**ABSTRACTS**

** **

**Title: Painlevé equations and conformal field theory**

Abstract: It will be explained how the Riemann-Hilbert problem associated to isomonodromic deformations of rank $2$ linear systems with
$n$ regular singular points on $\mathbb{P}^1$ can be
solved by taking suitable linear combinations of conformal blocks of the Virasoro algebra at $c=1$. This implies a similar
representation for the isomonodromic tau function. In
the case $n=4$, it provides the general solution of the Painlevé
VI equation in the form of combinatorial sum over pairs of Young diagrams.
Analogous solutions of Painlevé V and III may be
formulated in terms of irregular limits of conformal blocks corresponding to
decoupling of the matter hypermultiplets on the gauge
side of the AGT correspondence.

Global
Weyl groups and wild mapping class groups

In the classical theory of Painleve equations and isomonodromy there

are two types of discrete groups: braid/mapping class groups

controlling the global nonlinear monodromy, and
Okamoto's affine Weyl

group symmetries giving equivalences between Painleve
equations at

different values of the parameters. In this talk I will describe how

to generalise both of these groups. This involves for
example all the

G-braid groups, and many non-affine Kac-Moody Weyl groups.

**Title: The bosonic representation of tau-functions**

The Riemann-Hilbert problem to find multivalued analytic functions

with SL(2,C)-valued monodromy on Riemann surfaces of
genus zero

with n punctures can be solved by taking suitable linear combinations of the
conformal blocks of Liouville theory at c=1. This
implies a similar representation for the isomonodromic
tau-functions. In the case n=4 we thereby get a proof of the relation between
tau-functions and conformal blocks discovered by Gamayun,
Iorgov and Lisovyy. These
results can be understood as a bosonization of the fermionic constructions of isomonodromic
tau-function due to Sato, Jimbo and Miwa.

Dynamics in one dimension: from fractional excitations to new
out-of-equilibrium states of matter

This talk will review a number of integrability-based
methods to investigate the dynamics of low-dimensional systems such as
interacting atomic gases and quantum spin chains. In the equilibrium case,
space- and time-dependent correlations will be considered. For
out-of-equilibrium situations, a number of recent results will be reviewed,
including a new method for explicitly calculating the relaxation of observables
after a quantum quench. Exact solutions to the interaction turn-on quench in
the Lieb-Liniger model and to the Néel-to-XXZ
quench in spin chains will be presented. Particular emphasis will be
given to interesting open issues from a mathematical point of view.

Mobile
impurity propagation in a one-dimensional quantum gas

We
investigate the time evolution of an impurity atom injected into a gas

of
impenetrable bosons (Tonks-Girardeau gas). The
interaction between the

gas
and impurity is assumed to be contact. If masses of the impurity and gas

particles
coincide then the system is integrable by means of
Bethe Ansatz

and
we are able to obtain compact answer for the asymptotic momentum of

the
impurity. At weak coupling we develop kinetic theory that describes impurity

behavior
both in integrable and non-integrable
cases and find striking

differences
between them. The asymptotic momentum as a function of the

final
momentum is found explicitly and the case of the external force applied

to
the impurity is analyzed as well.

**Title: Boundary qKZB equations**.

Abstract: In this talk I introduce an explicit 9-parameter elliptic family of
boundary qKZB equations. They are expected to
describe the consistency conditions for correlation functions of the 8-vertex
solid-on-solid model with reflecting boundaries. The family is constructed
using a new 4-parameter elliptic family of solutions

of a dynamical reflection equation, obtained by computing wall crossing
formulas for the boundary qKZ equation associated to
the XXZ spin chain.

**Title: Painleve' equations and q-Askey
scheme**

Abstract: In this talk we will give a quantisation of
the monodromy manifold associated to the Painleve' equations. We will show that the so obtained
quantum algebras admit a representation in the space of (Laurent) polynomials
by q-difference operators. We will show that special elements of the q-Askey scheme span such representations.

**Title: Integrating the
AdS_5 x S^5 superstring**

Abstract: I review the integrability approach to the
energy spectrum of the AdS_5 x S^5 superstring and, via the gauge-string
correspondence,

to the spectrum of primary operators in planar N=4 super Yang-Mills theory. I
will start from classical integrability of the string
sigma-model

and end up with the construction of the mirror Thermodynamic Bethe Ansatz which encodes the spectrum of the corresponding
quantum theory.

# Bäcklund transformations for certain rational solutions of Painlevé VI

We introduce certain Bäcklund
transformations for rational solutions of the Painlevé
VI equation. These transformations act on a family of Painlevé
VI tau functions. They are obtained from reducing the Hirota
bilinear equations that describe the relation between certain points in the 3
component polynomial KP Grassmannian. In this way we
obtain transformations that act on the root lattice of *A*_{5}.
We also show that this *A*_{5} root lattice can be
related to the *F*_{4}^{(1)} root lattice. We
thus obtain Bäcklund transformations that relate Painlevé VI tau functions, parametrized
by the elements of this *F*_{4}^{(1)} root
lattice.
This is based on joined work with Henrik Aratyn.

**Title:
Bihamiltonian cohomology of
KdV Poisson brackets and spectral sequences.**

The dispersive deformations of a Poisson pencil of hydrodynamic type are
controlled by certain bihamiltonian cohomology groups. Liu and Zhang (2013) computed the first
three bihamiltonian cohomology
groups associated with the dispersionless KdV Poisson pencil and conjectured that all remaining bihamiltonian cohomology groups
vanish.

After reviewing some of the theory, we outline a new method to derive the bihamiltonian cohomology in the KdV case. This is based on the introduction of a filtration
of a related polynomial complex and on the computation of the associated
spectral sequence. Using the convergence theorem for spectral sequences we
obtain the cohomology of the polynomial complex and,
by a long exact sequence argument, the bihamiltonian cohomology for the KdV Poisson
pencil.

In particular we rederive the Liu-Zhang results and
show that the remaining cohomology groups vanish,
hence proving their conjecture.

Based on: G. Carlet, H. Posthuma,
S. Shadrin “Bihamiltonian

cohomology of KdV
brackets”, arXiv:1406.5595.