Marco Bertola (Concordia University, Montreal): "Special structures in integrable systems associated with poles of Painleve transcendents: semiclassical Focusing Nonlinear Schroedinger equation and Orthogonal Polynomials"

In the semiclassical ($\hbar \to 0$) limit of the fNLS equation, known to be modulationally unstable, it is known that for certain initial data a transition in the asymptotic behavior occurs at a certain point in space-time; I will explain how certain structures of scale $\hbar$ appear in the neighborhood of this point, within spatiotemporal scale $\hbar^{\frac 4 5}$ and how they are related to the location of poles of a certain solution of the first Painleve equation. A very similar phenomenon occurs in the asymptotic study of certain (non Hermitean) orthogonal polynomials appearing in the formal models of 2D quantum gravity. I will try to give a sense of the common thread between these two seemingly distant problems and -time permitting- an idea of the ingredients of the proof.

Andrea Brini (Imperial College London): "A gentle introduction to the Gromov-Witten/Integrable Systems correspondence"

This lecture is intended as an invitation the Gromov-Witten/Integrable Systems correspondence, which relates intersection numbers on moduli spaces of stable maps to tau functions of 1+1 dimensional integrable hierarchies, intended mainly for a non-specialist audience. I will introduce and discuss some aspects of the topology of moduli spaces of stable morphisms as well as their conjectural relation to classical integrable hierarchies (after Witten, Dubrovin, and Dubrovin-Zhang). I will also try to highlight some recent promising directions to further this recent program in the context of the local theory of Calabi-Yau threefolds.

Mattia Cafasso (Universite de Angers): "Random Point processes and Riemann-Hilbert problems"

Dyson, in 1962, described how to implement a dynamics into random matrices in such a way that the eigenvalues of the matrix behave like finitely many non-intersecting Brownian motions on the real line. In a suitable scaling regime we are lead to the study of certain interesting time-dependent determinantal random point processes generalizing the typical probability distributions appearing in random matrices. The Airy process, in particular, is relevant for the study of the models in the celebrated Kardar-Parisi-Zhang universality class. We will show how, as it is well known for the case of random matrices, also in the case of Dyson's non-intersecting Brownian motions the (joint) gap probability is equal to the tau function associated to a suitably chosen isomonodromic system of ordinary differential equations. The main tool we will use is the theory of integrable operators a la Its-Izergin-Korepin-Slavnov, hence the connection with Riemann-Hilbert problems. This result has been obtained in a joint work with M. Bertola.

Andrea Raimondo (SISSA Trieste): Hamiltonian perturbations of quasilinear PDEs: the Dubrovin's universality conjecture

The aim of this lecture is to present the conjecture, proposed by B. Dubrovin, concerning the analytic description of critical behaviour for solutions to Hamiltonian PDEs, near the points of phase transition from regular to oscillatory regime. The PDEs considered can be thought as Hamiltonian (dispersive) perturbations of first order quasilinear conservation laws (dispersionless systems). According to the conjecture, generic solutions to a large class of Hamiltonian PDEs manifest an integrable behavior at the phase transition points, described by certain Painleve transcendents. In the lecture, I will particularly focus on the one component case, providing an explicit derivation of the conjecture and stating the rigorous results obtained so far.

Paolo Rossi (Universitaet Zurich): "Hamiltonian Systems from Symplectic Field Theory"

As Boris Dubrovin has remarked in more than one occasion, Symplectic Field Theory [Eliashberg, Givental, Hofer] provides an approach to Hamiltonian systems from holomorphic curves which is both more general and more direct than Gromov-Witten theory. In this series of lectures we will review the construction which leads from holomorphic curves inside cylindrical symplectic manifolds to infinite dimensional Hamiltonian systems and we will describe what is known about the properties of such Hamiltonian systems. We will in particular focus on how much of the structure of a Frobenius manifold survives in situations that are more general than Gromov-Witten theory.

Gaetan Borot (Universite de Geneve): "Integrability, non-perturbative topological recursion, and an application to knot theory"

Given a compact Riemann surface S, the algebro-geometric construction of Krichever (77') associates an integrable system whose times are coordinates on the space of meromorphic forms on S. A interesting question is to build a "dispersionful" deformation of this system, where S is allowed to evolve slowly with the times. We have proposed a candidate for the Tau function and the wave function of such a system, based on a kind of Whitham averaging of the partition function obtained from the topological recursion. The aim of this talk is to present this construction, and describe a conjectural application to knot theory, namely to the computation of the all-order asymptotics of the Jones polynomial of hyperbolic knots. This is based on joint works with Bertrand Eynard.

