**AVAN Jean ** * Quantization of trace-brackets *

The quantization problem for the trace-bracket algebra, derived from double Poisson brackets, is discussed. We obtain a generalization of the boundary YBE for the quantization of the quadratic trace-brackets. A dynamical deformation is proposed on the lines of Gervais-Neveu-Felder quantum algebras.

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**BATCHELOR Murray ** * Integrability vs exact solvability in the quantum Rabi model and beyond *

The quantum Rabi model, which describes the simplest interaction between light and matter, is a fundamental model with widespread applicability in quantum physics. The applications include the interaction between light and trapped ions or quantum dots, and between microwaves and superconducting qubits. The model is also applicable to both cavity and circuit QED. The eigenspectrum of the quantum Rabi model has recently been obtained exactly and the model has been claimed to be integrable. This talk will discuss recent work with Huan-Qiang Zhou on examining the Yang-Baxter integrability of this model. In particular, it is found that the quantum Rabi model does not appear to be Yang-Baxter integrable in general. It is however, integrable in two special limits. These considerations can be extended to various physical generalisations of the Rabi model.

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**BELLIARD Samuel ** * Modified algebraic Bethe ansatz. *

We present a modified version of the algebraic Bethe ansatz (MABA) that allows to characterize the eigenvalues and the eigenstates of spin chains without U(1) symmetry [Belliard Crampé (2013)] and work in progress. The XXX and XXZ Heisenberg spin chains on the segment will be discussed. The solution involves the Baxter T-Q equation with an inhomogeneous term introduced by [Cao et al. (2013)] and used in the quantum separation of variable approach by [Niccoli et al. (2014)]. The MABA also produces the associated Bethe vectors.

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**BOROT Gaetan ** * Asymptotic expansion in 1d log-gas with many-body interactions *

1d log-gas are systems of N particles on the real line, subjected to a Coulomb repulsion in 2d, and possibly to other non-singular interactions. Their appear in random matrix theory, in Chern-Simons theory, in statistical physics on the 2d random lattice, ... I will give an overview of systematic methods based on large deviations and Schwinger-Dyson equations to obtain large N asymptotic expansion in such models, and give some examples of applications in integrable and non-integrable models. This talk is based on joint works with Alice Guionnet and Karol Kozlowski.

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**BROCKMANN Michael ** * Gaudin-like determinants for overlaps in integrable models and their applications to quench problems *

We study interaction quenches defined by releasing the ground state of a non-interacting theory into a system with finite interaction. We focus here on the spin-1/2 Heisenberg chain and the Lieb-Liniger Bose gas. Starting from exact overlaps between the initial state and Bethe eigenstates we show that an application of the quench action method allows to give an exact description of the post-quench steady state in the thermodynamic limit. This steady state in particular correctly reproduces expectation values of local correlators.

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**CANTINI Luigi ** * Inhomogenous Multi-TASEP (M-TASEP) on a ring with spectral parameters *

I will report on some ongoing work about a multispecies version of the TASEP, a model which describes the stochastic evolution of a system of particles of different species (labeled by an integer) on a periodic oriented one dimensional lattice, where two neighboring particles exchange their position with a rate which depends on their species. For some choice of these rates the Markov matrix turns out to be integrable and for the same choice the (unnormalized) stationary probability is conjectured to show remarkable positivity and combinatorial properties. By introducing spectral parameters in the problem we will uncover a novel algebraic structure underlying this problem which will allow us to provide exact formulas for the stationary probability of some configurations.

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**CAUDRELIER Vincent ** * Set-theoretical reflection equation in integrable field theories and fully discrete systems *

The Yang-Baxter equation (YBE) is central in the theory of integrable systems. It has been mainly studied and used in the quantum realm. But it was suggested by Drinfeld in 1990 that the general study of the so-called « set-theoretical YBE » is also important. It turns out that the dressing method in the theory of classical integrable field theories provides a means to construct solutions to this equation, called Yang-Baxter maps, by looking at soliton collisions. Using the vector nonlinear Schrödinger (NLS) equation as the main example, we will review this notion of "classical solutions of the quantum YBE". Then, we will show how the new concept of set-theoretical reflection equation naturally emerges by studying the vector NLS on the half-line. The study of soliton collisions on the half-line provides solutions to this equation, called reflection maps. It is the beauty of integrable systems that completely classical and nonlinear field theories share common fundamental structures with their quantum counterparts. Such notions have direct applications to discrete integrable systems and raise the question of the general geometric picture in the context of Poisson-Lie groups.

