Geometric and Algebraic Approach to Polytopes

多面体的几何和代数方法

基本信息

  • 批准号:
    RGPIN-2016-05354
  • 负责人:
  • 金额:
    $ 1.31万
  • 依托单位:
  • 依托单位国家:
    加拿大
  • 项目类别:
    Discovery Grants Program - Individual
  • 财政年份:
    2018
  • 资助国家:
    加拿大
  • 起止时间:
    2018-01-01 至 2019-12-31
  • 项目状态:
    已结题

项目摘要

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。***提出的研究中的几何问题是处理欧几里得和双曲三维空间中高度规则多面体结构的实现,这些结构满足某些特定的条件,这些条件应该使数学家以及化学家或结构工程师感兴趣。我打算研究一些具有高度对称性的离散多面体,目的是对它们进行分类。这实际上包括几个不同的项目,这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规则多面体(具有规则面的多面体,不一定是平面或有限的,以及同构的顶点图形);欧几里得三维空间中的均匀多面体(具有规则切面和同构顶点图形的多面体)和遗传多面体(从其切面继承所有对称性的多面体)。***我还建议继续研究最近发表的一系列关于紧致欧几里得空间形式(平闭3流形)上镶嵌分类的论文。更准确地说,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得镶嵌的商的轨道空间。规则的镶嵌只有在所有等级的分类都已完成的环面上才有可能。其他空间形式的工作只完成了一部分。然而,为了完成非定向流形的分类,还有很多工作要做。***在过去的25年里,人们对手性多面体产生了相当大的兴趣,手性多面体是具有经典多面体基本组合性质的抽象结构,通过旋转达到最大对称,但通过反射不对称。大部分出版的作品都集中在常规地图上。我和合作者最近将这一概念扩展到薄几何的工作已经提出了许多新问题,我建议至少部分回答这些问题。特别感兴趣的是环面残连薄几何的分类

项目成果

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Weiss, Asia其他文献

Weiss, Asia的其他文献

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{{ truncateString('Weiss, Asia', 18)}}的其他基金

Geometric and Algebraic Approach to Polytopes
多面体的几何和代数方法
  • 批准号:
    RGPIN-2016-05354
  • 财政年份:
    2021
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Geometric and Algebraic Approach to Polytopes
多面体的几何和代数方法
  • 批准号:
    RGPIN-2016-05354
  • 财政年份:
    2020
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Geometric and Algebraic Approach to Polytopes
多面体的几何和代数方法
  • 批准号:
    RGPIN-2016-05354
  • 财政年份:
    2019
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Geometric and Algebraic Approach to Polytopes
多面体的几何和代数方法
  • 批准号:
    RGPIN-2016-05354
  • 财政年份:
    2017
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Geometric and Algebraic Approach to Polytopes
多面体的几何和代数方法
  • 批准号:
    RGPIN-2016-05354
  • 财政年份:
    2016
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Regular and chiral polytopes and their realizations
规则和手性多胞体及其实现
  • 批准号:
    8857-1997
  • 财政年份:
    1998
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Regular and chiral polytopes and their realizations
规则和手性多胞体及其实现
  • 批准号:
    8857-1997
  • 财政年份:
    1997
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Combinatorial and group theoretic study of polytopes
多胞体的组合和群论研究
  • 批准号:
    8857-1993
  • 财政年份:
    1996
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Combinatorial and group theoretic study of polytopes
多胞体的组合和群论研究
  • 批准号:
    8857-1993
  • 财政年份:
    1995
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual
Combinatorial and group theoretic study of polytopes
多胞体的组合和群论研究
  • 批准号:
    8857-1993
  • 财政年份:
    1994
  • 资助金额:
    $ 1.31万
  • 项目类别:
    Discovery Grants Program - Individual

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同伦和Hodge理论的方法在Algebraic Cycle中的应用
  • 批准号:
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    RGPIN-2016-05354
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