Topological and Geometrical Problems in Soft Matter

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TECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education to advance and elucidate the theory of knotted fields. Though it has been well known for more than three decades that topological defects can act on each other through commutation, it is only in recent times that we can: a) knot defects by design; b) tie and manipulate knotted disclinations; and c) study biaxial nematics, a physical system with a non-Abelian fundamental group. The stream of experimental results that has already started will be ample fodder for studying the topological dynamics of these knots. What makes this most exciting is that these are now questions that can be probed directly through experiment.A second thrust of the research is the topological characterization and classification of defects in translationally ordered media. The PI will focus specially on smectic liquid crystals because translational order is only broken in one direction and thus there is only one associated Nambu-Goldstone mode. In addition to dislocations and disclinations, smectics enjoy a different class of geometrically precise configurations, focal conic domains. The interaction between focal conic domains and topological defects is largely underexplored. Finally, and most importantly, because smectics are so soft, dislocations, disclinations and focal conic domains are easy to observe and manipulate in experiment. Building on the work of the PI and his group over the past three years, the PI will formulate a theoretical description to combine the three types of building blocks in the case of three-dimensional smectics. This award also supports the PI's efforts to convey the science of liquid crystals effectively to K-12 and college students, high school teachers, and the general public.NONTECHNICAL SUMMARYThe Division of Mathematical Sciences and the Division of Materials Research contribute funds to this award. It supports theoretical research and education to advance the theory of soft matter. The PI has a track record of learning and applying modern mathematics and incorporating it into his research. In turn, these insights have been applied directly to experiments that employ traditional microscopy, polarizing filters, and hot plates. The focus of this research project is on liquid crystals, materials that pervade our modern life, from our phones and watches to our computers and cars. The optical properties that make them the backbone of a $100 billion/year industry arise from a beautiful interplay between chemistry, theoretical physics, geometry, and topology. The PI will study the generalization of the fluid vortices that shed off the tip of a canoe paddle or an airplane wing. To do this, the PI will integrate the fields of materials physics, statistical mechanics, topology, and geometry, challenging traditional disciplinary lines with the hope of substantial technological and intellectual progress. Visually compelling, the field of liquid crystals has and will allow the PI to reach out effectively to K-12 and college students, high school teachers, and the general public.