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**CAUX Jean-Sébastien ** * Exact solutions for quenches in the Lieb-Liniger Bose gas and Heisenberg quantum spin chains *

A number of recent results on the out-of-equilibrium dynamics of integrable models will be presented, based on a novel method for explicitly calculating the relaxation of observables following a quantum quench. Specifically, interaction quenches in the Lieb-Liniger model will be treated, as well as anisotropy quenches in quantum spin chains.

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**DEGUCHI Tetsuo ** * Formulas of form factors for the spin-1/2 and integrable spin-s XXZ chains and non-equilibrium dynamics *

We derive various expressions of the form factors for the spin-1/2 XXZ chain and the integrable spin-s XXZ chains, which are useful for studying the non-equilibrium dynamics of the integrable quantum many-body systems. We first show compact formulas for the form factors of arbitrary operators in the spin-1/2 XXZ chain and those of the integrable higher-spin XXZ chains. The compact formulas have several applications: One can evaluate the reduced density matrices systematically through them; One can also derive the multiple-integral representations of arbitrary correlation functions at zero temperature for the integrable higher-spin XXZ chains [1,2]. Furthermore, we transform the compact formulas into those which can be evaluated through the Fredholm determinants. We show that they are numerically and practically useful. For instance, we show the time evolution of physical operators such as the local magnetization in the XXZ spin chain through them.

[1] T. Deguchi and C. Matsui, Nucl. Phys. B831[FS] (2010) 359;

[2] T. Deguchi, JSTAT (2012) P04001.

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**DELFINO Gesualdo ** * Universal results for two-dimensional percolation *

We present recent universal results for percolation in two dimensions obtained using conformal and integrable quantum field theory. At criticality we point out a connection with time-like Liouville field theory and obtain the three-point connectivity constant. Away from criticality we determine the universal amplitude ratios as well as the asymptotic crossing probabilities. Our results are confirmed by high precision numerical simulations.

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**DUGAVE Maxime ** *Thermal form factors of the anisotropic Heisenberg chain *

We derive expressions for the form factors of the quantum transfer matrix of the spin‑1/2 XXZ chain which are suitable for taking the infinite Trotter number limit. These form factors determine the finitely many amplitudes in the leading asymptotics of the finite-temperature correlation functions of the model. We consider form factor expansions of the longitudinal and transversal two-point functions. Remarkably, the formulae for the amplitudes are in both cases of the same form. The usefulness of our novel formulae is demonstrated by working out explicit results in the low-temperature limit in the massless regime at finite magnetic field. By summing up the most relevant terms in the form factor series, we obtain the low-temperature large-distance asymptotics of the two-point functions, which is in accordance with conformal field theory predictions. Moreover, we obtain explicit expressions for the amplitudes which can be efficiently evaluated numerically. Finally, we present recent results for the massive regime at zero temperature.

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**EVANS Martin ** * The Asymmetric Exclusion Process: an exactly solvable nonequilibrium system *

The asymmetric exclusion process models the stochastic transport of a conserved quantity (mass, cars, molecular motors etc) through an open system. Since a current of mass always flows, the system is out of equilibrium but will nevertheless attain a nonequilibrium stationary state in the long time limit. In this talk I will give an overview of how the stationary state can be solved exactly by a matrix product ansatz resulting in a quadratic algebra. I will discuss the phase diagram and additional dynamical transitions. I shall also discuss generalisations to multispecies system resulting in higher rank tensor product states and richer algebraic relations.

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**FADDEEV Ludwig D ** * Zero modes for the Liouville model *

I propose to use for the definition of zero modes in the quantum Liouville Model the canonical variables, entering the monodromy matrix of the corresponding Lax operator introduced. This matrix was investigated by Takhtajan and me in the middle of 80-ties and was one of the sources for quantum groups and their role in CFT.

