Combinatorics, Algebra and Topology of simplicial complexes

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This proposal describes four projects aimed to deepenour understanding of the combinatorics, algebra and topology ofsimplicial complexes through the study of their size (or face numbers).This includes work on the Upper Bound Theorem that has its origins inthe theory of convex polytopes, and the study of the face numbers andBetti numbers of balanced simplicial complexes, flag complexes, andcomplexes with symmetry.Combinatorics is an active field of mathematics today that has closeconnections with many other subjects such asalgebra, geometry and topology, optimization, statistics,etc. inside mathematics, and computer science, high-energy physics, andbiology outside. Continuing exploring and developing new cross-pollinations and interactions between combinatorics and other areas ofmathematics is a central intellectual pursuit of this project.Specifically, this proposal concerns simplicial complexes --- finiteobjects that provide the easiest way to represent multi-dimensional shapesin a computer through their triangulations --- and various bounds on thesizes of such complexes. The minimal size of a triangulation has taken onpractical significance, since it directly affects the complexity ofcomputations involving a particular shape.