Mathematical Sciences: Quantum and Finite Type Invariants of Links in 3-Manifolds, Quasicrystals

9626404乐尚乐正在研究三维流形和准晶中的链接。关于环的研究涉及三维流形中纽结和环的量子和有限类型不变量，包括Witten-Reshetikhin-Turaev不变量的推广。准晶的研究涉及到确定欧几里得空间准周期平铺的局部规则。这个项目中的每个主题都与物理密切相关。目前已知的最强大的纽结和链环不变量(即最有可能区分不同但相似的物体)在量子场论中找到了它们的动机。该理论的代数方面是美丽的，结果也很丰富。然而，除了简单情况外，不变量的计算仍然很困难，其内在的拓扑意义仍然模糊不清。第二个主题，准晶，源于1984年发现的固体物质，其光谱具有尖锐的峰，享有一些真正晶体没有的对称性(例如，5倍对称性)。人们认为，准晶的数学模型是空间的准周期瓦片，比如物理学家罗杰·彭罗斯在1973年发现的著名的彭罗斯瓦片，但当时人们认为这只是一种数学上的好奇。该项目旨在阐明这些主题中的每一个。***

9626404 Le Thang Le is studying links in 3-manifolds as well as quasicrystals. The study of links concerns quantum and finite type invariants of knots and links in 3-manifolds, including generalizations of Witten-Reshetikhin-Turaev invariants. The study of quasicrystals involves local rules that determine quasiperiodic tilings of Euclidean space. Each of the topics in this project is intimately related to physics. The most powerful invariants of knots and links currently known (i.e., most likely to distinguish among different but similar objects) find their motivation in quantum field theory. The algebraic aspect of the theory is beautiful and rich in results. However, the invariants remain difficult to compute except in simple cases, and their intrinsic topological meaning remains obscure. The second topic, quasicrystals, stems from the discovery in 1984 of solid substances whose spectrum has sharp peaks enjoying some symmetries that real crystals do not (for example, 5-fold symmetry). It is believed that a mathematical model of a quasicrystal is a quasiperiodic tiling of space, such as the famous Penrose tilings of the plane discovered in 1973 by the physicist Roger Penrose but considered at that time to be only a mathematical curiosity. The project aims to shed light on each of these topics. ***