Bridging Persistent Homology and Discrete Morse Theory with Applications to Shape Reconstruction

Explore the intricate connections between persistent homology and discrete Morse theory in this 56-minute lecture by Fabian Roll. Delve into the unification of persistence pairs and gradient pairs within a common framework, gaining insights into how cycle reduction in persistent homology computations can be interpreted as a gradient flow in the algebraic generalization of discrete Morse theory. Discover the practical applications of these theoretical concepts in shape reconstruction, examining the relationship between lexicographically optimal homologous cycles in Delaunay complexes and Wrap complexes for given radius parameters. Enhance your understanding of applied algebraic topology and its potential for solving complex geometric problems.