If you find a seven-dimensional doughnut hiding in your data, that’s very interesting, but what do you do with that information?Robert’s chapter in the book I’m reviewing has a nice introductory paragraph that helps put TDA in context.
The section heading for the paragraph is “When is Homology Useful?”Homological methods are, almost by definition, robust, relying on neither precise coordinates nor careful estimates for efficiency.
As such, they are most useful in settings where geometric precision fails.
With great robustness comes both great flexibility and great weakness.
Topological data analysis is more fundamental than revolutionary: such techniques are not intended to supplant analytic, probabilistic, or spectral techniques.
They can however reveal a deeper basis for why some data sets and systems behave the way they do.
It is unwise to wield topological techniques in isolation, assuming that the weapons of unfamiliar “higher” mathematics are clad with incorruptible silver.
Robert’s background was in engineering and more conventional applied mathematics before he turned to applications of topology, and so he brings a broader perspective to TDA than someone trained in topology looking for ways to make topology useful.
He also has a decade more experience applying TDA than when I interviewed him here.
I’m looking forward to reading his new chapter carefully.
As I wrote about the other day, apparently the US Army believes that topological data analysis can be useful, presumably in combination with more quantitative methods.
 More generally, it seems the Army is interested in mathematical models that are complementary to traditional models, models that are robust and flexible.
The quote above cautions that with robustness and flexibility comes weakness, though ideally weakness that is offset by other models.
Related postsInterview with Robert GhristUS Army applies new mathNext areas of math to be applied Algebraic topology is quantitative in one sense and qualitative in another.
It aims to describe qualitative properties using algebraic invariants.
It’s quantitative in the sense of computing homology groups, but it’s not as directly quantitative as more traditional mathematical models.
It’s quantitative at a higher level of abstraction.