The computation of multidimensional persistent homology is one of the major open problems in topological data analysis.
One can define r-dimensional persistent homology to be a functor from the poset category N^r, where N is the poset of natural numbers, to the category of modules over a commutative ring with identity. While 1-dimensional persistent homology is theoretically well-understood and has been successfully applied to many real-world problems, the theory of r-dimensional persistent homology is much harder, as it amounts to understanding representations of quivers of wild type.
In this talk I will introduce persistent homology, give some motivation for how it is related to the study of data, and present recent results related to the classification of multidimensional persistent homology.
- Junior Algebra and Representation Seminar