Geometric and Algebraic Approach to Polytopes

对称性是数学中经常出现的一个主题，也是这一提议的核心。在过去的三十年里，人们对几何结构和组合结构及其对称性的兴趣显著复苏。拟议研究的一般领域是离散和组合几何以及几何和代数之间的相互作用。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。我对拟议的研究的贡献是使用经典的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。 拟议研究中的几何问题是关于高度正则多面体结构在欧几里得和双曲三维空间中的实现，这些结构满足某些特定条件，这些条件应该使数学家以及例如化学家或结构工程师感兴趣。我建议研究，目的是对某些具有高度对称性的离散多面体进行分类。这实际上包括几个不同的项目，取决于对称的类型和程度以及周围空间。一些例子是双曲3-空间中的正多面体(具有正则面的多面体，不一定是平面或有限的，并且同构的顶点图形)；均匀多面体(具有正则面片和同构的顶点图形的多面体)，以及欧几里德3-空间中的遗传多面体(继承其面上的所有对称性的多面体)。 我还建议继续最近发表的一系列论文中包含的关于紧欧氏空间形式(平坦的闭3-流形)上的镶嵌分类的研究。更确切地说，我们用欧几里得等距的一个不动点自由子群来分析欧几里德格子的商的轨道空间。规则镶嵌只能在所有等级都已完成分类的环面上进行。其他空间形式的工作只完成了部分。然而，为了完成对不可定向流形的分类，仍有许多工作要做。 在过去的25年里，人们对手性多面体产生了相当大的兴趣，手性多面体是一种抽象结构，具有经典多面体的基本组合性质，通过旋转是最大对称的，但通过反射不是对称的。许多已出版的作品都集中在常规地图上。最近，我和合作者将这一概念扩展到薄几何的工作打开了许多新问题，我建议至少部分回答这些问题。特别感兴趣的是环形剩余连接的薄几何的分类。

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.