toposys2013

Toposys Project Meeting, Hayfa, Israel

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 that can be described as a complete Heyting algebra. Unlike the lattice of subspaces of a vector space, these lattice operations are constructed using equalizers and coequalizers that guarantee distributivity. We will discuss the further study of the structure properties of these objects of great interest, and their interpretation within the framework of persistence. This is a joint work with Primoz Skraba (Jozef Stefan Institut, Ljubljana).