For $G$ connected, reductive algebraic group defined over $\mathbb{C}$ the Springer Correspondence gives a bijection between the irreducible representations of the Weyl group $W$ of $G$ and certain pairs comprising a $G$-orbit on the nilpotent cone of the Lie algebra of $G$ and an irreducible local system attached to that $G$-orbit. These irreducible representations can be concretely realised as a W-action on the top degree homology of the fibres of the Springer resolution. These Springer fibres are geometrically very rich and provide interesting Weyl group combinatorics: for instance, the irreducible components of these Springer fibres form a basis for the corresponding irreducible representation of $W$. In this talk, I'll give a general survey of the Springer Correspondence and then discuss recent joint projects with Daniele Rosso, Vinoth Nandakumar and Arik Wilbert on Kato's Exotic Springer correspondence.

I will explain the notion of a singular ring and sketch how singular rings provide field and vertex algebras introduced by Borcherds and Kac. All of these notions make sense in general symmetric monoidal categories and behave nicely with respect to symmetric lax monoidal functors. I will provide a complete classification of singular rings if the tensor product is a cartesian product. This applies in particular to categories of topological spaces or (algebraic) stacks equipped with the usual cartesian product. Moduli spaces provide a rich source of examples of singular rings. By combining these ideas, we obtain vertex and field algebras for each reasonable moduli space and each choice of an orientable homology theory. This generalizes a recent construction of vertex algebras by Dominic Joyce.

When the linear system in Newton’s method is approximately solved using an iterative method we have an inexact or truncated Newton method. The outer method is Newton’s method and the inner iterations will be the iterative method. The Inexact Newton framework is now close to 30 years old and is widely used and given names like Newton-Arnoldi, Newton-CG depending on the inner iterative method. In this talk we will explore convergence properties when the outer iterative method is Gauss-Newton, the Halley method or an interior point method for linear programming problems.

Einstein metrics are fixed points (up to scaling) of Hamilton's Ricci flow. A natural question to ask is whether a given metric is stable in the sense that the flow returns to the Einstein metric under a small perturbation. I'll give a brief survey of this area focussing on the case when the Einstein constant is positive. An interesting class of metrics where this question is not completely resolved are the compact symmetric spaces. I'll report on some recent progress with Tommy Murphy and James Waldron where we have been able to use a criterion due to Kroencke to show the Kaehler-Einstein metric on some Grassmannians and the bi-invariant metric on the Lie group G_2 are unstable.

Driven by its numerous applications in computational science, the approximation of smooth, high-dimensional functions via sparse polynomial expansions has received significant attention in the last five to ten years.  In the first part of this talk, I will give a brief survey of recent progress in this area.  In particular, I will demonstrate how the proper use of compressed sensing tools leads to new techniques for high-dimensional approximation which can mitigate the curse of dimensionality to a substantial extent.  The rest of the talk is devoted to approximating functions defined on irregular domains.  The vast majority of works on high-dimensional approximation assume the function in question is defined over a tensor-product domain.  Yet this assumption is often unrealistic.  I will introduce a method, known as polynomial frame approximation, suitable for broad classes of irregular domains and present theoretical guarantees for its approximation error, stability, and sample complexity.  These results show the suitability of this approach for high-dimensional approximation through the independence (or weak dependence) of the various guarantees on the ambient dimension d.  Time permitting, I will also discuss several extensions.

Chemical reaction network (CRN) theory focusses on making claims about dynamical behaviours of reaction networks which are, as far as possible, dependent on the network structure but independent of model details such as functions chosen and parameter values. The claims are generally about the existence, nature and stability of limit sets, and the possibility of bifurcations, in models of CRNs with particular structural features. The methodologies developed can often be applied to large classes of models occurring in biology and engineering, including models whose origins are not chemical in nature. Many results have a natural algorithmic formulation. Apart from the potential for application, the results are often pleasing mathematically for their power and generality. 

This talk will concern some recent themes in CRN theory, particularly focussed on how the presence or absence of particular subnetworks ("motifs") influences allowed dynamical behaviours in ODE models of a CRN. A number of recent results take the form: "a CRN containing no subnetworks satisfying condition X cannot display behaviour of type Y"; but also, in the opposite direction, "if a CRN contains a subnetwork satisfying condition X, then some model of this CRN from class C admits behaviour of type Y". The proofs of such results draw on a variety of techniques from analysis, algebra, combinatorics, and convex geometry. I'll describe some of these results, outline their proofs, and sketch some current challenges in this area. 
 

Cracks in many soft solids behave very differently to the classical picture of fracture, where cracks are long and thin, with damage localised to a crack tip. In particular, small cracks in soft solids become highly rounded — almost circular — before they start to extend. However, despite being commonplace, this is still not well understood. We use a phase-separation technique in soft, stretched solids to controllably nucleate and grow small, nascent cracks. These give insight into the soft failure process. In particular, our results suggest fracture occurs in two regimes. When a crack is large, it obeys classical linear-elastic fracture mechanics, but when it is small it grows in a new, scale-free way at a constant driving stress.