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**FALDELLA Simone ** *Spin chains by separation of variables method*

I'll show that the Separation of Variables (SoV) method leads to the construction of the full spectrum of the open XXZ and XYZ spin-1/2 chains with the most generic boundary conditions. The eigenstates, in the inhomogeneous case, are constructed in terms of solutions of a system of quadratic equations. The SoV representation permits to compute scalar products as well and can be useful for the eventual calculation of correlation functions.

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**FEHER Laszlo ** *Recent developments in the reduction approach to action-angle dualities of integrable many-body systems*

Key properties of integrable Hamiltonian systems can be often viewed as consequences of their realizations as reductions of "canonical free systems" having rich symmetries on higher dimensional phase spaces. We first summarize results of the last few years towards explaining known action-angle duality relations between one-dimensional classical integrable many-body systems in terms of Hamiltonian reduction. We then present our recent results such as for example the construction of new self-dual compact forms of the trigonometric Ruijsenaars-Schneider system and the description of a novel dual pair involving the trigonometric BC(n) Sutherland system.

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**FOERSTER Angela ** * Integrability in ultracold physics *

The impressive development in cooling and trapping atoms in 1D tubes brought the area of integrable systems to a new audience, when it became clear that these models can be realized in the lab. Prominent examples include the Lieb- Liniger model for spinless bosons and the Gaudin-Yang model for two-component fermions. In this work we give an overview of these ongoing developments. In addition, new results, such as the Wilson ratio of the 1D attractive Fermi gas with spin imbalance [1] and a geometric ansatz for a few particles system [2] will be presented. Finally we discuss how the breakdown of integrability in the fundamental 1D model of bosons with contact interactions affects the stationary correlation properties of the system [3]. In principle, all these results can be detected by state-of-the art experiments.

[1] X. W. Guan, X. G. Yin, A. Foerster, M. T. Batchelor, C. H. Lee, H. Q. Lin, Phys. Rev. Lett. 111 (2013) 130401.

[2] B. Wilson, A. Foerster, C. C. N. Kuhn, I. Roditi, D. Rubeni, Phys. Lett. A 378 (2014) 1065.

[3] I. Brouzos and A. Foerster, Broken Integrability Trace on Stationary Correlation Properties, arXiv:1405.0430, to appear in PRA.

**GAINUTDINOV Azat ** * The periodic sl(2|1) alternating spin chain and Logarithmic Conformal Field Theory at c = 0 *

Statistical properties of hulls of 2D percolation clusters on a torus can be encoded by periodic sl(2|1) alternating spin chain with Heisenberg-like interaction. The algebra of local Hamiltonian densities in the chain is provided by a representation of the affine or periodic Temperley-Lieb (PTL) algebra at the primitive 6th root of unity. I will discuss our recent results (with H. Saleur, N. Read and R. Vasseur) on the lattice side: the full decomposition of the spin chain onto (indecomposable but reducible tilting) modules of the PTL and the structure of Jordan blocks for the Hamiltonian. Using a connection with the spectrum of twisted XXZ spin chains we can analyze the continuum limit - a logarithmic conformal field theory (LCFT) at central charge c=0. We then analyze how the finite-size algebraic properties carry over to the continuum limit and deduce the structure of the space of states as a representation over the product of left and right Virasoro algebras. Our main result is the full structure of the vacuum module of the LCFT, which exhibits Jordan cells of arbitrary rank for the Hamiltonian.

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**GOEHMANN Frank ** * Form factor approach to correlation functions of the XXZ chain for delta larger than one *

Abstract TBA

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**GOLDSTEIN Garry ** * GGE and applications for integrable systems *

We study the nonequilibrium properties of 1-D interacting systems. For the 1-D Lieb Liniger gas we introduce a new version of the Yudson representation applicable to finite size systems. We provide a formalism to compute various correlation functions for highly nonequilibrium initial states. In the strong coupling regime we are able to find explicit expressions for the density, density-density and related correlation functions. We are able to show that for initial sates with smooth correlations the gas equilibrates at long times. For nearly translationaly invariant states the gas equilibrates to the GGE. We show that for other integrable models, ones with bound states, the GGE fails. We also apply our results on equilibration and the GGE to studying initial states with small Yang-Yang entropies in the Tonks Girardeau regime. We show that at long time such states have universal power law density-density correlations.

