On some coarse geometric notions inspired by topology and category theory

Coarse geometry is the study of the large scale properties of spaces. The interest in large scale properties is mainly motivated by applications to geometric group theory and index theory, as well as to important open problems such as the Novikov Conjecture. In this thesis, we introduce and study coarse versions of the following classical topological notions: connectedness, monotone-light factorizations, extension theorems, and quotients by properly discontinuous group actions. We will draw on the analogy between large scale geometry and topology as well as on the perspective of category theory using Roe\u27s coarse category. In the first of four research chapters, we look at a large scale connectedness condition arising from the coarse category and show that it coincides with the topological connectedness of the Higson corona. In the second, we introduce coarse versions of monotone and light maps (calling them coarsely monotone and coarsely light maps respectively) and show that these maps constitute a factorization system on the coarse category. We also show that coarsely light maps preserve some important large scale properties. In the third research chapter, we unify the proof of three extension theorems: the classical Tietze Extension Theorem from topology, Katetov\u27s extension theorem for uniform spaces, and an extension theorem for slowly oscillating functions (an important class of functions in coarse geometry). The unification is achieved via a general extension theorem for neighbourhood operators. In the final research chapter, we study warped spaces associated to group actions on metric spaces, focussing in particular on coarsely discontinuous actions which we introduce as large scale analogues of properly discontinuous actions in topology. For such actions, we relate the (maximal) Roe algebra of the warped space with the crossed product of the (maximal) Roe algebra of the original space and the group, and prove a deck transformation result

Extension Theorems for Large Scale Spaces via Neighbourhood Operators

Coarse geometry is the study of the large scale behaviour of spaces. The motivation for studying such behaviour comes mainly from index theory and geometric group theory. In this talk we introduce the notion of (hybrid) large scale normality for large scale spaces and prove analogues of Urysohn’s Lemma and the Tietze Extension Theorem for spaces with this property, where continuous maps are replaced by (continuous and) slowly oscillating maps. To do so, we first prove a general form of each of these results in the context of a set equipped with a neighbourhood operator satisfying certain axioms, from which we obtain both the classical topological results and the (hybrid) large scale results as corollaries. We prove that all metric spaces are large scale normal, and give some examples of spaces which are not hybrid large scale normal. Finally, we look at some properties of Higson coronas of hybrid large scale normal spaces

Coarse embeddability of Wasserstein space and the space of persistence diagrams

We prove an equivalence between open questions about the embeddability of the space of persistence diagrams and the space of probability distributions (i.e.~Wasserstein space). It is known that for many natural metrics, no coarse embedding of either of these two spaces into Hilbert space exists. Some cases remain open, however. In particular, whether coarse embeddings exist with respect to the $p$-Wasserstein distance for $1\leq p\leq 2$ remains an open question for the space of persistence diagrams and for Wasserstein space on the plane. In this paper, we show that embeddability for persistence diagrams implies embeddability for Wasserstein space on $\mathbb{R}^2$, with the converse holding when $p > 1$. To prove this, we show that finite subsets of Wasserstein space uniformly coarsely embed into the space of persistence diagrams, and vice versa (when $p>1$).Comment: 11 pages, 1 figur

Coarse infinite-dimensionality of hyperspaces of finite subsets

We consider infinite-dimensional properties in coarse geometry for hyperspaces consisting of finite subsets of metric spaces with the Hausdorff metric. We see that several infinite-dimensional properties are preserved by taking the hyperspace of subsets with at most n points. On the other hand, we prove that, if a metric space contains a sequence of long intervals coarsely, then its hyperspace of finite subsets is not coarsely embeddable into any uniformly convex Banach space. As a corollary, the hyperspace of finite subsets of the real line is not coarsely embeddable into any uniformly convex Banach space. It is also shown that every (not necessarily bounded geometry) metric space with straight finite decomposition complexity has metric sparsification property