A Topological Framework for Local Structure Analysis in Large Point Sets
Emanuel A. Lazar, University of Pennsylvania

Many systems in our everyday lives can be abstracted as large sets of points in space -- the Milky Way galaxy, the coffee in your mug, and the aluminum housing of your iPhone can all be viewed as large collections of point-like objects.  Understanding how these points are arranged is thus an interesting and natural problem, though aside from perfect crystals and ideal gases, describing this structure can be a very thorny problem.  We present a unified framework for characterizing local structure in ordered and disordered point systems; this framework is built on a semi-algebraic stratification of a relevant configuration space by the complete topology of a particle’s Voronoi cell.  Some relevant theorems, as well as applications to computational materials science, are considered.