Liquid crystals are a phase of matter with properties somewhere between liquids and crystalline solids, with applications in display technologies, optical devices, and biosensors. This project studies theoretical frameworks for liquid crystal systems which require the development of advanced mathematical techniques. The research focuses on two areas: ferroelectric nematic liquid crystals, which exhibit switchable electric polarization and show promise for improved electro-optical devices, and liquid crystals with very high symmetries that require mathematical tools like tensors for an accurate description. The mathematical methods proposed in this project should improve computational simulations and help understand structural ordering and defects in these materials. By developing new theoretical models and computational methods, this work supports advances in sensing and simulation technologies. This project also contributes to education and workforce development in science and engineering, involving the training of both graduate and undergraduate students in this research area. This proposal presents two research directions investigating novel structures in nematic liquid crystals. The first focuses on ferroelectric nematic liquid crystals, a recently observed phase that was long predicted theoretically and enables polarity switching. These materials develop singular structures similar to domain walls in ferromagnetic systems, occurring in regions where polarity transiti