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**GOMES Jose Francisco ** * Backlund Transformation for mKdV Hierarchy *

A systematic construction of Integrable hierarchies is proposed in terms of an Afﬁne Kac-Moody algebra. In particular we shall discuss the mKdV hierarchy and its higher nonlinear solitonic equations. We explicitly construct the Backlund transformation for the mKdV equation as well as for the ﬁrst few higher members of the hierarchy

**KAROWSKI Michael ** * O(N) Bethe Ansatz and exact form factors of the O(N) sigma and Gross-Neveu models *

General form factor formulas for the O(N) and Gross-Neveu models are constructed and applied to several operators. The large N limits of these form factors are computed and compared with the 1/N expansion of the models in terms of Feynman graphs and full agreement is found. In particular for the O(3)-model several low particle form factors are calculated explicitly.

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**KHARLAMOV Mikhail ** * Some integrable systems with algebraic separation of variables *

We consider integrable systems of mechanical origin with a compact configuration space. Such a system is called irreducible if it does not have any continuous symmetry group and therefore does not reduce, globally, to a family of systems with two degrees of freedom. One of the known examples of an irreducible system was found by A.G. Reyman and M.A. Semenov-Tian-Shansky [1]. It is a widest of known generalizations of the classical Kowalevski case and the Kowalevski--Yehia gyrostat [2]. The integrable systems with two degrees of freedom appear in an irreducible system as invariant almost symplectic submanifolds of dimension 4. They are called critical subsystems. First, we present the complete classification of critical subsystems for the system found in [1]. If the gyrostatic momentum in this problem is zero, there exist exactly three critical subsystems. For two of them we present an algebraic separation of variables, i.e., two auxiliary variables satisfy the equations of the Kowalevski type and all initial phase variables are expressed in terms of the separated ones as rational functions of simple radicals with polynomial coefficients. One separation is integrated in elliptic functions, another is hyperelliptic. Considering the phase space of the separated variables as a 4-dimensional Euclidean space of two coordinates and two momenta, we derive the polynomial equations for the Hamilton function H. In the elliptic case it is of degree 4 with respect to H. The work is partially supported by the RFBF (research project No. 14-01-00119).

[1] A.G. Reyman, M.A. Semenov-Tian-Shansky. Lax representation with a spectral parameter for the Kowalewski top and its generalizations. Lett. Math. Phys. 14, 1987, 55-61.

[2] H.M. Yehia. New integrable cases in the dynamics of rigid bodies. Mech. Res. Commun. 13, 1986, 169-172.

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**KLUEMPER Andreas ** * Non-linear integral equation approach to sl(2|1) integrable network models *

Abstract TBA

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**KONNO Hitoshi ** * Elliptic Quantum Groups, Drinfeld Coproduct and Deformation of W-algebras *

We first discuss a quantum Z-algebra structure of the elliptic algebra U_{q,p}(g) associated with an untwisted affine Lie algebra g, and show that the irreducibility of the level-k representation of the U_{q,p}(g)-module is governed by the corresponding Z-algebra module. The level-1 examples for g=A_{l}^{(1)}, B_{l}^{(1)}, D_{l}^{(1)} show that the irreducible U_{q,p}(g)-modules are decomposed as a direct sum of the irreducible W-algebra modules. We secondly introduce the Drinfeld coproduct to U_{q,p}(g) and discuss the intertwining operators (vertex operators) with respect to this new coproduct. Constructing the vertex operators for the level-1 U_{q,p}(g)-modules with g=A_{l}^{(1)}, B_{l}^{(1)}, D_{l}^{(1)} explicitely, we show that the these vertex operators factor the generating functions of the known deformed W-algebras associated with A_{l}^{(1)}, D_{l}^{(1)}, and further obtain a conjectural expression for the B_{l}^{(1)} case corresponding to a deformation of Fateev-Lukyanov's WB_{l}-algebra.

