| > I feel the current marketing of topological data analysis and similar is directed towards people who have at least taken several graduate math courses. Oh I don't know about that. Lots of TDA is accessible at an undergrad level, as for example the book by Ghrist that you mention shows. I may be biased as I did my PhD in the field, but I also supervised several master's students, and some came from engineering or science backgrounds with little more math than undergrad level under their belts. They mostly did well and learned the basics of the field enough to make contributions during their thesis work. I would say that the field spans a vast range of applied-ness (from straight up data analysis with persistent homology as merely a tool, to fully theoretical work on e.g. multi-persistence). The more applied areas of the field tends to be far more accessible to people without a deep math background. > Are there really important topology ideas in topological learning, or it can all be phrased combinatorially? I think topological machine learning is still in its (very exciting!) infancy, and there's nothing I'd like more than to be able to answer that question! I think there's lots of potential for an extremely interesting fusion of two disciplines. And I never forget something the above-mentioned Ghrist once told me (during a discussion unrelated to ML): Never underestimate the treasure-hunting way of doing research – there's lots of be had from bringing together ideas from seemingly unrelated fields, especially when combining old and young ones. (That's me paraphrasing from memory, obviously, Ghrist is a lot more eloquent than that!) TDA is in some sense just that: the very old of algebraic topology meets the very new of computation, data analysis, etc. And I think the meeting of topology and machine learning has potential to be very exciting too! PS: When you say "or can it all be phrased combinatorially?", I wanna point out that topology and combinatorics are not mutually exclusive. This is a nice book that straddles the realms of TDA, topology and combinatorics: https://www.springer.com/gp/book/9783540719618 . See also Matt Kahle's fabulous works on showing the probabilistic properties of certain randomly generated combinatorial topological spaces (in some sense generalizing the classical work of Erdős and Rényi pertaining to properties of graphs that you may know, https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93R%C3%A9nyi_...). |