In the classical theory of fluid mechanics, a linear relationship between the stress and rate of strain is often assumed. Even when this relationship is non-linear, it is typically formulated in terms of an explicit relation. Implicit constitutive theories provide a theoretical framework that generalises this, allowing a, possibly multi-valued, implicit constitutive relation. Since it is not possible to solve explicitly for the stress in the constitutive relation, a more natural approach would be to include the stress as a fundamental unknown in the formulation of the problem. In this talk I will present a formulation with this feature and a proof of convergence of the finite element approximations to a solution of the original problem.

The celebrated localisation theorem of Beilinson-Bernstein asserts that there is an equivalence between representations of a Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. In this talk we discuss certain analogues of this result in various contexts. Namely, there is a localisation theorem for quantum groups due to Backelin and Kremnizer and, more recently, Ardakov and Wadsley also proved a localisation theorem working with certain completed enveloping algebras of p-adic Lie algebras. We then explain how to combine the ideas involved in these results to construct
a p-adic analytic quantum flag variety and a category of D-modules on it, and we show that the global section functor on these D-modules yields an equivalence of categories.

 I will describe recent progress in the study of scattering amplitudes in gauge theory and gravity at loop level, using the formalism of the scattering equations. The scattering equations relate the kinematics of the scattering of massless particles to the moduli space of the sphere. Underpinned by ambitwistor string theory, this formalism provides new insights into the relation between tree-level and loop-level contributions to scattering amplitudes. In this talk, I will describe results up to two loops on how loop integrands can be constructed as forward-limits of trees. One application is the loop-level understanding of the colour-kinematics duality, a symmetry of perturbative gauge theory which relates it to perturbative gravity.

Superstring scattering amplitudes in genus one have a low-energy expansion in terms of certain real analytic modular forms, called modular graph functions (D'Hoger, Green, Gürdogan and Vanhove). I will sketch the proof that these functions belong to a family of iterated integrals of modular forms (a generalization of Eichler integrals), recently introduced by Francis Brown, which explains many of their properties. The main tools are elliptic multiple polylogarithms (Brown and Levin), single-valued versions thereof, and elliptic multiple zeta values (Enriquez).

This is a survey on recent advances in classical decidability issues for local and global fields and for some canonical infinite extensions of those.