**KORFF Christian ** * Solving the Bethe ansatz equations: generalised quantum Schubert calculus*

One of the main obstacles in obtaining the exact solution of exactly solvable lattice models is solving the Bethe ansatz equations. This problem is usually considered to be intractable. In this talk I will focus on two 5-vertex degenerations of the asymmetric six-vertex model and present the solution for arbitrary anisotropy parameter Δ. By using a special parametrisation of the Boltzmann weights, the problem of solving the equations reduces to the explicit description of a coordinate ring (a polynomial algebra modulo certain relations). For special values of the anisotropy parameter this ring has been of great interest in the mathematics literature. At the free fermion point Δ=0 one obtains the quantum cohomology and at the combinatorial point Δ=1/2 the quantum K-theory of Grassmannians. Both objects appear in the context of the study of moduli spaces of curves and have their origins in CFT and string theory. The combinatorics which describes the multiplication in these rings is known as quantum Schubert calculus. Our work allows us to use the six-vertex model to define a generalised complex oriented cohomology for the Grassmann varieties and use it to explicitly compute K-theoretic Gromov-Witten invariants. This is joint work with Vassily Gorbounov, Aberdeen.

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**KOSTOV Ivan ** * Quasiclassical expansion of the Slavnov determinant *

We propose a systematic way to compute the quasi-classical expansion of the on-shell/off-shell scalar product of multi-magnon states in the generalised SU(2) model. We give explicit expressions for the first two terms of the expansion in terms of contour integrals.

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**LOEBBERT Florian ** * Spin Chains and Three-Point Functions in N=4 SYM Theory *

We propose a new method for the computation of quantum three-point functions for operators in su(2) sectors of N=4 super Yang-Mills theory. The method is based on the existence of a unitary transformation relating inhomogeneous and long-range spin chains. This transformation can be traced back to a combination of boost operators and an inhomogeneous version of Baxter's corner transfer matrix. We reproduce the existing results for the one-loop structure constants in a simplified form and indicate how to use the method at higher loop orders. Then we evaluate the one-loop structure constants in the quasiclassical limit and compare them with the recent strong coupling computation.

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**MATVEEV Vladimir ** * From continous to the discrete Fourier transform: classical and quantum aspects *

We explain and generalize certain results by Mehta relating the eigenfunctions of quantum harmonic oscillator with the eigenvectors of the discrete Fourier transform (DFT). Using far more general construction based on a kind of Poisson summation, we associate the eigenvectors of the DFT with the sums of any absolutely convergent series. This construction in particular provides the overcomplete bases of the DFT eigenvectors by means of Jacobian theta functions or ν theta-functions and leads, in a quite natural way to some new identities and addition theorems in the theory of special and q-special functions

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**NEPOMECHIE Rafael ** * On the completeness of solutions of Bethe's equations *

Ever since Bethe solved the spin-1/2 Heisenberg model, the nagging question of completeness has persisted: namely, whether the Bethe equations (BE) have too many, too few, or just the right number of solutions to account for all the eigenstates of the Hamiltonian. We shall argue that, in some sense, there are too many: the BE have many "singular" solutions, of which only a small subset (the "physical singular" solutions) correspond to eigenstates of the Hamiltonian. We find a condition for identifying the physical singular solutions. We formulate a conjecture for the number of solutions of the BE with pairwise distinct roots, in terms of numbers of singular solutions. Using homotopy continuation methods, we find all such solutions of the BE for chains of length up to 14, and observe perfect agreement with the conjecture. We briefly discuss the generalization to higher spin.

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**NICCOLI Giuliano ** * SOV approach for the exact solution of integrable quantum models: the open XXZ spin chain *

Abstract TBA

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**NIJHOFF Frank ** * Lagrangian Multiform Theory and Integrability *

Recent developments in the theory of discrete integrable systems, in particular the notion of multidimensional consistency and its consequences, has led to the formulation of a novel variational principle based on the notion of Lagrangian multiforms, introduced by Lobb and Nijhoff in 2009. The talk reviews the main ideas and recent results, including the quantum aspects.

