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

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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. ***