Some statistical challenges of topological inference in the 1D case

In this talk I will present some thoughts on challenges of inferring topological features from noisy data in the perhaps most simple setting: estimating the number of modes (i.e., local maxima) of a 1D signal. Already this simple setup illustrates some of the statistical obstacles when trying to infer topological information. A particular focus will be on inference without presmoothing the data. In particular, I will discuss some aspects of persistent topology in the 1D setting and present a somewhat alternative approach based on the Kolmogorov instead of the sup norm. I will highlight some of the advantages and disadvantages of these approaches from a statistical perspective.