Joao Pita Costa: The Persistence Lattice

 
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).