Organisation des séminaires et calendrier

Over the past decade integrability for rational maps (but also for lattice equations) has come to be defined through a « zero algebraic entropy » requirement, and a lot of research effort has gone into the development of novel rigorous methods to calculate the entropy for a given mapping or equation. For mappings on the complex (projective) plane this definition fits very well with a more analytic notion of « integrability », in terms of the existence or non-existence of conserved quantities. In higher dimensions, or for lattice equations, such a link is far from clear and one could argue that the zero algebraic entropy requirement is perhaps too coarse as a meaningful integrability criterion and that a more refined definition is needed.

In the first part of the talk I will give an overview of the different notions involved in detecting integrability in rational mappings, in particular algebraic entropy and singularity confinement, and I will present a novel and quite spectacular link between both notions for mappings of the plane. In the latter part of the talk I will comment on the paucity of results for mappings in higher dimensions and I will discuss several problems with the entropy approach to defining integrability in the higher dimensional (or lattice) case.

This talk is partly based on

• A. Stokes, T. Mase, R. Willox and B. Grammaticos, Deautonomisation by singularity confinement and degree growth, The Journal of Geometric Analysis 35, 65 (2025)
  and
• A Ramani, B Grammaticos, A S Carstea and R Willox, Obtaining the growth of higher order mapping through the study of singularities, Journal of Physics A: Mathematical and Theoretical 58(11) 115201 (2025)

Sebastian Harris (DESY)

Abstract: AdS/CFT is a strong–weak duality: it allows the study of strongly coupled gauge theories through classical geometry, and highly stringy regimes of quantum gravity through perturbative gauge theory. This complementarity makes the correspondence powerful but also difficult to prove, since there is usually no regime in which both sides are simultaneously under computational control. AdS3 offers a remarkable exception to this. There exists an exactly solvable AdS3 string background dual to a free 2d gauge theory. In this context, AdS/CFT can be fully derived: It was shown that all closed string scattering amplitudes match precisely the local correlation functions of the dual gauge theory. The talk reviews this remarkable derivation and presents recent progress extending it to encompass branes and open strings, as well as the dual interfaces and their local excitations.

Overview : https://indico.cern.ch/event/1514154/overview

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