals with "quantum spectral curves'' of this type of KP tau functions. These non-commutative curves are closely related to the notion of the Kac-Schwarz operators.  q-difference analogues of the Kac-Schwartz operators emerge in the case of the open string amplitudes. 
The contents of these lectures are based on joint work with Toshio Nakatsu.

-1 hour lecture

Speaker: Alexey Kokotov (Concordia University)
Title: DtN isospectrality, flat metrics with non-trivial holonomy and comparison formula for determinants of Laplacian
Abstract: We study comparison formulas for ζ-regularized determinants of self-adjoint extensions of the Laplacian on flat conical surfaces of genus g≥2. The cases of trivial and non-trivial holonomy of the metric turn out to differ significantly. This is the joint work with Luc Hillairet (Orleans).