Related papers: Rips Complexes of Planar Point Sets

This is a sequel to a previous paper, developing an intrinsic, combinatorial homotopy theory for simplicial complexes; the latter form the cartesian closed subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial sets,…

We study the relationship between metric thickenings and simplicial complexes associated to coverings of metric spaces. Let $\mathcal{U}$ be a cover of a separable metric space $X$ by open sets with a uniform diameter bound. The Vietoris…

This paper makes some preliminary observations towards an extension of current work on graphs defined on groups to simplicial complexes. I define a variety of simplicial complexes on a group which are preserved by automorphisms of the…

The computational cost of persistent homology is often dominated by the growth of the underlying simplicial filtrations. Many different filtrations exist, each with its own assumptions and trade-offs, but all face some form of this growth…

Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…

The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…

The edge group of a simplicial complex is a well-known, combinatorial version of the fundamental group. It is a group associated to a simplicial complex that consists of equivalence classes of edge loops and that is isomorphic to the…

Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent…

We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing…

Flip graphs of combinatorial and geometric objects are at the heart of many deep structural insights and connections between different branches of discrete mathematics and computer science. They also provide a natural framework for the…

This paper introduces a method to detect each geometrically significant loop that is a geodesic circle (an isometric embedding of $S^1$) and a bottleneck loop (meaning that each of its perturbations increases the length) in a geodesic space…

Many practical applications in topological data analysis arise from data in the form of point clouds, which then yield simplicial complexes. The combinatorial structure of simplicial complexes captures the topological relationships between…

An outerplanar graph is a planar graph that has a planar drawing with all vertices on the unbounded face. The matching complex of a graph is the simplicial complex whose faces are subsets of disjoint edges of the graph. In this paper we…

A polygonal complex in euclidean 3-space is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r of faces surround each edge. It is said to be regular…

Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be…

In a seminal paper, Erdos and Renyi identified the threshold for connectivity of the random graph G(n,p). In particular, they showed that if p >> log(n)/n then G(n,p) is almost always connected, and if p << log(n)/n then G(n,p) is almost…

A topological group is constructed which is homotopy equivalent to the pointed loop space of a path-connected Riemannian manifold $M$ and which is given in terms of "composable small geodesics" on $M$. This model is analogous to J. Milnor's…

For any given natural $d\ge 1$ we provide examples of rational self-maps of complex projective plane $\pp^2$ of degree $d$ without (holomorphic) fixed points. This makes a contrast with the situation in one dimension. We also prove that the…

Many simplicial complexes arising in practice have an associated metric space structure on the vertex set but not on the complex, e.g. the Vietoris-Rips complex in applied topology. We formalize a remedy by introducing a category of…

For a simplicial complex $X$, the $d$-clique complex $\Delta_d(X)$ is the simplicial complex having all subsets of vertices whose $(d + 1)$-subsets are contained by $X$ as its faces. We prove that if $p = n^{\alpha}$, with $\alpha <…