Erlangen Seminar Series: Studying the Geometry of Loops on the Möbius Band

Date: Friday 23 January, from 3–4pm

Location:  Online

Title:  Studying the Geometry of Loops on the Mobius Band

From contours of 2D shapes to knotted proteins, many areas of statistics and physical sciences contend with data in the form of geometric loops — an embedding of S1 into a Euclidean space. Similar to many problems in geometric deep learning, we desire a representation of the geometry of the loop that is invariant under Euclidean symmetries, and indifferent to how the loop is parametrised. In this joint work with James Binnie (Cardiff) and Otto Sumray (MPI-Dresden), we propose a novel representation of a loop’s geometry by representing the distance matrix of the loop as a Morse function on a Möbius band — the two point configuration space of S1. By considering the persistent homology of this function, we obtain a representation of the loop’s geometry that has the desired symmetry invariance. We show that this feature map is stable with respect to perturbations, and give a geometric interpretation of the features obtained.

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