This paper introduces the key concepts and problems of the new research area of Periodic Geometry and Topology for applications in Materials Science. Periodic structures such as solid crystalline materials or textiles were previously studied as isolated structures without taking into account the continuity of their configuration spaces. The key new problem in Periodic Geometry is an isometry classification of periodic point sets. A required complete invariant should continuously change under point perturbations, because atoms always vibrate in real crystals. The main objects of Periodic Topology are embeddings of curves in a thickened plane that are invariant under lattice translations. Such periodic knots were classified in the past up to continuous deformations (isotopies) that keep a fixed lattice structure, hence are realized in a fixed thickened torus. The more practical equivalence is a periodic isotopy in a thickened plane without fixing a lattice basis. The paper states the first results in the new area and proposes further problems and directions.

中文翻译：

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周期几何和拓扑简介

本文介绍了在材料科学中应用的周期性几何和拓扑学新研究领域的关键概念和问题。以前将诸如固态晶体材料或纺织品之类的周期性结构作为隔离结构进行了研究，而没有考虑其配置空间的连续性。周期几何中的关键新问题是周期点集的等距分类。所需的完全不变性应在点扰动下不断变化，因为原子总是在真实晶体中振动。周期性拓扑的主要对象是在晶格平移下不变的，在增厚平面中的曲线嵌入。过去，此类周期性结被分类为保持固定晶格结构的连续变形（同位素），因此可以在固定的加厚圆环中实现。更实际的当量是在不固定晶格基础的情况下在加厚平面中的周期性同位素。本文陈述了新领域的第一个结果，并提出了进一步的问题和方向。