This proposal describes four projects aimed to deepen our understanding of the combinatorics, algebra and topology of simplicial complexes through the study of their size (or face numbers). This includes work on the Upper Bound Theorem that has its origins in the theory of convex polytopes, and the study of the face numbers and Betti numbers of balanced simplicial complexes, flag complexes, and complexes with symmetry.
Combinatorics is an active field of mathematics today that has close connections with many other subjects such as algebra, geometry and topology, optimization, statistics, etc. inside mathematics, and computer science, high-energy physics, and biology outside. Continuing exploring and developing new cross- pollinations and interactions between combinatorics and other areas of mathematics is a central intellectual pursuit of this project. Specifically, this proposal concerns simplicial complexes --- finite objects that provide the easiest way to represent multi-dimensional shapes in a computer through their triangulations --- and various bounds on the sizes of such complexes. The minimal size of a triangulation has taken on practical significance, since it directly affects the complexity of computations involving a particular shape.
