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The Question: Are there any ways that "applied" mathematicians can use Homology theory? Have you seen any good applications of it to the "real world" either directly or indirectly?

Why do I care? Topology has appealed to me since beginning it in undergrad where my university was more into pure math. I'm currently in a program where the mathematics program is geared towards more applied mathematics and I am constantly asked, "Yeah, that's cool, but what can you use it for in the real world?" I'd like to have some kind of a stock answer for this.

Full Disclosure. I am a first year graduate student and have worked through most of Hatcher, though I am not by any means an expert at any topic in the book. This is also my first post on here, so if I've done something wrong just tell me and I'll try to fix it.

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A keyword I've heard here is "persistent homology." –  Qiaochu Yuan Dec 9 '10 at 9:22
See: comptop.stanford.edu also the book "Computational Topology" by Edelsbrunner and Harer, and this thread: mathoverflow.net/questions/2556/… –  Ryan Budney Dec 9 '10 at 9:32
Also cloud and/or blob homology? I'm not sure if those are different, or different from persistent homology, but they sound sort of applied-ish. –  Aaron Mazel-Gee Dec 9 '10 at 9:41
Homology is also related to electromagnetism. Whether "applied" mathematicians actually use it that way, I don't know... –  Qiaochu Yuan Dec 9 '10 at 9:42
Also, when people ask you about applications of topology you can always just say "oh you know, theoretical physics, string theory, that sort of stuff..." :) –  Aaron Mazel-Gee Dec 9 '10 at 9:54

4 Answers 4

up vote 10 down vote accepted

There are definite real world applications. I would look at the website/work of Gunnar Carlsson (http://comptop.stanford.edu/) and Robert Ghrist (http://www.math.upenn.edu/~ghrist/). Both are excellent mathematicians.

The following could be completely wrong: Carlsson is one of the main proponents of Persistent Homology which is about looking at what homology can tell you about large data sets, clouds, as well as applications of category theory to computer science. Ghrist works on stuff like sensor networks. I don't understand any of the math behind these things.

Also there are some preprints by Phillipe Gaucher you might want to check out. Peter Bubenik at cleveland state might also have some fun stuff on his website.

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As this comment is 2 1/2 years old a lot of stuff has happened in the applied topology scene. For an introduction to persistent homology, I would recommend Robert Ghrist's Chauvenet Prize winning paper. –  M.B. Jun 4 '13 at 12:59

As an additional source of ideas, may I point out that vol. 157 of the Springer Applied Mathematical Sciences series is entitled Computational Homology (it is by Kaczinski, Mischaikow and Mrozek). This was related to the CHOMP project which is well worth checking out.

These relate more to applied mathematics than to data analysis. Both threads probably deserve more attention by the mathematical community.

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Thanks for these pointers! I made a minor edit so as to make the link to the CHOMP project clickable, I hope you don't mind. Making links in markup is very easy: simply write (link description)[http://...]. –  t.b. Apr 24 '11 at 10:19
Thanks. I should have guessed as we use that syntax on the nLab as well. The CHOMP project stuff is great fun. –  Tim Porter Apr 24 '11 at 11:31
It looks like that! Sorry: I did it the wrong way around in my previous comment. It's [link description](http://...). –  t.b. Apr 24 '11 at 11:41

There is a very nice application of homology in data mining and computer science called "topological data analysis". It is mainly based on the computation of a homology theory called "persistent homology" which describes those topological features that are "persistent" while varying the parameter which is used in the clustering analysis (for example, the radius of balls around the points in the data cloud and giving classical topological complexes).

This homological theory gives a qualitative description of the topology of the underlying data cloud. In this sense it is a new approach to clustering, which is classically seen as an optimization problem subject to the choice of a metric or metric like function for the data cloud under examination.

Topological data analysis is particularly interesting in presence of "Big Data", i.e. extremely high numbers of often uncorrelated data, such as those from medical sciences, social networks or meteorology.

You can check Gunnar Carlsson's et al. works on the arxiv to get a better idea of it. A great introduction to the topic is http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf

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The answer overlaps to Sean Tilson's but provides some more detail on topological data analysis vs. classical clustering –  Avitus Jun 4 '13 at 12:18
The study of multiparameter persistent homology, or, multiparameter clustering is of great interest. In general one does not have a finite invariant (such as barcodes) for multidimensional persistence but there might be cases which is not that general with nice invariants. E.g. persistent homology is just multipersistence where all but one parameter are fixed. One immediate application of this would be clustering both on distance and density (see the work of Memoli and Carlsson). There is also "mapper" a clustering algorithm based on topological ideas. –  M.B. Jun 4 '13 at 12:55
And a cool video –  M.B. Jun 4 '13 at 12:56

You may want to check out "Topological and Statistical Behavior Classifiers for Tracking Applications" abstract preprint

This has the first unified theory for target tracking using Multiple Hypothesis Tracking, Topological Data Analysis, and machine learning. Our string of innovations are 1) robust topological features are used to encode behavioral information (from persistent homology), 2) statistical models are fitted to distributions over these topological features, and 3) the target type classification methods of Wigren and Bar Shalom et al. are employed to exploit the resulting likelihoods for topological features inside of the tracking procedure.

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