Bjørnar Gullikstad Hem

Publications

The discrete flow category: structure and computation (2024)

Abstract: In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Preprints

Poset functor cocalculus and applications to topological data analysis (2025)

Abstract: We introduce a new flavor of functor cocalculus, named poset cocalculus, as a tool for studying approximations in topological data analysis. Given a functor from a distributive lattice to a model category, poset cocalculus produces a Taylor telescope of degree n approximations of the functor, where a degree n functor takes strongly bicartesian (n plus 1)-cubes to homotopy cocartesian (n plus 1)-cubes. We give several applications of this new functor cocalculus. We prove that the degree n approximation of a multipersistence module is stable under an appropriate notion of interleaving distance. We show that the Vietoris-Rips filtration is precisely the degree 2 approximation of the Čech filtration, and we draw connections between poset cocalculus and discrete Morse theory. We demonstrate that the degree 1 approximation of the space of simplicial maps between two simplicial complexes is in some sense the space of continuous maps between their realizations, and that this statement can be made precise.