I was skeptical about the “applied” part, but glancing through it, the examples are really interesting! Discussing magnetic fields in the same chapter as lambda calculus, very cool! Seems a bit light on details though...
and he explains in the preface why it is seemingly light on details. it's because the details (the math) are elsewhere. the applications (as isolated entities) are also out there to a degree. what isn't out there is the bridge connecting the math to the applications, and so that is what his book sets out to do.
> Discussing magnetic fields in the same chapter as lambda calculus
where did you see this? i don't remember seeing it and couln't find it.
I see a lot of mathematics in the book, but from skimming it, I can't see a 'killer application' as such. I'd like to see an example of a practical problem that can be solved using reasonably deep topolgical methods (beyond, say the Euler characteristic) that can't be solved in any other way.
Ghrist was championing persistent homology for network analysis as such an application a few years ago, but I'm not sure if it ever progressed beyond 'toy' problems to give state-of-the-art results.
But did I miss any heavier stuff? Is this all people mean when they talk about topological data analysis?
My favorite recent application: two-parameter persistent homology for drug discovery (link to pdf of talk: https://www.ima.umn.edu/materials/2017-2018/SW8.13-15.18/274...). Frankly I find two-parameter persistence very hard to interpret, though I hope to spend some time on it this summer. But it's undeniable that the work described in the link is an application that can't be reduced to clustering.
I also feel that describing Mapper as 'just a clustering tool' is not really accurate; I'm getting good results from using Mapper for feature discovery in some specific domains where clustering has truly failed because traits lie on continua in a way that renders usual clustering muddy and useless.
Edited to add: here's another interesting paper, about subgroups of type 2 diabetes patients: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4780757/ They use Mapper to find 'clusters' and then go back to more traditional statistics to test significance. I don't have the data so I don't know if clustering alone would have worked.
From your position of expertise, can you see any major problems off the top of your head?
[1] https://media.springernature.com/lw785/springer-static/image... - from the open-access paper https://link.springer.com/article/10.1186/s13104-018-3482-7
[2] quote: "Step III... we applied the Vietoris–Rips filtration to determine which particular sensors (thus, which areas of the brain) are more “involved”∖“significant” concerning the spreading of epileptic seizures"
Would also be curious to hear of practical applications of heavy-duty topology, if anyone can think of some.