Mathematics of Planet Earth 2013

Liquid crystals represent a vast and diverse class of anisotropic soft matter materials which are intermediate between isotropic liquids and crystalline solids. Liquid crystal ordering is found in a wide variety of systems, ranging from fluids made up of simple rods, polymers and amphiphile solutions, to elastomers and biological organisms. Liquid crystals have a multitude of applications, notably those in flat panel display technology, which has fundamentally impacted modern life. From a theoretical point of view, liquid crystals offer a unique opportunity for the study of partial order, as complex liquid crystal phases represent the most well-organised known states of soft matter. At present, the theory of liquid crystals is developed by two groups of researchers which do not currently interact effectively with each other. A first group consists of physicists, chemists and engineers with extensive experimental and theoretical experience in the field. A second, rapidly expanding group, comprises mathematicians who have started to work on the theory of liquid crystals, attracted by its intriguing mathematical structure and connections to different branches of mathematics. The aim of the programme is to bring together experts from these two groups to facilitate an exchange of ideas, the exploration of different approaches and the sharing of knowledge. One important goal is the identification of key mathematical problems arising from experiment, computer simulations and device applications.

The programme will facilitate knowledge transfer, attract mathematicians to the field and help establish long term collaborations which will enrich both groups of researchers. A wide spectrum of mathematical problems will be considered, related to the modeling of liquid crystals with varying detail and at different length scales. Models will range from continuum descriptions, where the symmetry group of the ordered phase plays a key role, to computer simulations on an atomistic level. One important set of open problems to be considered is the relationship between these different levels of modeling; for example how one can make a rigorous passage from molecular/statistical descriptions to continuum theories and how the results of computer simulations can be used to test the validity of statistical and continuum models. Emphasis will be placed on the use of symmetry, bifurcation and group theory approaches to study the relationship between the symmetry of individual molecules and interaction potentials, the symmetry of the resulting phases and the structure of the order parameter space. On the continuum level, the programme will address the dynamics of various liquid crystal phases and the systematic derivation of various equations of motion, and will aim to clarify the accuracy of different descriptions and their applicability to different liquid crystal systems. Special consideration will be given to the existence, uniqueness and regularity of the solutions of these equations, as well as the bifurcation and properties of equilibrium and periodic solutions, solitary waves, and dynamic switching, and to how well the equations describe phase transitions.

An important target is to find effective approximations which can be used to overcome the challenges posed by the complexity of liquid crystal phases with biaxial and more complicated order, which are becoming important in emerging applications. A strong effort will be made to identify and explore mathematical problems arising from novel liquid crystal systems, such as liquid crystal elastomers and nano-particle liquid crystal metamaterials. The programme will include four workshops covering all major aspects of the subject.