Geometric and Algebraic Approach to Polytopes

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。

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.