Geometric and Algebraic Approach to Polytopes

多面体的几何和代数方法

• 批准号: RGPIN-2016-05354
• 负责人: Weiss, Asia
• 金额: $ 1.31万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=676880
• 关键词: Geometric; Algebraic; Approach; Polytopes

Symmetry, a frequently recurring theme in mathematics, is at core of this proposal. The last three decades have seen a remarkable revival of interest in geometric and combinatorial structures and their symmetry. The general area of the proposed research is in discrete and combinatorial geometry and interaction between geometry and algebra. The questions posed and the methods proposed will capitalize on the recent rapid developments of three areas that have largely developed independently: the study of maps, the theory of geometric and abstract polytopes (combinatorial objects that locally have structure of classical polytopes or tessellations) and thin geometries. My contribution to the proposed research is in the use of classical euclidean and hyperbolic geometry and algebra. Many of the proposed projects will involve junior researchers.***The geometric problems in the proposed research are dealing with realizations of highly regular polyhedral structures in euclidean and hyperbolic 3-spaces which satisfy certain specific conditions that should make them interesting to mathematicians as well as to, for example, chemists or structural engineers. I propose to investigate, with the aim to classify, certain discrete polyhedra with high degree of symmetry. This in fact comprises several different projects depending on the type and degree of symmetry and the ambient space. Some examples are regular polyhedra (polyhedra with regular faces, which are not necessarily planar or finite, and isomorphic vertex-figures) in hyperbolic 3-space; uniform polyhedra (those that have regular facets and isomorphic vertex-figures), and hereditary polyhedra (polyhedra that inherit all symmetries from its facets) in euclidean 3-space.***I also propose to continue the research contained in a series of recently published papers on the classification of tessellations on compact euclidean space-forms (flat closed 3-manifolds). More precisely, we analyze the orbit-space of quotients of euclidean tessellations by a fixed-point free subgroup of euclidean isometries. The regular tessellations are only possible on torus for which the classification has been completed for all ranks. The work on other space-forms has been only partially completed. However, much of the work remains to be done in order to complete the classification for non-orientable manifolds.***Over the past 25 years there has been considerable interest in chiral polytopes, the abstract structures with basic combinatorial properties of classical polytopes that are maximally symmetric by rotations but are not symmetric by reflections. Much of the published work has centred on regular maps. Recent work by myself and collaborators in extending this concept to thin geometries has opened numerous new questions that I propose to at least partially answer. Of particular interest is the classification of toroidal residually connected thin geometries.**

对称性是数学中一个经常出现的主题，也是这个提议的核心。 在过去的三十年里，人们对几何结构、组合结构及其对称性的兴趣显著复苏。建议研究的一般领域是在离散和组合几何和几何与代数之间的相互作用。所提出的问题和提出的方法将利用最近的快速发展的三个领域，在很大程度上独立发展：研究地图，理论的几何和抽象多面体（组合对象，局部具有结构的经典多面体或镶嵌）和薄几何。 我的贡献，拟议的研究是在使用古典欧几里德和双曲几何和代数。 许多拟议的项目将涉及初级研究人员。拟议研究中的几何问题涉及欧几里得和双曲三维空间中高度规则的多面体结构的实现，这些结构满足某些特定条件，这应该会让数学家以及化学家或结构工程师等感兴趣。我建议调查，目的是分类，某些离散多面体的高度对称性。 这实际上包括几个不同的项目，具体取决于对称的类型和程度以及周围的空间。 一些例子是双曲三维空间中的正多面体（具有规则面的多面体，这些面不一定是平面或有限的，并且同构顶点图形）;均匀多面体（具有规则面和同构顶点图形的多面体），以及欧几里得三维空间中的遗传多面体（从其面继承所有对称性的多面体）。我还建议继续研究包含在最近发表的一系列论文的分类上紧欧几里德空间形式（平坦封闭三维流形）的镶嵌。 更确切地说，我们通过欧氏等距的不动点自由子群来分析欧氏镶嵌的子空间。 常规镶嵌仅可在已完成所有等级分类的环面上进行。 关于其他空间形式的工作只完成了一部分。然而，为了完成不可定向流形的分类，还有许多工作要做。在过去的25年中，人们对手征多面体有相当大的兴趣，手征多面体是具有经典多面体的基本组合性质的抽象结构，其通过旋转最大对称，但通过反射不对称。 出版的大部分作品都集中在常规地图上。 我和我的合作者最近将这一概念扩展到薄几何形状的工作已经提出了许多新的问题，我建议至少部分回答。特别令人感兴趣的是环形剩余连通薄几何的分类。