Geometry and Dynamics of Topological Solitons

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Field theory is the natural mathematical language to explain fundamental physics, from particle physics, to condensed matter, to gravitation and cosmology. In many field theories it is possible for the dynamical field, a map from spacetime into some manifold, to wrap itself up rather like a high-dimensional knot. Such a field traps large amounts of energy in a smooth, spatially localized lump, called a topological soliton, which can move around and interact with other solitons in remarkably particle-like fashion. Solitons are natural candidates for fundamental particles (e.g. magnetic monopoles, protons, neutrons), but they also model large scale structures in condensed matter physics (e.g. vortices in superconductors) and cosmology (e.g. cosmic strings). There is a beautiful geometric theory of the dynamics of solitons which describes their motion in terms of the moduli space of static multisolitons, a mathematically rich and fascinating object in its own right. The key challenge in the theory is to compute, or understand in the absence of explicit formulae, the canonical metric on the soliton moduli space. The aim of this project is to study the geometry of topological solitons and connect it to their dynamics, in the specific context of gauge theory.Specific goals are:1. The compuation of the canonical metric on vortex moduli spaces in certain tractable limits.2. The construction of instantons on spaces of dimension higher than 4.This is a pure mathematical project, and is not expected to have any technological applications.