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Computational Topology - an Introduction
TLDR
This book is ideal for teaching a graduate or advanced undergraduate course in computational topology, as it develops all the background of both the mathematical and algorithmic aspects of the subject from first principles.
Stability of persistence diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Persistent Homology — a Survey
Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical
Combinatorics of Train Tracks.
Measured geodesic laminations are a natural generalization of simple closed curves in surfaces, and they play a decisive role in various developments in two-and three-dimensional topology, geometry
Stability of the homology of the mapping class groups of orientable surfaces
The mapping class group of F = Fgs r is F = rgs = wo(A) where A is the topological group of orientation preserving diffeomorphisms of F which are the identity on dF and fix the s punctures. When r =
The virtual cohomological dimension of the mapping class group of an orientable surface
Let F = F ~ r be the mapping class group of a surface F of genus g with s punctures and r boundary components. The purpose of this paper is to establish cohomology properties of F parallel to those
The second homology group of the mapping class group of an orientable surface
In I-7] Mumford shows that the Picard group P ic (~ ' ) is isomorphic to H2(F; 2~) and conjectures the latter is rank one, g>3 . We prove this below for g>5 . Another interpretation of this theorem
Stability of Persistence Diagrams
The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram
Hierarchical morse complexes for piecewise linear 2-manifolds
TLDR
While Morse complexes are defined only in the smooth category, the construction is extended to the piecewise linear category by ensuring structural integrity and simulating differentiability and then simplified by cancelling pairs of critical points in order of increasing persistence.
Hierarchical Morse—Smale Complexes for Piecewise Linear 2-Manifolds
TLDR
Algorithm for constructing a hierarchy of increasingly coarse Morse—Smale complexes that decompose a piecewise linear 2-manifold by canceling pairs of critical points in order of increasing persistence is presented.
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