Strange Random Topology of the Circle

Despite intense research in stochastic topology, little is known about the case when the connectivity threshold doesn't shrink with sample size. In this talk I will characterise high-dimensional topology arising from a simple low-dimensional manifold: Circle. We observe a fascinating phase transition going from 1-sphere, bouquet of 2-sphere, 3-sphere, bouquet of 4-sphere, and so on. Our main tool is the expected Euler characteristic, which is computed exactly for any fixed point count. These systematic behaviour cannot be regarded as "topological noise", and calls for deeper investigations from the TDA community.