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**PALMAI Tamas ** * Quench echo and work statistics in integrable quantum field theories *

We study integrable quantum field theories in the context of out of equilibrium situations, in particular from a thermodynamic point of view. Such studies so far have only been carried out for non-interacting models and those equivalent to such. To take into account genuine interactions we develop a boundary thermodynamic Bethe ansatz technique for the computation of the statistics of the work done when a global quench is performed in the elementary mass scale. We find analytic expressions for the low energy part of the statistics and we also present numerical results for the simplest relativistic integrable QFT, the sinh-Gordon model, where we study in particular the effects of turning on the interaction relative to the case of the free boson theory.

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**PASQUIER Vincent ** *On Bethe Ansatz for the open XXX chain *

Abstract TBA

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**PEARCE Paul ** *Exact solution of critical dense polymers with Robin boundary conditions *

Solvable critical dense polymers is a loop model on the square lattice with central charge c=-2 and loop fugacity β=0 so that closed loops are forbidden. We solve this model exactly on a strip with general Robin boundary conditions which are given as a linear combination of Neumann and Dirichlet boundary conditions on the loops allowing loops to terminate on the boundary. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitized characters related to spaces of coinvariants of Z_{4} fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights given by a Kac table with half-integer Kac labels: Δ_{r,s-1/2}=(L^{2}-4)/32 where L=2s-1-4r, r∈ **Z**, s∈ **N**.

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**PIMENTA Rodrigo ** * Algebraic Bethe ansatz for 19-vertex models with upper triangular K-matrices *

By means of an algebraic Bethe ansatz approach we study the Zamolodchikov-Fateev and Izergin-Korepin vertex models with non-diagonal boundaries, characterized by reflection matrices with an upper triangular form. Generalized Bethe vectors are used to diagonalize the associated transfer matrix. The eigenvalues as well as the Bethe equations are presented.

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**REGELSKIS Vidas ** * Twisted Yangians for symmetric pairs of types B, C, D *

I will present a class of quantized enveloping algebras, called twisted Yangians, associated with the symmetric pairs of types B, C, D in terms of the Cartan classification of compact Riemannian symmetric spaces. These algebras play an important role in quantum integrable systems with open boundary conditions. They are regarded as coideal subalgebras of the extended Yangian for orthogonal or symplectic Lie algebras, whose defining relations are written in an R-matrix form. On the other hand, these algebras can be presented as quotients of the reflection equation algebra by additional symmetry relations.

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**RUIJSENAARS Simon ** * A recursive construction of joint eigenfunctions for the commuting hyperbolic Calogero-Moser Hamiltonians *

We present a recursive scheme involving so-called kernel functions, which yields an explicit diagonalization of the hyperbolic N-particle Calogero-Moser quantum systems of nonrelativistic and relativistic type. For the nonrelativistic case, the first step of the scheme yields the well-known 2-particle eigenfunction, basically a specialization of the Gauss hypergeometric function. The work on the N>2 case (together with Martin HallnŠs) gives rise to an elementary representation of the joint eigenfunctions, which were previously obtained in a quite different form by Heckman and Opdam. Last but not least, we sketch a similar recursive construction for the joint eigenfunctions of the 2N commuting relativistic hyperbolic Calogero-Moser Hamiltonians.

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**RYABOV Pavel ** * Phase topology of generalized two-field gyrostat *

We consider the integrable system with three degrees of freedom for which Sokolov and Tsiganov specified a Lax representation. This representation generalizes the L--A pair of the Kowalevski gyrostat in two constant fields found by Reyman and Semenov-Tian-Shansky. We give explicit formulas for two almost everywhere independent additional first integrals K and G. These integrals are functionally connected with the coefficients of the spectral curve of the L--A pair by Sokolov and Tsiganov. In the talk, we suggest an approach to describe phase topology of a new integrable system with three degrees of freedom, using the method of critical subsystems. The notion of a critical subsystem was formed in the beginning of 2000-ies in the problem of study of phase topology of irreducible systems with three degrees of freedom. Due to the obtained form of additional integrals K, G and parametric reduction, we managed to find analytically invariant four-dimensional submanifolds on which the induced dynamic system is almost everywhere Hamiltonian with two degrees of freedom. The system of equations that describes one of these invariant submanifolds is a generalization of the invariant relations of the corresponding integrable Bogoyavlensky case for the rotation of a magnetized rigid body with a fixed point in homogeneous magnetic and gravitational fields. For each subsystem, we construct the bifurcation diagrams and specify the bifurcations of Liouville tori both in subsystems, and in the system as a whole. We also present new net diagrams on isoenergetic surfaces (analogues of Fomenko's nets), which are not in the list of net invariants in the problem of the Kowalevski top motion in the case of a double force field. The work is partially supported by RFBR, research project No. 14-01-00119.

