On MSE I am working on a tag excerpt/wiki for a currently non-existing computational topology tag. Please see this post on MSE Meta:
I would like to invite members of MO with an interest in / expertise in computational topology to visit MSE Meta to comment on my efforts, or post alternatives, as I hope the final result can also be used for the already-existing computational topology tag on MO. Please note that while the tag does exist on MO, the tag excerpt and tag wiki are empty at the time of writing.
Edit: For the convenience of those who do not wish to make an account at MSE, here are my thoughts so far, with some minor editing. However, the various comments by MSE members are missing.
One possibility is a modified version of the text for computational geometry: (I've substituted "topological" for "geometric.")
The study of computer algorithms which admit topological descriptions, and topological problems arising in association with such algorithms. The two major classes of problems are (a) efficient design of algorithms and data classes using topological concepts and (b) representation and modelling of curves and surfaces.
However, I'm not sure if "(b) representation and modelling of curves and surfaces" is entirely the right emphasis for topology. Perhaps an improvement would be "(b) topologically accurate representation of curves, surfaces and higher dimensional manifolds" (influenced by quote from Blackmore & Mileyko below).
I also wonder if it should be "efficent design of algorithms" or "design of efficient algorithms." In an ideal world, I suppose it would be "efficient design of efficient algorithms and efficient data classes."
I note that Computational Topology: An Introduction by Edelsbrunner & Harer is divided into three parts:
- Computational Geometric Topology
- Computational Algebraic Topology
- Computational Persistent Topology
In the preface, the authors mention the obtaining of global insights by meaningful integration of local information (p. xi). They also refer to removing "the burdens of size to focus on the phenomenon of connectivity" (p. x).
I would say the key material includes, but is not limited to, simplicial complexes, homology, Morse functions and persistence. I am aware that such a short list could be much too restrictive. In private correspondence, Robert Ghrist wrote: "there are computational issues in all branches of topology, including combinatorial, geometric, symplectic, algebraic, differential, ..." (quote included with permission).
There's also Rote & Vegter's definition from Chapter 7 of Effective Computational Geometry for Curves and Surfaces, Boissonaud & Teillaud (Eds.), Springer 2006. For convenience, I quote it here: (If this is in breach of copyright, I will remove it and just leave the link. The same goes for any other quotes.)
Computational topology deals with the complexity of topological problems, and with the design of efficient algorithms for their solution, in case these problems are tractable. These algorithms can deal only with spaces and maps that have a finite representation. To this end we restrict ourselves to simplicial complexes and maps. In particular we study algebraic invariants of topological spaces like Euler characteristics and Betti numbers, which are in general easier to compute than topological invariants.
In Blackmore and Mileyko's article, Computational Differential Topology, p. 3, Applied General Topology, Volume 1, No. 1, 2007, the authors state that
...the fundamental goal of Computational Topology is to algorithmically guarantee that a computer generated representation of an object is equivalent to the actual object in an appropriate topological sense.
The overview at http://comptop.stanford.edu/ is also potentially of interest.
Two suggested tag excerpts combining elements of the above are:
The study of the computational complexity of topological problems and the design of efficient and robust algorithms for their solution.
The study of computer algorithms which admit topological descriptions, and topological problems arising in association with such algorithms. The two major classes of problems are (a) design of efficient and robust algorithms and data classes using topological concepts and (b) topologically accurate representation of curves, surfaces and higher dimensional manifolds.