Studies in geometry and topology of translationally ordered materials.

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Despite many attempts, there is still no coherent theory of defects and goldstones theorem for translationally ordered materials, as viewed through the paradigm of spontaneous symmetry breaking.This project will study the geometry and topology of translationally ordered materials, using smectic-A liquid crystals as a prototype.The first part of the project will be an examination of defects in two dimensional translationally order materials through understanding the decomposition of paths in the groundstate manifold (GSM) of the system, and their relation to the symmetry group of the groundstate. The aim is to show that a novel quantisation phenomena on the GSM accounts for many observed material properties. This will be in collaboration with Prof. J.H. Hannay (Bristol) and Prof. R.D. Kamien (UPenn).This perspective will then be used to inform the development of accurate continuum models of these systems (most notably smectic liquid crystals) that successfully capture the fully symmetry of the material, which current theories do not.The second part of the PhD project will aim to develop a version of Goldstone's theorem for fields on general homogeneous (symmetric) spaces, by understanding the role of Lie algebra homorphisms between special symmetries and field symmetries, allowing for an elegant and unifying description of low-energy deformation of translationally ordered systems. This work will attempt to unify and extend results from condensed matter as well high energy theory.