Realizations, symmetry and monodromy groups of abstract polytopes

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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.