Dr. Yuriy Mileyko, Assistant Professor at the Mathematics Department at Duke University
The talk will focus on the problem of recovering homology groups of a subspace of a high-dimensional Euclidean space given only a finite number of points lying on or near the subspace. We approach this problem by computing persistent homology groups of a specific family of nested simplicial complexes -- alpha-beta witness complexes. This 2-parameter family is sensitive to differences in sampling density and thus amenable to detecting patterns within the data set. It also lends itself to theoretical analysis. For example, we can prove that in the limit, when the witnesses cover the entire domain, witness complexes in the family that share the first, scale parameter have the same homotopy type. We shall also show a nice relation among alpha-beta witness complexes, alpha shapes, and witness complexes of Carlsson and de Silva. An overview of persistent homology as well as several computational results will also be presented.