I recently read about Riemann's Mapping Theorem, which guarantees the existence of a conformal map between two open disks on the plane and I saw this interesting use of circle mappings to quickly approximate such a function. A natural question then came to me - how do mathematicians in general specify explicitly maps between arbitrary spaces?
For example, I would have no idea where to start in defining explicitly a function from the plane to a torus, as I don't know of any coordinate-system for the surface of a torus that I could use a reference point. However, even if someone created one, I cannot see how one could explicitly define functions between arbitrary spaces, such as situations where you have a highly abstract conception of the nature of the spaces (i.e. you know "Space A satisfies X property").
How is this done? Do people not generally explicitly do such things in favour of existence proofs?
I apologise in advance if the question seems terribly vague or misinformed, as I am only an enthusiast who learns in his spare time.
Define "explicitly". Every structure in maths is built on top of smaller/simplier structures. For example, if your set is equiped with an addition and a multiplication (i.e. a ring) then you already have polynomial functions. So are polynomials explicit? Sure, as long as you know addition and multiplication.
So in order to define a function from a plane to a torus you need three things: an understanding what plane is (and its mathematical model), an understanding what torus is (an its mathematical model as well) and a formula that can be derived from these structures.
So for example if by plane you understand $\mathbb{R}^2$ and by torus $S^1\times S^1\subseteq\mathbb{R}^4$ then this already opens a path for explicit definitions
$$\mathbb{R}^2\to S^1\times S^1$$ $$(x,y)\mapsto \big(f_1(x), f_2(x), f_3(x), f_4(x)\big)$$ $$f_i:\mathbb{R}\to\mathbb{R}$$ where functions above have to satisfy
$$\sqrt{f_1(x)^2+f_2(x)^2}=1$$ $$\sqrt{f_3(x)^2+f_4(x)^2}=1$$ Without a coorinate system or any other deep understanding about plane and/or torus there is nothing you can do.
Often in order to construct a map between spaces you first make the abstract space more concrete. For example lets say that you deal with a $1$-dimensional compact manifold $M$. It may be a difficult task to create any sensible continuous function $M\to\mathbb{R}$ until you realize that $M$ has to be homeomorphic to a disjoint union of $1$-dimensional spheres $M\simeq\bigsqcup S^1$. Now constructing such a function is a lot easier, after all $S^1\subseteq\mathbb{R}^2$, you can use addition, multiplication, function limits/series, etc. Once you do it and you prove stuff about it you go back and say "every $1$-dimensional compact manifold $M$ has this and that property".
But how to show that an abstract $1$-dimensional compact manifold has to be homeomorphic to a disjoint union of spheres? The proof of that does not have to be (and often is not) by explicit construction. The existence is enough. So as you can see existential and explicit constructions/proofs often mix making the distinction harder.
Some may argue that explicit constructions are better then the existence proofs. But at the end of the day if they serve the same purpose (i.e. proving a statement) then it doesn't really matter.