The (Co)isoperimetric Problem in (Random) Polyhedra

We consider some aspects of the global geometry of cellular complexes. Motivated by techniques in graph theory, we develop combinatorial versions of isoperimetric and Poincare inequalities, and use them to derive various geometric and topological estimates. This has a progression of three major topics: 1. We define isoperimetric inequalities for normed chain complexes. In the graph case, these quantities boil down to various notions of graph expansion. We also develop some randomized algorithms which provide (in expectation) solutions to these isoperimetric problems. 2. We use these isoperimetric inequalities to derive topological and geometric estimates for certain models of random simplicial complexes. These models are generalizations of the well-known models of random graphs. 3. Using these random complexes as examples, we show that there are simplicial complexes which cannot be embedded into Euclidean space while faithfully preserving the areas of minimal surfaces.Ph