Related papers: Morse Theoretic Templates for High Dimensional Hom…
Different cell types aggregate and sort into hierarchical architectures during the formation of animal tissues. The resulting spatial organization depends (in part) on the strength of adhesion of one cell type to itself relative to other…
A theory of modules over posets is developed to define computationally feasible, topologically interpretable data structures, in terms of birth and death of homology classes, for persistent homology with multiple real parameters. To replace…
In this paper, we introduce a novel persistence framework for Morse decompositions in Markov chains using combinatorial multivector fields. Our approach provides a structured method to analyze recurrence and stability in finite-state…
For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then…
We present a distributed algorithm to compute the first homology of a simplicial complex. Such algorithms are very useful in topological analysis of sensor networks, such as its coverage properties. We employ spanning trees to compute a…
Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances…
We study the categorical framework for the computation of persistent homology, without reliance on a particular computational algorithm. The computation of persistent homology is commonly summarized as a matrix theorem, which we call the…
Homological algebra of modules over posets is developed, as closely parallel as possible to that of finitely generated modules over noetherian commutative rings, in the direction of finite presentations and resolutions. Centrally at issue…
Discrete cubical homology arose as the homology theory associated with discrete cubical homotopy theory. Despite the combinatorial nature of this homology, its computation has posed a significant challenge to the researchers in the field.…
We propose a novel graph clustering method guided by additional information on the underlying structure of the clusters (or communities). The problem is formulated as the matching of a graph to a template with smaller dimension, hence…
In this paper we survey three approaches to computing the homology of a finite dimensional compact smooth closed manifold using a Morse-Bott function and discuss relationships among the three approaches. The first approach is to perturb the…
We introduce a new topological invariant of complex line arrangements in the complex projective plane, derived from the interaction between their complement and the boundary of a regular neighbourhood. The motivation is to identify Zariski…
We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…
Topological invariants, including the Chern numbers, can topologically classify parameterized Hamiltonians. We find that topological invariants can be properly defined and calculated even if the parameter space is discrete, which is done by…
The Witten index for certain supersymmetric lattice models treated by de Boer, van Eerten, Fendley, and Schoutens, can be formulated as a topological invariant of simplicial complexes arising as independence complexes of graphs. We prove a…
The analysis of global dynamics, particularly the identification and characterization of attractors and their regions of attraction, is essential for complex nonlinear and hybrid systems. Combinatorial methods based on Conley's index theory…
We introduce a filtration on the simplicial homology of a finite simplicial complex X using bi-colourings of its vertices. This yields two dual homology theories closely related to discrete Morse matchings on X. We give an explicit…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
Our main objective is to demonstrate how homological perturbation theory (HPT) results over the last 40 years immediately or with little extra work give some of the Koszul duality results that have appeared in the last decade. Higher…