" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

We present a thorough study of the theoretical properties and devise efficient algorithms for
the persistent Laplacian, an extension of the standard combinatorial Laplacian to the setting …

Propelled by a fast evolving landscape of techniques and datasets, data science is growing
rapidly. Against this background, topological data analysis (TDA) has carved itself a niche …

Simplicial complexes can be viewed as high dimensional generalizations of graphs that
explicitly encode multi-way ordered relations between vertices at different resolutions, all at …

Algorithms for persistent homology are well-studied where homomorphisms are induced by
inclusion maps. In this paper, we propose a practical algorithm for computing persistence …

In recent years, algebraic topology and its modern development, the theory of persistent
homology, has shown great potential in graph representation learning. In this paper, based …

A special family of non-trivial loops on a surface called handle and tunnel loops associates
closely to geometric features of" handles" and" tunnels" respectively in a 3D model. The …

In cardiac image analysis, it is important yet challenging to reconstruct the trabeculae,
namely, fine muscle columns whose ends are attached to the ventricular walls. To extract …

Resting state functional magnetic resonance imaging (rfMRI) can be used to measure
functional connectivity and then identify brain networks and related brain disorders and …

Persistent homology with coefficients in a field F coincides with the same for cohomology
because of duality. We propose an implementation of a recently introduced algorithm for …