Around the theory of f-vectors

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The primary aim of this proposal is to deepen our understanding of several classes of regular cell complexes through the study of their face numbers. Specifically, research on this project will involve the use of algebraic, geometric, topological, and combinatorial methods to attack fundamental enumerative questions involving balanced and flag simplicial complexes, cubical complexes, simplicial manifolds and pseudomanifolds.Cell complexes provide a convenient way to build various shapes from smaller and simpler blocks, such as from pyramids or cubes. A natural question is: What restrictions on the number of building blocks are imposed by specific shapes? Variations of this question (e.g., What is the minimum number of blocks required?) have practical importance when one wants to represent or store such a structure in a computer.