Geometric theory of topological defects: methodological developments and new trends

ABSTRACT Liquid crystals generally support orientational singularities of the director field known as topological defects. These latter modify transport properties in their vicinity as if the geometry was non-Euclidean. We present a state-of-the-art of differential geometry of nematic liquid crystals, with a special emphasis on linear defects. We then discuss unexpected but deep connections with cosmology and high-energy-physics and conclude with a review on defect engineering for transport phenomena.