Contemporary Ways of Integrability
Lectures

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 spacetime; 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
GromovWitten/Integrable Systems correspondence"
This lecture is intended as an invitation the GromovWitten/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 nonspecialist 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 DubrovinZhang). I will also try to highlight some recent promising
directions to further this recent program in the context of the local theory of
CalabiYau threefolds.

Mattia Cafasso (Universite de Angers): "Random Point processes and
RiemannHilbert 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 nonintersecting Brownian motions on the real line. In a
suitable scaling regime we are lead to the study of certain interesting
timedependent 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
KardarParisiZhang universality class. We will show how, as it is well
known for the case of random matrices, also in the case of Dyson's
nonintersecting 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 ItsIzerginKorepinSlavnov, hence the connection
with RiemannHilbert 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 GromovWitten 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 GromovWitten theory.
Talks

Gaetan Borot (Universite de Geneve): "Integrability, nonperturbative topological recursion, and an
application to knot theory"
Given a compact Riemann surface S, the algebrogeometric 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 allorder 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 ramication 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 innite 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 TracyWidom 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 TracyWidom 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 twomatrix 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 nonintersecting 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 nonhydrodynamic 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
BouchardKlemmMarinoPasquetti conjecture"

Stefano Romano (SISSA Trieste): "4dimensional Frobenius manifolds and
Painleve VI"
Following a work of Pavlov and Tsarev, I describe a class of evendimensional 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 3dimensional Frobenius manifolds: in both cases the WDVV equations are reduced to (a special version of) PVI. This leads to an explicit procedure recostructing the 4dimensional manifold from 3dimensional data.

Daniele Sepe (IST Lisbon): Integral affine geometry of noncommutatively 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 noncommutatively 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.