KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Chachólski, Wojciech

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Lundman, Anders

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Ramanujam, Ryan

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Öberg, Sebastian

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

Multidimensional Persistence and Noise2017In: Foundations of Computational Mathematics, ISSN 1615-3375, E-ISSN 1615-3383, Vol. 17, no 6, p. 1367-1406Article in journal (Refereed)

Abstract [en]

In this paper, we study multidimensional persistence modules (Carlsson and Zomorodian in Discrete Comput Geom 42(1):71–93, 2009; Lesnick in Found Comput Math 15(3):613–650, 2015) via what we call tame functors and noise systems. A noise system leads to a pseudometric topology on the category of tame functors. We show how this pseudometric can be used to identify persistent features of compact multidimensional persistence modules. To count such features, we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For one-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

This thesis contains three papers. Paper A and Paper B deal with homotopy theory and Paper C deals with Topological Data Analysis. All three papers are written from a categorical point of view.

In Paper A we construct categories of short hammocks and show that their weak homotopy type is that of mapping spaces. While doing this we tackle the problem of applying the nerve to large categories without the use of multiple universes. The main tool in showing the connection between hammocks and mapping spaces is the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding Borel constructions.

In Paper B we investigate the notion of homotopy commutativity. We show that the fundamental category of a simplicial set is the localization of a subset of the face maps in the corresponding simplex category. This is used to define ∞-homotopy commutative diagrams as functors that send these face maps to weak equivalences. We show that if the simplicial set is the nerve of a small category then such functors are weakly equivalent to functors sending the face maps to isomorphisms. Lastly we show a connection between ∞-homotopy commutative diagrams and mapping spaces of model categories via hammock localization.

In Paper C we study multidimensional persistence modules via tame functors. By defining noise systems in the category of tame functors we get a pseudo-metric topology on these functors. We show how this pseudo-metric can be used to identify persistent features of compact multidimensional persistence modules. To count such features we introduce the feature counting invariant and prove that assigning this invariant to compact tame functors is a 1-Lipschitz operation. For 1-dimensional persistence, we explain how, by choosing an appropriate noise system, the feature counting invariant identifies the same persistent features as the classical barcode construction.

We construct a category of short hammocks and show that it has the weak homotopy type of mapping spaces. In doing so we tackle the problem of applying the nerve to large categories without the use of multiple universes. We also explore what the mapping space is. The main tool in showing the connection between hammocks and mapping spaces will be the use of homotopy groupoids, homotopy groupoid actions and the homotopy fiber of their corresponding bar constructions.

In this paper we investigate functors indexed by simplex categories that send certain face maps to weak equivalences. We explain why such functors can be regarded as homotopy commutative diagrams. The key question we consider is related to rigidifications of such functors: under what circumstances is such a functor weakly equivalent to a functor that send these face maps to isomorphisms? We show that if the simplicial set is the nerve of a small category then such an homotopy commutative diagram can indeed be rigidified. We conjecture that this is also true whenever the simplicial set is a quasi-category. Lastly we show a connection between our homotopy commutative diagrams and mapping spaces of model categories via hammock localization.