General principles which lead to good questions in many concrete situations [closed]

Question

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such principles. Below is my own list.

1) Given a Banach space, one may ask what is its dual. In many concrete situations this question has interesting and important answers. (For example the dual space of the space of continuous functions on a compact space is the space of measures on it.)

2) Given a functor between two categories, one may ask whether is has right/ left adjoint. One may also ask whether it has right/ left derived functors. (For example the simple operation of push-forward of sheaves of abelian groups has left adjoint - pull back- which is somewhat less obvious.)

3) Given a metric space, one may ask what is its completion. (Concrete example which I like: consider the set of isometry classes of $n$-dimensional closed Riemannian manifolds of diameter at most $D$ and sectional curvature at least $\kappa$ equipped with the Gromov-Hausdorff metric. Points of its completion are compact metric spaces which are so called Alexandrov spaces with curvature bounded below.)

Answer 1

I suggest two "general principles which lead to good questions in many concrete situations": making "abstract" results more explicit, and making explicit results more abstract.

I think these things are well-known to researchers, in particular I make no claim of originality or novelty; but, the question has been asked. And perhaps some student will read this. So, here is more detail.

I can't remember where I read/heard this, but there is a pretty well-known scale of explicitness in mathematical results, something like the following:

1. A non-constructive existence proof of $X$ (e.g., using a Noetherian hypothesis)
2. A "theoretically" constructive existence proof of $X$, or an "effective" theorem (in the sense of having some bounds or explicit information about $X$)
3. A method for constructing $X$
4. An explicit algorithm for $X$
5. A computer implementation of an algorithm for $X$
6. A usable computer program for $X$, meaning that it can be used by non-specialists

Of course not all of these are applicable in all situations, perhaps I left out some levels, perhaps some of the levels I included are a little silly... Anyway, the point is that with a scale like that, we can extract at least two "general principles" for generating questions in concrete situations:

Explicitization: Increase your position on the scale. If you have a nonconstructive existence proof, try to find a constructive one. If you have an algorithm, try implementing it. (It was a humbling experience to see that my beautiful theorem would take forever to actually compute a trivial case.)

Abstraction: Decrease your position on the scale (hopefully leading to generalizations or new ideas).

If I may use myself as an example, I've spotted a few proofs that used more or less explicit manipulations of big matrices, or coordinate charts, and I was able to simplify and generalize them by replacing matrices with linear transformations, coordinate charts with some basic topology, and so on. There's nothing special about my case. (And I'm not criticizing the earlier authors.)

Answer 2

A good principle would be to apply this piece of advice from Hermann Weyl : to understand well a mathematical object, determine and study the structure of its group of automorphisms.

Answer 3

Look for situations where two effects are competing, and study which effect "wins," or whether they balance each other in some sense.

For example, people talk about certain dispersive PDEs being subcritical (dispersion wins over the nonlinear effects) or supercritical (nonlinear effects win). The critical/borderline case is often the most interesting. Other examples arise in combinatorics, where some structure is guaranteed to appear if the size of the problem is sufficiently large, and one may ask where this guarantee begins, i.e. at what size the complexity/unlikeliness of that structure is exactly balanced with the pigeonhole principle.