Aleksandr Buryak (Universiteit van Amsterdam): Integrals of $\Psi$-classes over double ramification cycles

Double ramication cycles are certain codimension g cycles in the moduli space Mg;n of stable genus g curves with n marked points. They have proved to be very useful in the study of the intersection theory of Mg;n. In my talk I will explain that integrals of arbitrary monomials in -classes over double ramification cycles have an elegant expression in terms of vacuum expectations of certain operators that act in the innite wedge space. The talk is based on a joint work with S. Shadrin, L. Spitz and D. Zvonkine

Tom Claeys (UC Louvain): Higher order analogues of the Tracy-Widom distribution: analytic and numerical results

We study a family of distributions that arise in critical unitary random matrix ensembles. They are expressed as Fredholm determinants and describe the limiting distribution of the largest eigenvalue when the dimension of the random matrices tends to infinity. The family contains the Tracy-Widom distribution and higher order analogues of it, which can be expressed in terms of the second Painlev\'e hierarchy. We study analytic properties of the distributions, show plots of them, and discuss several properties that they appear to exhibit.

Maurice Duits (KTH Stockholm): "Critical phenomena in the two-matrix model with one quartic potential"

After a short introduction to the Hermitian two matrix model, I will report on some recent joint work with Dries Geudens on a new critical phenomenon appearing in the two matrix model with one quartic and one quadratic potential. If time permits, I will also discuss a connection with a critical phenomenon appearing near a tacnode in a model of non-intersecting paths.

Antonio Moro (SISSA Trieste): "Thermodynamic phase transitions and shock singularities"

We show that, under rather general assumptions on the form of the entropy function, the energy balance equation for a system in thermodynamic equilibrium is equivalent to a set of nonlinear equations of hydrodynamic type. This set of equations is integrable via the method of characteristics and it provides the equation of state for the gas. The shock wave catastrophe set identifies the phase transition. A family of explicitly solvable models of non-hydrodynamic type such as the classical plasma and the ideal Bose gas is also discussed. This is a joint work with Giuseppe De Nittis.

Nicolas Orantin (IST Lisbon): "A proof of the Bouchard-Klemm-Marino-Pasquetti conjecture"

Stefano Romano (SISSA Trieste): "4-dimensional Frobenius manifolds and Painleve VI"

Following a work of Pavlov and Tsarev, I describe a class of even-dimensional semisimple Frobenius manifold possessing a particular symmetry, which results in the existence of a third compatible flat metric. Several concrete examples can be constructed within the framework of Hurwitz spaces, where the additional symmetry has a simple geometric interpretation. I focus on the simplest nontrivial case, where the dimension is 4, and establish a correspondence between manifolds in the above class and arbitrary semisimple 3-dimensional Frobenius manifolds: in both cases the WDVV equations are reduced to (a special version of) PVI. This leads to an explicit procedure recostructing the 4-dimensional manifold from 3-dimensional data.

Daniele Sepe (IST Lisbon): Integral affine geometry of non-commutatively integrable Hamiltonian systems

In mechanics there are many Hamiltonian systems which admit more integrals of motion (with suitable properties) than degrees of freedom (e.g. the Euler top). These systems are known as superintegrable or non-commutatively integrable and have been studied extensively since the pioneering work of Nekhoroshev, Mischenko and Fomenko. In this talk, we will give an overview of how to classify (topologically and symplectically) and how to construct (in some sense) these systems using integral affine geometry, which studies manifolds equipped with an atlas whose changes of coordinates are affine transformations of Euclidean space whose linear part is invertible over the integers. Time permitting, we shall illustrate how these questions relate to some problems in Poisson geometry

Yang Shi (The University of Sydney): Isomonodromy deformation for discrete Painleve equations

We present a new method of deducing infinite sequences of exact solutions of $q$-discrete Painleve equations by using their associated linear problems. The specific equation we consider here is a $q$-discrete version of the second Painleve equation ($q$-P$_{\rm {II}}$) with affine Weyl group symmetry of type $(A_2+A_1)^{(1)}$. We show, for the first time, how to use the $q$-discrete linear problem associated with $q$-P$_{\rm {II}}$ to find an infinite sequence of exact rational and hypergeometric type solutions and also show how to find their representation as determinants by using the linear problem. The method, while demonstrated for $q$-P$_{\rm {II}}$ here, is also applicable to other discrete Painlev\'e equations.