Supersymmetric gauge theories in five dimensions, although power counting non-renormalizable, are known to be in some cases UV completed by a superconformal field theory. Many tools, such as M-theory compactification and pq-web constructions, were used in recent years in order to deepen our understanding of these theories. This framework gives us a concrete way in which we can try to search for additional IR conformal field theory via deformations of these well-known superconformal fixed points. Recently, the authors of 2001.00023 proposed a supersymmetry breaking mass deformation of the E_1theory which, at weak gauge coupling, leads to pure SU(2) Yang-Mills and which was conjectured to lead to an interacting CFT at strong coupling. During this talk, I will provide an explicit geometric construction of the deformation using brane-web techniques and show that for large enough gauge coupling a global symmetry is spontaneously broken and the theory enters a new phase which, at infinite coupling, displays an instability. The Yang-Mills and the symmetry broken phases are separated by a phase transition. Quantum corrections to this analysis are discussed, as well as possible outlooks. Based on arXiv: 2109.02662.

Climate- and weather prediction centres such as the Met Office rely on efficient numerical methods for simulating large scale atmospheric flow. One computational bottleneck in many models is the repeated solution of a large sparse system of linear equations. Preconditioning this system is particularly challenging for state-of-the-art discretisations, such as (mimetic) finite elements or Discontinuous Galerkin (DG) methods. In this talk I will present recent work on developing efficient multigrid preconditioners for practically relevant modelling codes. As reported in a REF2021 Industrial Impact Case Study, multigrid has already led to runtime savings of around 10%-15% for operational global forecasts with the Unified Model. Multigrid also shows superior performance in the Met Office next-generation LFRic model, which is based on a non-trivial finite element discretisation.

A topological insulator has anomalous boundary spectrum which completely fills up gaps in the bulk spectrum. This ``topologically protected’’ spectral property is a physical manifestation of coarse geometry and index theory ideas. Special examples involve spectral flow and gerbes, related to Hamiltonian anomalies, and they arise experimentally in quantum Hall systems, time-reversal invariant mod-2 insulators, and shallow-water waves.

Symmetry protected topological (SPT) phases have recently attracted a lot of
attention from physicists and mathematicians as a topological classification
scheme for gapped ground states. In this talk I will briefly introduce the
operator algebraic approach to SPT phases in the infinite-volume limit. In
particular, I will focus on the quasifree (free-fermionic) setting, where we

can adapt tools from algebraic quantum field theory to describe phases of
gapped ground states using K-homology and the coarse index.