Realizations, symmetry and monodromy groups of abstract polytopes

普通立方体是规则多面体的常见示例：它有各种正方形(上、下、前、后、左、右)，由几条边(12条边)分隔，这些边在八个角处以特定的方式以三相交。通常，这样的立方体给人的想象是一个非常对称和刚性的物体，即欧几里得的实现；但如果我们相反地考虑一个橡胶副本，那么我们可以想象以奇妙的方式扭曲和扭曲立方体，这破坏了它的对称性。然而，立方体的“组合特征”将在这次破坏企图中幸存下来；仍然会有6个正方形的面，12个边和8个角，像以前一样相互连接。 “抽象规则多面体”就是这种类型的数学对象--它具有立方体或其在任何维度上的任何亲属所暗示的基本结构特征。然而，这些组合属性本身并不总是意味着对象应该有任何类型的真实模型，因为在仅仅的组合对称性(想一想‘汇编指令’)和实际空间之间可能存在一些微妙的不相容 对称性(想一想‘能够实际建造一个很好的模型’)。 这项研究项目的主旨是探索描述和分类抽象规则多面体及其不太对称的亲属所需的数学机制。在数学中，几何和代数的任何不同分支之间的任何这样的相互作用都将是令人着迷的，也是值得的。

An ordinary cube is a familiar example of a regular polytope: it has various square faces (top, bottom, front, back, left, right), separated by several edges (twelve of them), which meet by threes in a particular way at the eight corners. Usually such a cube suggests to the imagination a very symmetrical and rigid object, i.e. a Euclidean realization; but if we consider instead a rubber copy, then we can imagine twisting and distorting the cube in fantastic ways which destroy its symmetry. Nevertheless, the `combinatorial features' of the cube would survive this attempt at destruction; there would still be 6 square faces, 12 edges and 8 corners, interconnected as before. An `abstract regular polytope' is a mathematical object of just this sort - it has the basic structural features suggested by a cube or any of its relatives in any dimension. However, these combinatorial properties do not of themselves always imply that there should be any sort of real model for the object, since there could be some subtle incompatibility between mere combinatorial symmetry (think `assembly instructions') and actual spatial symmetry (think `able to actually build a nice model'). The thrust of this research project is to explore the mathematical machinery needed to describe and classify abstract regular polytopes and their less symmetrical kin. In mathematics, any such interplay between disparate branches of geometry and algebra promises to be both fascinating and worthwhile.