Multidimensional consistency is nowadays considered to be one of the hallmarks of integrable equations on the lattice, i.e. partial difference equations in two or more dimensions. However, it also applies also to certain families of continuous equations, namely partial differential equations associated with integrable hierarchies. The new notion of Lagrangian multiform structures, introduced by Lobb and Nijhoff in 2009, is a manifestation of multidimensional consistency on the level of the Lagrange structures and least-action principles. In the talk I will describe this property and show how it emerges in various examples of discrete and continuous equations. (This is work in collaboration with Sarah Lobb and Pavlos Xenitidis.)
Critical points of the bending energy, the so-called Willmore surfaces, have already been studied by Blaschke and Thomsen in the 1920s and Willmore rekindled the interest forty years later by conjecturing the minimal value of the bending energy for tori. A lot of progress has been made since then towards solving the Willmore conjecture by connecting Willmore surfaces to integrable systems. In my talk I will discuss the geometric interpretation of the spectral curve of a Willmore torus in terms of Darboux transformations.
It is interesting to study topological solitons on d-dimensional spaces R^(d-1) x S^1. Surprising things happen when the circle S^1 is small compared to the typical size of a soliton. Here I focus on the case of sigma model lumps on the cylinder R x S^1: I discuss the topological charges present in this case and describe what the solutions look like. I argue that the structure of the solutions can be understood using root systems of loop groups.
I consider superstring sigma models on semi-symmetric superspaces. Such sigma models are integrable and arise naturally in the context of gauge theory/string theory dualities. I will explain how they can be understood in terms of a particular eight-dimensional generalisation of self-dual Yang-Mills theory. I will also explain that this generalised theory admits a twistorial re-interpretation. Such a relationship might help shed light on the explicit construction of solutions to the superstring equations and their hidden symmetry structures.
We consider ways of constructing Ricci solitons by reducing the equations to dynamical systems. In some cases the resulting equations are integrable, while in others this is an open question.
I will present some results, obtained with Martin Svensson, on existence and stability of critical points of a natural energy functional for maps from a Riemannian to a symplectic manifold, which arises as a certain limit of the Faddeev-Skyrme model. I will concentrate particularly on the case where the domain is two-dimensional. In this case, the model supports "plastic" topological solitons: global energy minimizers which can be deformed into any shape, and have extremely unusual spatial decay properties.
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Last updated 11th May, 2010