Dr Panagiota Adamopoulou

Prof. Sir Michael Berry

Prof. Andrew Green

Marianne Hoogeveen

Fabrizio Nieri

__Title:__Aspects of the ODE/IM correspondence__Abstract:__We discuss the so-called ODE/IM correspondence, which is a link between certain linear ordinary differential equations (ODEs) and quantum integrable models (IMs). Furthermore, we present some recent results concerning a certain generalisation, to include certain partial differential equations, as well as other aspects of the correspondence.Prof. Sir Michael Berry

__Title:__Divergent series: from Thomas Bayes’s bewilderment to today’s resurgence via the rainbow__Abstract:__Following the discovery by Bayes in 1747 that Stirling’s series for the factorial is divergent, the study of asymptotic series has today reached the stage of enabling summation of the divergent tails of many series with an accuracy far beyond that of the smallest term. Several of these advances sprang from developments of Airy’s theory of waves near optical caustics such as the rainbow. Key understandings by Euler, Stokes, Dingle and Écalle unify the different series corresponding to different parameter domains, culminating in the concept of resurgence: quantifying the way in which the low orders of such series reappear in the high orders.Prof. Andrew Green

__Title:__ Towards a Field Theory over Tensor Network States__Abstract:__Several works have pointed out the similarities between properties of hierarchical tensor network descriptions of critical systems and their gravitational duals. Though the body of circumstantial evidence for this link is compelling, it is difficult to make these notions concrete due to the different language in which they are naturally described. Here, I outline a first step towards bridging this language barrier, by explicitly constructing a field theory over the simplest tensor network states - matrix product states. I develop a functional integral representation of the partition function of a spin chain, where the measure of integration is explicitly over matrix product tensors. I will discuss the potential applications of such a field theory and the challenge presented by extending these ideas to critical systems, to higher dimensions and to hierarchical tensor networks.Marianne Hoogeveen

__Title:__Entanglement negativity in one-dimensional critical systems in the presence of an energy current__Abstract:__In recent years, finding ways to quantify the entanglement of quantum many body systems has attracted great interest. An interesting problem is studying the dynamics of the entanglement in various out-of-equilibrium situations. An important result by Calabrese and Cardy was finding exact results for the the time evolution of the entanglement entropy after a local quench, in which two one-dimensional critical systems at zero temperature are connected at a point and left to evolve unitarily. One can generalize this to the situation where the two critical systems are each thermalized independently at a different temperature. In that case, one can study the entanglement as energy starts to flow between the systems. Since the systems under consideration are infinite and critical, the behaviour after long times is that of a steady state, with a nonzero energy current. I will show that the logarithmic negativity, which is a good measure of entanglement for systems in a mixed state, can be computed exactly in the non-equilibrium steady state, and in certain time regimes leading up to that. I will explain the real-time method used, and discuss their validity. The results are then compared to earlier numerical and analytical results by Eisler and Zimboras for the harmonic chain, and shown to agree.Fabrizio Nieri

__Title:__ 3d & 5d gauge theory partition functions as q-deformed CFT correlators

__Abstract:__ In recent years, due to the method of supersymmetric localization, many exact results have been achieved in the study of supersymmetric gauge theories on compact spaces of various dimension and topology, leading to the discovery of surpraising structures. An important example is provided by the correspondence introduced by Alday, Gaiotto and Tachikawa, relating the partition functions of a large class of supersymmetric gauge theories on S4 and S2 to correlators in Liouville CFT. In this talk, I will explain how this picture can be lifted to higher dimensional gauge theories via the correspondence of partition functions on S5, S4xS1, S3 and S2xS1 to correlators in theories whose underlying symmetry is given by a quantum deformation of the Virasoro algebra. In particular, I will discuss how 3-point functions can be derived by the bootstrap approach and used to define this novel class of q-deformed CFTs. I will also discuss some aspects related to integrable structures in these models, such as reflection coefficients, as well as possible generalisation.