Abstract: Large scale geometry is almost the dual of topology, and a large scale structure is almost dual to a topology (large scale structures are dual to uniform structures, and every uniform structure induces a topology). Topology is concerned with properties holding for smaller and smaller scales, while large scale geometry is concerned with properties holding for larger and larger scales. Persistent homology uses algebra to measure topological properties. Topological properties which persist through multiple scales are considered important features of the space, while those appearing in few scales are considered noise.
This talk will discuss the relationship between large scale geometry and persistent homology. In particular, I will show how large scale geometry induces a filtration of complexes for a space; this filtration is then used to compute the persistent homology. Both theory and examples will be discussed.