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Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For
by joppy 8y ago
Solution sets to more general equations are so difficult and pathological that a lot of modern mathematics just entirely rules them out as objects of study. For example, any time you hear the word “manifold”, it refers to a space which has none of these pathologies and is entirely smooth. So the entire theory of differentiable manifolds will never encourage anything which “pinches”, or drops down a dimension, etc.
On the other hand, by restricting ourselves only to polynomials and their solution sets, it turns out that the singularities which arise are not too bad, and we can study them in detail. In other words, restricting to polynomials is the _only_ restriction we have to make, pretty much every solution set can be studied from there on.
- aaaaaaaaaaab 8y agoBut aren’t we missing out on something important by glossing over these “pathological” cases? Like how we kept ignoring nonlinear differential equations, because they aren’t that well-behaving as their linear counterparts... And then when we eventually looked into it, we found a completely new paradigm: chaos theory.
- joppy 8y agoFor sure we are missing something. But what hope do we have of understanding the more complicated case, before we figure out what we can know in simpler cases? The eventual goal is to move back to the more complicated cases.
- phkahler 8y agoWhat if the approach to the simpler cases is not applicable to the harder ones. Studying the simple may shape ones thoughts in a way that prevents seeing the way to harder problems. Maybe? Sometimes new things are discovered because someone didn't know any better and tried something they were not ready for. Innovation sometimes comes from the self taught.
- deleted 8y ago[deleted]
- kaitai 8y agoThere 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.