Persistence modules, symplectic Banach–Mazur distance and Riemannian metrics

We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a nonlinear version of the classical Banach–Mazur distance, called symplectic Banach–Mazur distance and denoted by [Formula: see text] The relevant persistence modules come from filtered symplectic homology and are stable with respect to [Formula: see text] Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped with Riemannian metrics. We consider some questions about large-scale geometry of this space and in particular we give a construction of a quasi-isometric embedding of [Formula: see text] into this space for all [Formula: see text] On the other hand, in the case of domains in [Formula: see text], we can show that the corresponding metric space has infinite diameter. Finally, we discuss the existence of closed geodesics whose energies can be controlled.

On the Stability of Interval Decomposable Persistence Modules

The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ 2 n - 1 that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ n = 2 . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.