Let me first say, that I consider links to papers referenced in MO posts a useful things. And so do other users, judging by an older discussion on this topic: Is it worth editing old posts to add links for references? Occasionally you can find links which contain part related to proxy of a speci...

Here are a few remarks on more specialized situations; this is not a complete answer. If $m=1$ and $a$ has additional smoothness (say $a\in C^2$, but I can really get away with $a'\in BV$ and probably somewhat less than that), I can run a Kummer-Liouville transformation to bring your equation $-...

In answer to your last paragraph, a good starting point for deformation theory, not specifically of quantum groups, is the first order deformation theory of associative algebras. Good references for this have been mentioned by Mariano (Gerstenhaber's papers) and Kevin Lin (Kontsevich's notes), bu...

Sard's Theorem, which is foundational, may be an example of a theorem of the sort you are looking for when the differentiability class of the function is low. Let's recall its classical statement: Sard's Theorem: Let $f\colon\, \mathbb{R}^n\to\mathbb{R}^m$ be a $k$ times continuously differenti...

Very much so. There are a number of small industries centred around studying equivalence classes of knot diagrams generated by a set of moves. The study of claspers. For example, $C_k$-moves are a special type of clasper surgeries. MathSciNet indicates 123 citations for Habiro's fundamental pap...