Faltings' almost purity theorem asserts that, in p-adic geometry, things become simpler when passing to certain highly ramified, largely non-geometric, coverings. The coverings in question are somewhat akin to covering an interval by a Cantor set. I will show how embracing this idea has led us (with Dustin Clausen) to recast the basic notions of topology, leading to a better setup in which to do algebra with topological groups.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/9682
[4] https://www.mpim-bonn.mpg.de/sites/default/files/videos/download/20190807_scholze.mp4