I am a master student and will apply for PhD soon. I like geometry and topology, especially knot theory and three/four manifolds, and have taken some courses on them.
It's hard to survive in my country if doing pure math, but I dislike typical applied math such as computational PDEs, so I am asking whether there are any applications of geometric topology.
I know topological data analysis is a popular area, but it seems that it's far from "geometric", and just requires very basic algebraic topology. Geometric visualization seems fun. Are there any other applications of geometry and topology to the real life? Could you introduce some references? Which places/universities in Europe have such research areas? Thanks in advance.
Topological data analysis definitely has some overlap with geometry, especially metric and computational geometry, and there are some people who have introduced more advanced algebraic and topological techniques into TDA, but the more sophisticated they are the more they tend to be impractical computationally and have a limited scope of application. Anyhow, there are some actual applications of geometric topology proper, I'll provide a sampler below.
As a whole, applied geometric topology is not a developed field like applied topology/topological data analysis; there are many reasons for this but this is not the place for such a ramble. Still, there are definitely many interesting lines of research, as can be seen in the papers above. If you're looking for a research group in Europe, try looking at the authors/references of these papers, I'm not sure which are in Europe and accepting students. However, if you are fine with computational geometry and/or computational topology, Europe has many fantastic universities and research groups.