Open Collections

Stratifying multi-parameter persistent homology

Banff International Research Station for Mathematical Innovation and Discovery

In their paper "The theory of multidimensional persistence", Carlsson and Zomorodian write "Our study of multigraded objects shows that no complete discrete invariant exists for multidimensional persistence. We still desire a discriminating invariant that captures persistent information, that is, homology classes with large persistence." In this talk I will discuss how tools from commutative algebra give computable invariants able to capture homology classes with large persistence. Specifically, multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.

Mathematics; Algebraic topology; Probability theory and stochastic processes; Data mining

video/mp4

Author affiliation: University of Oxford

2018-01-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64489

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/