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**SEMENOV-TIAN-CHANSKI Michel ** * Difference Lax operators, Poisson Lie groups and differential/difference Galois theory*

I discuss the lift of the Poisson structure in the space of differential/difference Lax operators to the space of solutions of the associated auxiliary linear problem (the space of wave functions). The key phenomenon is a peculiar symmetry breaking which makes the associated differential/difference Galois group a Poisson group. I shall describe a new class of classical r-matrices associated with generalized exchange-algebras. Reduction to the case of higher order scalar difference Lax operators is particularly rigid; in this case all Poisson structures are defined in a unique way.

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**SLAVNOV Nikita ** * Form factors in quantum integrable models with Gl(3)-invariant R-matrix *

We consider form factors of the monodromy matrix entries in quantum integrable models with Gl(3)-invariant R-matrix and find detreminant representations for them. We show that all determinant representations are related with each other by a set transformations in such a way, that it is enough to find the determinant formula for only form factors. All others follow from this initial representation.

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**SUZUKI Junji ** * Correlations in massive phase *

The quantum correlation functions of spin chains in massive phase will be addressed. (Joint work with M. Dugave, K. Kozlowski and F. Goehmann)

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**TERRAS Veronique ** *Form factor approach to correlation functions in massless quantum integrable models*

Abstract TBA

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**VLJIM Rogier ** * Applications of algebraic Bethe ansatz matrix elements to spin chains*

Algebraic Bethe ansatz based techniques at finite size afford a way into computing observables for integrable models. Dealing with Bethe roots explicitly as deviated string-solutions is necessary. Dynamical correlations of the Babujan-Tahktajan spin-1 chain are obtained by a higher spin generalisation of this method. The obtained real-space spin–spin correlation displays asymptotics fitting predictions from conformal field theory. Moreover, prepared out-of-equilibrium initial states for the anisotropic spin chain can be addressed as well. For the Néel state in particular as an initial state, contributions from coinciding deviated strings at zero are challenging to implement.

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**WANG Yupeng ** *The inhomogeneous T-Q relation and solutions of integrable models without reference state *

The off-diagonal Bethe ansatz method will be introduced. By constructing the operator product identities, the inhomogeneous T-Q relation can be constructed. This method overcomes the difficulty of the absence of a reference state for models without U(1) symmetry. Applications of this method on several models will be introduced.

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**YANG Wen-Li ** * Off-diagonal Bethe ansatz for the Izergin-Korepin model with non-diagonal boundary terms *

The Izergin-Korepin model with general non-diagonal boundary terms, a typical integrable model beyond A-type and without U(1)-symmetry, is studied via the off-diagonal Bethe ansatz method. Based on some intrinsic properties of the R-matrix and the K-matrices, certain operator product identities of the transfer matrix are obtained at some special points of the spectral parameter. These identities and the asymptotic behaviors of the transfer matrix together allow us to construct the inhomogeneous T-Q relation and the associated Bethe ansatz equations. In the diagonal boundary limit, the reduced results coincide exactly with those obtained via other methods.

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**ZULLO Federico ** * Backlund transformations and a q-difference Baxter's equation for the Ablowitz-Ladik chain. *

We shall give the Baxter's operator and the corresponding Baxter's equation for a quantum version of the Ablowitz Ladik model. The result is achieved in two different ways: by using the well-known Bethe ansatz technique and looking at the quantum analogue of the classical Backlund transformations. General results about integrable models governed by the same r-matrix algebra will be given. The Baxter's equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. We will show how the spectrality property of the classical Backlund transformations gives a trace formula representing the classical analogue of the Baxter's equation. Finally we will discuss a q-integral representation of the Baxter's operator.

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