I will review analytic SU(2) Yang-Mills solutions with finite action on de Sitter space from a new perspective. As a byproduct, all abelian solutions are classified and related with rational electromagnetic knots. In the Yang-Mills case, the gravitational backreaction is easily taken in to account as well.

We will argue that superstring theory on ${\rm AdS}_3\times {\rm S}^3\times \mathbb{T}^4$ with the smallest amount of NS-NS flux (``$k=1$'') is dual to the spacetime CFT given by the large $N$ limit of the free symmetric product orbifold $\mathrm{Sym}^N(\mathbb{T}^4)$. The worldsheet theory, at $k=1$, is defined using the hybrid formalism in which the ${\rm AdS}_3\times {\rm S}^3$ part is described by a $\mathfrak{psu}(1,1|2)_1$ WZW model (which is well defined). Unlike the case for $k\geq 2$, it turns out that the string spectrum at $k=1$ does not exhibit a long string continuum, and perfectly matches with the large $N$ limit of the symmetric product. The fusion rules of the symmetric orbifold are also reproduced from the worldsheet perspective. This proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory.

Why do some systems organise themselves into well ordered patterns with astonishing symmetry and regularity, while other superficially similar systems produce defects and disorder? In systems where two different length scales are unstable, the nonlinear interaction between the different modes is key: steady complex patterns can be stabilised when the modes act together to reinforce each other. But, if the two types of pattern compete with each other, the outcome can be considerably more complicated: a time-dependent disordered mixture of patterns constantly shifting and changing. In a small domain, the nature of the interaction between a small number of modes on each length scale can readily be computed. In a large domain, each mode can interact with hundreds of other modes, but the overall behaviour still appears to be guided by small-domain considerations.

I plan to discuss how Q-operators for rational spin chains can be constructed in the framework of the quantum inverse scattering method. The presentation will include open and closed XXX type spin chains with compact and non-compact representations of sl(2) in the quantum space. Further I plan to elaborate on the generalisation to spin chains of higher rank and in particular u(2,2|4) which underlies N=4 super Yang-Mills theory at weak coupling. Finally I will discuss the classification of the oscillator type solutions to the Yang-Baxter equation that are relevant to build Q-operators and give an outlook.

The infrared fixed point of graphene under the renormalization group flow is a relatively under studied yet important example of a boundary conformal field theory with a number of remarkable properties. It has a close relationship with three dimensional QED. It maps to itself under electric-magnetic duality. Moreover, it along with its supersymmetric cousins all possess an exactly marginal coupling -- the charge of the electron -- which allows for straightforward perturbative calculations in the weak coupling limit. I will review past work on this model and also discuss my own contributions, which focus on understanding the boundary contributions to the anomalous trace of the stress tensor and their role in helping to understand the structure of boundary conformal field theory.

The topological string is a simplified version of physical string theory. It is of interest because it computes the BPS spectrum of relevant string theory compactifications, but also because it shares structural properties of physical string theory, Dualities and symmetries which often must be argued for arduously in the physical string can often be verified by computation in the topological setting. The central observable of the theory is the topological string partition function Z_top. This quantity has an eerie habit of making surprise appearances in many areas of mathematical physics. Numerous techniques exist for its computation in various expansions in parameters of the theory, yet to date, no satisfactory closed form for this quantity is known. In this talk, after reviewing notions of topological string theory with an emphasis on the interplay between worldsheet and target space physics (one of the structural similarities between the physical and the topological string alluded to above), I will report on progress in computing Z_top in settings where it is related to enigmatic 6d theories.

Recently a new formulation for supergravity has emerged inspired by the presence of duality symmetries in reduced theories. These new theories generalise ideas of Riemannian geometry and lead to new ways of looking at string and M-theory.

The hydrodynamic approximation is an extremely powerful tool to describe the behavior of many-body systems such as gases. At the Euler scale (that is, when variations of densities and currents occur only on large space-time scales), the approximation is based on the idea of local thermodynamic equilibrium: locally, within fluid cells, the system is in a Galilean or relativistic boost of a Gibbs equilibrium state. This is expected to arise in conventional gases thanks to ergodicity and Gibbs thermalization, which in the quantum case is embodied by the eigenstate thermalization hypothesis. However, integrable systems are well known not to thermalize in the standard fashion. The presence of infinitely-many conservation laws preclude Gibbs thermalization, and instead generalized Gibbs ensembles emerge. In this talk I will introduce the associated theory of generalized hydrodynamics (GHD), which applies the hydrodynamic ideas to systems with infinitely-many conservation laws. It describes the dynamics from inhomogeneous states and in inhomogeneous force fields, and is valid both for quantum systems such as experimentally realized one-dimensional interacting Bose gases and quantum Heisenberg chains, and classical ones such as soliton gases and classical field theory. I will give an overview of what GHD is, how its main equations are derived and its relation to quantum and classical integrable systems. If time permits I will touch on the geometry that lies at its core, how it reproduces the effects seen in the famous quantum Newton cradle experiment, and how it leads to exact results in transport problems such as Drude weights and non-equilibrium currents. This is based on various collaborations with Alvise Bastianello, Olalla Castro Alvaredo, Jean-Sébastien Caux, Jérôme Dubail, Robert Konik, Herbert Spohn, Gerard Watts and my student Takato Yoshimura, and strongly inspired by previous collaborations with Denis Bernard, M. Joe Bhaseen, Andrew Lucas and Koenraad Schalm.

We study non-planar corrections to gluon scattering amplitudes in N = 4 SYM theory. In this talk, we focus on the first correction. It is computed by the double trace amplitude and is suppressed by one power of 1/Nc with respect to the leading single trace contribution. We extend the duality between planar scattering amplitudes and null polygonal Wilson loops to the double trace amplitude. The new duality allows us to extend the notion of loop integrand beyond the planar limit and to determine it using recursion relation. It also allows us to apply the integrability pentagon approach to the first non-planar order. We shortly discuss higher orders in the 't Hooft 1/Nc expansion.

The extremal limit of single R-charged AdS5 black holes in type IIB is known to be described by a system of N free fermions in a one dimensional harmonic oscillator potential. Since the quantum mechanical problem is solvable and its phase space formulation appears in the gravity dual (LLM geometries), it allows us to explore the relation between entanglement, quantum correlation design and connectivity in space in this set-up, both in a single and a two boundary situation.

The AdS/CFT duality maps supersymmetric heavy operators with conformal dimension of the order of the central charge to asymptotically AdS supergravity solutions. I'll show how by studying the quadratic fluctuations around such backgrounds it is possible to derive the 4-point correlators of two light and two heavy states in the supergravity approximation. I'll present some explicit examples in the AdS3 setup relevant for the duality with the D1-D5 CFT and discuss what we can learn in general about correlators among heavy pure states.