Topological similarity of random cell complexes, and applications

Although random cell complexes occur throughout the physical sciences, there does not appear to be a standard way to quantify their statistical similarities and differences. I'll introduce the notions of a 'swatch' and a 'cloth', which provide a description of the local topology of cell complexes which is both general (any physical system that may be represented as a regular cell complex is admissible) and complete (any statistical question about the local topology may be answered from the cloth). This approach also allows a distance to be defined that measures the similarity of the local topology of two cell complexes. The distance can be used in many applications, for example to compare simulation with experiment, to test the convergence of an evolution process to a steady state, or to iteratively modify a structure to reach a desired state. In my talk, I'll employ it to identify a steady state of a simple model dislocation network evolving by energy minimization, and then to quantify the approach of the simulation to this steady state.