We will describe the duality between two integrable systems: the 2D Sine-Gordon model and the 2D Thirring model. We will spend some time describing the classical and quantum Sine-Gordon model, in particular its spectrum, S-matrices and underlying quantum-group symmetry. We will then present the duality with the Thirring model as originally stated by Coleman and refined in subsequent literature. All the basic elements will be provided without relying on too many pre-requisites beyond standard graduate-level quantum field theory. The notes comprise a series of exercises.

We will describe the duality between two integrable systems: the 2D Sine-Gordon model and the 2D Thirring model. We will spend some time describing the classical and quantum Sine-Gordon model, in particular its spectrum, S-matrices and underlying quantum-group symmetry. We will then present the duality with the Thirring model as originally stated by Coleman and refined in subsequent literature. All the basic elements will be provided without relying on too many pre-requisites beyond standard graduate-level quantum field theory. The notes comprise a series of exercises.

We will describe the duality between two integrable systems: the 2D Sine-Gordon model and the 2D Thirring model. We will spend some time describing the classical and quantum Sine-Gordon model, in particular its spectrum, S-matrices and underlying quantum-group symmetry. We will then present the duality with the Thirring model as originally stated by Coleman and refined in subsequent literature. All the basic elements will be provided without relying on too many pre-requisites beyond standard graduate-level quantum field theory. The notes comprise a series of exercises.

After a brief introduction to some of the impact which integrable methods and the Bethe ansatz have had on the study of the AdS/CFT correspondence in string theory, we will focus on the axiomatic approach to S-matrix theory in 1+1 dimensions. We will highlight the issues that arise when the particles are massless, and how this is in fact connected to Zamolodchikov's way of describing two-dimensional conformal field theories by means of integrability techniques. We will then mention how the axiomatic approach extends to form-factors, which are the gate to access the n-point functions of the theory. If time permits, we will briefly depict how this finds a contemporary application in the area of the AdS_3/CFT_2 correspondence; part of London TQFT Journal Club (please register at https://london-tqft.vercel.app);

We will review a series of algebraic techniques to describe classically integrable systems, with an eye to their quantisation. The topics we shall attempt to cover will be: 1. General introduction to classical integrability and notion of Lax pair 2. Classical r-matrices and exchange relations 3. Belavin-Drinfeld theorems and classifications 4. Quantisation and elements of algebraic Bethe ansatz

In recent years, the discovery of integrable sectors of string theory has expanded the range of models which are solvable by the techniques of the Bethe ansatz and by the general theory of quantum groups. In this talk, we will give an overview of the S-matrix theory involved in the solution of these models, with mention to the problem of massless particles emerging in a particular instance of the AdS/CFT correspondence.