Francesca Tombari, KTH Royal Institute of Technology

Abstract

Persistent homology is commonly encoded by vector space-valued functors indexed by posets. These functors are called tame, or persistence modules, and capture the life-span of homological features in a dataset. Every poset can be used to index a persistence module, however some posets are particularly well suited. 
We introduce a new construction called realisation, which transforms posets into posets. Intuitively, it associates a continuous structure to a locally discrete poset by filling in empty spaces. Realisations share several properties with upper semi-lattices. They behave similarly with respect to certain notions of dimension for posets that we introduce. Moreover, as indexing posets of persistence modules, they allow for good discretisations and effective computation of homological invariants via Koszul complexes.

Presented in association with the DATA-INSPIRE TRIPODS Institute.