Topological and Geometrical Problems in Soft Matter

数学科学部和材料研究部为该奖项提供资金。它支持理论研究和教育，以推进和阐明打结领域的理论。 虽然人们已经知道了三十多年，拓扑缺陷可以通过交换相互作用，但直到最近，我们才能：a）通过设计将缺陷打结; B）捆绑和操纵打结的向错; c）研究双轴向列，一种具有非阿贝尔基本群的物理系统。 已经开始的一系列实验结果将为研究这些结的拓扑动力学提供充足的素材。最令人兴奋的是，这些问题现在可以直接通过实验来探索。研究的第二个重点是有序介质中缺陷的拓扑表征和分类。PI将特别关注近晶液晶，因为平移顺序仅在一个方向上被打破，因此只有一个相关的南武-戈德斯通模式。 除了位错和向错，近晶还具有另一类几何上精确的结构，即焦锥域。 焦锥域和拓扑缺陷之间的相互作用在很大程度上是探索不足。 最后，也是最重要的一点，由于近晶是如此柔软，位错、向错和焦锥畴在实验中很容易观察和操作。 在PI及其小组过去三年工作的基础上，PI将制定一个理论描述，以便在三维smectics的情况下将三种类型的构建块联合收割机结合起来。 该奖项也支持PI的努力，有效地传达液晶科学的K-12和大学生，高中教师，和一般公众。非技术性总结数学科学部和材料研究部为该奖项提供资金。它支持理论研究和教育，以推进软物质理论。PI有学习和应用现代数学的记录，并将其纳入他的研究。 反过来，这些见解已被直接应用到实验中，采用传统的显微镜，偏振过滤器和热板。 这个研究项目的重点是液晶，这种材料渗透到我们的现代生活中，从我们的手机和手表到我们的电脑和汽车。光学特性使它们成为每年1000亿美元产业的支柱，这源于化学、理论物理、几何和拓扑之间的美丽相互作用。 PI将研究从独木舟桨或飞机机翼尖端脱落的流体涡流的一般化。 为此，PI将整合材料物理，统计力学，拓扑学和几何学领域，挑战传统学科，希望取得实质性的技术和知识进步。 视觉上引人注目的是，液晶领域已经并将使PI能够有效地接触到K-12和大学生，高中教师和公众。

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.