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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708510
• 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.

对称性是数学中经常出现的一个主题，是这个提议的核心。近三十年来，人们对几何和组合结构及其对称性的兴趣显著复苏。提出的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在很大程度上独立发展的三个领域的快速发展：地图研究、几何和抽象多面体理论（局部具有经典多面体或镶嵌结构的组合对象）和薄几何。我对提议的研究的贡献是在使用经典欧几里得和双曲几何和代数。许多拟议的项目将涉及初级研究人员。