There are many other tools for studying these pathological cases: symplectic geometry, different flavors of algebraic geometry that look at orbifolds (manifolds

There are many other tools for studying these pathological cases: symplectic geometry, different flavors of algebraic geometry that look at orbifolds (manifolds with corners), schemes (lots of nice singularities), deformation theory... There's a fun paper that I think of as the "bad as you wanna be" paper, about deformations of various mathematical objects: https://arxiv.org/abs/math/0411469 https://arxiv.org/abs/math/0411469 All the different areas of math just start at different places to reach toward the same important questions. It's like martial arts: aikido, shotokan, capoeira, krav maga -- very different approaches and understandings, but the masters are all reaching for the same goals, coming from different foundations and directions.