There are integrable (solvable) spin systems related to quantum groups (quasi-triangular Hopf algebras), with simple structure of space of states. The integrals of motion belong to a quotient A of the braid group, which is a centralizer of the symmetry quantum group of the system. The space of states is a direct sum of of tensor products of corresponding irreducible representations of A and the quantum group. This decomposition is multiplicity free.

The exact anomalous dimensions of the single trace composite operators in the planar limit of $N=4$ super Yang-Mills can be studied in the framework of the Thermodynamic Bethe Ansatz method (TBA). In this talk, I will discuss the analytic properties of the TBA solutions -the Y-functions- and the associated set of functional relations -the Y-system-. Contrary to the more studied relativistic invariant cases, the $AdS_5/CFT_4$-related Y-functions live on multi-sheeted coverings of the complex plane with an infinite number of square-root branch points. As a direct consequence of this fact, the TBA equations and in particular the dressing kernel cannot be obtained from the Y-system without extra information on the square-root discontinuities across semi-infinite segments in the complex plane. It is shown that the discontinuity functions fulfill an independent simple set of local functional equations. The Y-system extended by the discontinuity relations forms a fundamental set of local non-linear constraints that can be easily transformed into integral form through Cauchy's theorem.

After reviewing the R-matrix approach to classical integrable systems, we formulate the classical integrable structure of sigma-models relevant to the AdS/CFT correspondence.

In the first part I will give an overview of the recent results relating 4d N=2 quiver gauge theories, 2d conformal Toda field theories, topological string theories and quiver matrix models. In the second part surface operators in the gauge theories and their dual interpretations in the other classes of theories will be discussed

In recent years it has become clear that gauge theory and gravity amplitudes exhibit remarkable simplicity and beautiful structures. Here I will discuss one such structure: a duality between color and kinematics in gauge theory amplitudes. The duality implies nontrivial relations between color-ordered partial tree amplitudes. Remarkably the duality dictates the construction of gravity amplitudes as a double copy of gauge theory kinematic factors, clarifying the Kawai-Lewellen-Tye relations of open and closed strings. Originally considered as a classical (tree-level) duality, here I will present evidence that the duality is valid at the quantum level as well.

We discuss the realization of dual and conventional superconformal symmetries on the scattering amplitudes in N=4 SYM. While these symmetries are exact at tree level, they have to be complemented with analytic properties to fully fix the tree. At loop level both symmetries are broken by different mechanisms. We describe the formulation of amplitudes in twistor space and in terms of Grassmannian invariants.

The mirror Thermodynamic Bethe Ansatz (TBA) approach offers a tool to investigate the spectrum of AdS/CFT in the planar limit. The TBA equations governing the spectrum are based on several conjectures this is why it is important to test them carefully. In this talk we will focus on the analytical tests of the AdS/CFT TBA system in the weak coupling limit and show that the 5-loop anomalous dimensions of the twist-2 operators of the sl(2) sector derived from the TBA equations agree with the results obtained previously from the generalized Luscher formulae.