Since their discovery at the end of the nineteenth century, liquid crystals have been the source of important technological advances.
Liquid crystals consist of elongated and rigid molecules.
From a different point of view, the nature of the surface interaction between the liquid crystal and the surrounding media is at the center of most technological applications of such materials. A great deal of mathematical effort in liquid crystal research has been devoted to the study of the nematic phase. New mathematical research addresses questions such as the modeling of chiral smectic phases, and the role of the boundary conditions and surface interactions of the liquid crystal with neighboring materials. Such properties are at the core of important new applications such as large video display devices and organic semiconductors. This talk will deal with both such questions, the modeling of chiral smectic phases as well as free boundary problems for nematic droplets.