This presentation will focus on recent applications of persistent homology to the study of curvature measures, objects that provide a unified description of the curvatures of submanifolds of Euclidean spaces as well as many singular sets.
We introduce two new distances on the space of compact subsets of a Euclidean space, similar to the Fréchet distance but weaker, and show how tools from persistence theory yield continuity results for curvature measures with respect to these distances on a large class of compact sets, possibly non-smooth. More generally, we prove the continuity of the normal cycle, a current associated with sufficiently regular compact sets from which curvature measures can be defined. As a corollary, we show that every compact set definable in an o-minimal structure admits a normal cycle, and provide a new proof of this fact for sublevel sets of differences of convex functions at weakly regular values, which answers a question raised by Fu on the normal cycles of sublevel sets of differences of convex functions.
The Morse-Barannikov complex is a canonical reduced form of the Morse complex, from which the persistent homology of the sublevel filtration -equivalently, the associated barcode - can be read off directly.
In this talk, I will first recall the Morse-Barannikov complex on a smooth manifold. I will then give a brief introduction to intersection homology, before discussing Ludwig’s combinatorial complex for anti-radial Morse functions on spaces with isolated conical singularities. Afterwards, I will present a Barannikov-type reduced form of this complex, giving a singular analogue of the classical construction. Time permitting, I will discuss work in progress towards an Eyring–Kramers law for anti-radial Morse functions on such singular spaces.
This is part of a PhD project supervised by Ursula Ludwig.