Operators, functions, and functionals are combined in many problems of computational science in a fashion that has the same logical structure as is familiar for block matrices and vectors.  It is proposed that the explicit consideration of such block structures at the continuous as opposed to discrete level can be a useful tool.  In particular, block operator diagrams provide templates for spectral discretization by the rectangular differentiation, integration, and identity matrices introduced by Driscoll and Hale.  The notion of the rectangular shape of a linear operator can be made rigorous by the theory of Fredholm operators and their indices, and the block operator formulations apply to nonlinear problems too, where the convention is proposed of representing nonlinear blocks as shaded.  At each step of a Newton iteration, the structure is linearized and the blocks become unshaded, representing Fréchet derivative operators, square or rectangular.  The use of block operator diagrams makes it possible to precisely specify discretizations of even rather complicated problems with just a few lines of pseudocode.

[Joint work with Nick Trefethen]

The topological Fukaya category is a combinatorial model of the Fukaya category of exact symplectic manifolds which was first proposed by Kontsevich. In this talk I will explain work in progress (joint with J. Pascaleff and S. Scherotzke) on gluing techniques for the topological Fukaya category that are closely related to Viterbo functoriality. I will emphasize applications to homological mirror symmetry for three-dimensional CY LG models, and to Bondal's and Fang-Liu-Treumann-Zaslow's coherent constructible correspondence for toric varieties.  

The formal degree is a fundamental invariant of a discrete series representation which generalizes the notion of dimension from finite dimensional representations. For discrete series with unipotent cuspidal support, a formula for formal degrees, conjectured by Hiraga-Ichino-Ikeda, was verified by Opdam (2015). For split exceptional groups, this formula was previously known from the work of Reeder (2000). I will present a different interpretation of the formal degrees of unipotent discrete series in terms of the nonabelian Fourier transform (introduced by Lusztig in the character theory of finite groups of Lie type) and certain invariants arising in the elliptic theory of the affine Weyl group. This interpretation relates to recent conjectures of Lusztig about `almost characters' of p-adic groups. The talk is based on joint work with Eric Opdam.

The Faraday cage effect is the phenomenon whereby electrostatic and electromagnetic fields are shielded by a wire mesh "cage". Nick Trefethen, Jon Chapman and I recently carried out a mathematical analysis of the two-dimensional electrostatic problem with thin circular wires, demonstrating that the shielding effect is not as strong as one might infer from the physics literature. In this talk I will present new results generalising the previous analysis to the electromagnetic case, and to wires of arbitrary shape. The main analytical tool is the asymptotic method of multiple scales, which is used to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. In the electromagnetic case one observes interesting resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells. This is joint work with Ian Hewitt.

The accumulation of surface meltwater on ice shelves can lead to the formation of melt lakes. These structures have been implicated in crevasse propagation and ice-shelf collapse; the Larsen B ice shelf was observed to have a large amount of melt lakes present on its surface just before its collapse in 2002. Through modelling the transport of heat through the surface of the Larsen C ice shelf, where melt lakes have also been observed, this work aims to provide new insights into the ways in which melt lakes are forming and the effect that meltwater filling crevasses on the ice shelf will have. This will enable an assessment of the role of meltwater in triggering ice-shelf collapse. The Antarctic Peninsula, where Larsen C is situated, has warmed several times the global average over the last century and this ice shelf has been suggested as a candidate for becoming fully saturated with meltwater by the end of the current century. Here we present results of a 1-D mathematical model of heat transfer through an idealized ice shelf. When forced with automatic weather station data from Larsen C, surface melting and the subsequent meltwater accumulation, melt lake development and refreezing are demonstrated through the modelled results. Furthermore, the effect of lateral meltwater transport upon melt lakes and the effect of the lakes upon the surface energy balance are examined. Investigating the role of meltwater in ice-shelf stability is key as collapse can affect ocean circulation and temperature, and cause a loss of habitat. Additionally, it can cause a loss of the buttressing effect that ice shelves can have on their tributary glaciers, thus allowing the glaciers to accelerate, contributing to sea-level rise.