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

小行星9626404 Thang Le正在研究3-流形和准晶体中的链接。 链接的研究涉及量子和有限型不变量的纽结和链接在3-流形，包括推广Witten-Reshetikhin-Turaev不变量。 准晶的研究涉及确定欧氏空间的准周期镶嵌的局部规则。 这个项目中的每一个主题都与物理学密切相关。 目前已知的结和链接的最强大的不变量（即，最有可能区分不同但相似的物体） 在量子场论中找到他们的动机 代数方面的理论是美丽的和丰富的结果。 然而，不变量 仍然难以计算，除了在简单的情况下，其内在的拓扑意义仍然模糊。 第二个主题，准晶体，源于1984年发现的固体物质，其光谱具有 尖峰具有真实的晶体所不具有的对称性（例如，5重对称性）。 据信， 准晶体是空间的准周期性平铺，例如著名的 1973年，物理学家罗杰· 彭罗斯，但认为在当时只是一个数学好奇心。 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. ***