Topological Similarity of Cell Complexes

Benjamin Schweinhart (May 16, 2016)

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Abstract

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 method of swatches, which describes the local topology of a cell complex in terms
of probability distributions of local configurations. It allows a distance to be defined which
measures the similarity of the local topology of cell complexes. Convergence in this distance is
related to the notion of a Benjamini Schramm graph limit. In my talk, I will use this to state
universality conjectures about the long-term behavior of graphs evolving under curvature flow,
and to test these conjectures computationally. This system is of both mathematical and physical
interest.
If time permits, I will discuss other applications of computationally topology to curvature flow
on graphs, and describe recent work on a new notion of geometric graph limit.