Topics in Persistent Homology: From Morse Theory for Minimal Surfaces to Efficient Computation of Image Persistence

We study some problems and develop some theory related to persistent homology, separated into two lines of investigation.

In the first part, we introduce lifespan functors, which are endofunctors on the category of persistence modules that filter out intervals from barcodes according to their boundedness properties. They can be used to classify injective and projective objects in the category of barcodes and the category of pointwise finite-dimensional persistence modules. They also naturally appear in duality results for absolute and relative versions of persistent

(co)homology, generalizing previous results in terms of barcodes by de Silva, Morozov, and Vejdemo-Johansson. Due to their functoriality, we can apply these results to morphisms in persistent homology that are induced by morphisms between filtrations. This lays the groundwork for an efficient algorithm to compute barcodes of images and induced matchings of such morphisms, which performs computations in terms of relative cohomology and then translates to absolute homology via the aforementioned dualities. Our method is based on a previous algorithm by Cohen-Steiner, Edelsbrunner, Harer, and Morozov that did not make use of relative cohomology. Using it is crucial, however, because our algorithm applies the clearing optimization introduced by Chen and Kerber, which works particularly well in the context of relative cohomology. We provide an implementation of our algorithm for inclusions of filtrations of Vietoris–Rips complexes in the framework of the software Ripser by Ulrich Bauer.

In the second part, we introduce local connectedness conditions on a broad class of functionals that ensure that the persistent homology of their associated sublevel set filtration is q-tame, which, in particular, implies that they satisfy generalized Morse inequalities. We illustrate the applicability of these results by recasting the original proof of the unstable minimal surface theorem given by Morse and Tompkins in terms of persistent Čech homology in a modern and rigorous framework. Moreover, we show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a filtration consisting of paracompact Hausdorff spaces is 0 if it satisfies a similar local connectedness condition to the one used to ensure q-tameness, generalizing a result by Mardešić for locally connected spaces to the setting of filtrations. In contrast to singular homology, the persistent Čech homology of a compact filtration is always upper semi-continuous, which has structural implications in the q-tame case: using a result by Chazal, Crawley-Boevey, and de Silva concerning radicals of persistence modules, we show that every lower semi-continuous q-tame persistence module can be decomposed as a direct sum of interval modules and that every upper semi-continuous q-tame persistence module can be decomposed as a product of interval modules.