17 kwietnia 2024
 
 Niklas Hellmer  (IMPAN)
 
 Dowker Complexes: From Relations to Topological Data Analysis

After briefly introducing the main ideas of persistent homology and topological data analysis (TDA), I will review a classical construction of
Dowker. This Dowker complex associates to any relation an abstract simplicial complex in a functorial way. Moreover, this complex is endowed
with a canonical filtration, which we study. In the context of TDA, this filtration can be interpreted as encoding the density of the data.
Specifically, we address the problem of homological inference for metric measure spaces by a law of large numbers: We recover the true
underlying homology from the one of finite samples as the sample size goes to infinity. Time permitting, I will also go through some
applications and computations. This is joint work with Jan Spaliński.