# Tag creation request - tag for computational topology

What is the best way to request the creation of a new tag, assuming one doesn't have the rep to do so? I'm guessing Meta is the best place to ask, so here goes...

In my specific case, I was wondering if a new tag for computational topology could be created, as distinct from the existing algebraic topology tag. Other possibilities could be topological data analysis or applied topology, although the former might be too narrow and the latter too broad (or better handled using applications alongside another topology tag).

See questions such as:

• I think it is worth mentioning, that there already is (computational-geometry) tag, I suppose that the two areas are related. Do you think you would be able to make brief explanation what would be in this tag? (I.e., can you suggest tag-excerpt and tag-wiki for the proposed tag?) – Martin Sleziak Apr 27 '14 at 16:24
• Just for comparison: computational topology tag at MO has 6 questions at the moment. (And empty tag wiki, so we cannot simply copy it from there.) – Martin Sleziak Apr 27 '14 at 16:26
• Computational topology has some overlap with computational geometry, but I wouldn't call one a subset of the other (cf. differential topology and differential geometry). I've checked the Wikipedia page (en.wikipedia.org/wiki/Computational_topology), but currently its definition is somewhat vague. My impression these days is that computational topology is often used to refer to computational/persistent homology, as used in topological data analysis. – J W Apr 27 '14 at 18:07
• ...There's also the work of Robert Ghrist (and others), which he refers to as applied (algebraic) topology (see, for instance, math.upenn.edu/~ghrist/notes.html). As far as I can tell, applied topology is broader than computational topology, but there is significant overlap. – J W Apr 27 '14 at 18:17
• I support the creation of this tag. I like "computational topology" better than the other options. I do not think the overlap with computational-geometry is great enough to constitute a synonym. – Alexander Gruber Apr 27 '14 at 18:29
• "computational topology" sounds pretty clear what it supposed to be even for me who knows nothing about the subject. I'll support its creation provided someone have a sensible tag-wiki plus identify more questions suitable for this tag. 5 seems a little bit too few to me, may be 10? – achille hui Apr 27 '14 at 19:43
• @achillehui: I've added a few more questions. – J W Apr 28 '14 at 8:23
• @JW: I also support the creation. Please post an answer to this question containing a proposed "tag wiki" (even just one for the excerpt) for this tag. – Willie Wong Apr 28 '14 at 8:59
• @MartinSleziak: On MO, the computational topology tag was created in response to tagging mathoverflow.net/questions/79321/computational-topology-paper computational geometry. If we can come up with a reasonable tag excerpt/wiki here, perhaps MO will be interested in using it. – J W May 1 '14 at 11:01
• The term "computational topology" is used for some other un-listed subjects. For example, the work of people like Matveev and Ben Burton would usually be called computational topology but it's about fairly abstract techniques in 3-manifold theory, rather than applications. Similarly, Joel Hass applies hyperbolic geometry to brain imaging and cosmetic surgery. It would probably be useful to have a name that doesn't presume the non-existence of these other fields. Perhaps "applications of topology to statistics"? – Ryan Budney May 3 '14 at 0:40
• @RyanBudney: Thanks for your thoughts. I also see on en.wikipedia.org/wiki/Computational_topology that there is more out there than just, say, computational homology. Your suggestion, "applications of topology to statistics," reminds me of topological data analysis, which is one aspect of computational topology, but not the whole story. Be that as it may, I seem to have opened a can of worms. MO does have a computational topology tag, but, as Martin Sleziak says above, a [currently] empty tag wiki. – J W May 4 '14 at 10:31

As Willie Wong suggests, I am posting an answer to my question with a proposed tag excerpt (perhaps suitable for the tag wiki too). Please feel free to edit this answer or post your own suggestions.

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:

1. Computational Geometric Topology
2. Computational Algebraic Topology
3. 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.

A suggested tag excerpt combining elements of the above, without being too restrictive or wordy, is:

The study of the complexity of topological problems and the design of efficient algorithms for their solution.

## Updated suggestions:

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.

• IMHO, $(b)$ doesn't sound right. representation and modelling of curves and surfaces is usually geometrical in nature. Only the global aspect of them are topological. – achille hui Apr 28 '14 at 15:50
• @achillehui: I tend to agree. What do you think of "The study of the complexity of topological problems and the design of efficient algorithms for their solution."? Is it too concise? – J W Apr 28 '14 at 19:54
• Appeared alone, the term "study of complexity" isn't very clear to me. If I see the word "complexity", the first thing come to mind is something purely computer science, the next thing is stuff related to fractals. I can't see how is this related to topology. The second part "design of efficient algorithms" sounds good. – achille hui Apr 28 '14 at 20:02
• @achillehui: Would one of "algorithmic complexity" or "computational complexity" be better, or are you suggesting omitting all mention of complexity? Perhaps I should include the phrase about the problems being tractable. – J W Apr 28 '14 at 20:17
• Either "algorithmic complexity" or "computational complexity" is fine. There will be no confusion for what they refer to. – achille hui Apr 28 '14 at 20:40
• Do you want to differentiate between computational and computable topology? (In other words, should we also use this tag for the study of topological aspects of computation/algorithms?) – Willie Wong May 1 '14 at 8:07
• @WillieWong: I had in mind computational topology, rather than computable topology, but the dividing line is not entirely clear, at least not to me. – J W May 1 '14 at 8:23
• @JW: I am just pre-emptively bringing up the question of whether the two should or should not be separate tags. Since for some of the questions you listed above it seems both concepts would apply. – Willie Wong May 1 '14 at 8:36
• The updated suggestions on tag-wiki look pretty good. – achille hui May 2 '14 at 5:49
• @WillieWong: I now have a slightly better idea what computable topology is after reading blog.sigfpe.com/2008/01/what-does-topology-have-to-do-with.html and math.andrej.com/2006/03/27/…. I lean towards computable topology being a separate tag (or part of another tag), but I'm open to arguments to the contrary. – J W May 2 '14 at 20:07