The Persistence Lattice
Presented by Prof. Joao PITA COSTA
Type: Oral presentation
The intrinsic relation between lattice theory and topology has been reestablished with the works of M. H. Stone. Persistent homology is a recent addition to topology, where it has been applied to a variety of problems including to data analysis. It has been in the center of the interest of computational topology for the past twenty years. In this talk we will introduce a generalized version of persistence based on lattice theory, unveiling universal rules and reaching deeper levels of understanding. Its algorithmic construction leads to two operations on homology groups which describe a diagram of spaces as a complete Heyting algebra, a generalization of a Boolean algebra which also suffices a topological dual space. Unlike the lattice of subspaces of a vector space, these lattice operations are constructed using equalizers and coequalizers that guarantee distributivity. This interpretation reduces to known definitions of persistence in the cases of standard persistence, zigzag persistence and multi-dimensional persistence. We will further discuss some of the properties of this lattice, the algorithmic implications of it, and possible applications.