I am an engineer, so maybe this question is naive.
In my view, transformation is a kind of map that transforms space points into new ones. To preserve space structure, this map must be a function, injective, and subjective. Some examples are translate, rotate, etc. Also, I see Fourier transform, Laplace transform, etc. in this category.
Transformers are my interest. Therefore, I started to study about them. I spent some time in Topology (I can't find any difference between algebraic Topology and general Topology), Differential geometry, and Functional analysis. After that, I was a little confused. I think none of them can't completely answer my questions.
What am I trying to find about transformers?
Characteristics of transformers, applications of them, affective spaces, structure of new space, can we design them, can transform affect just on a structure from space not on whole, etc.
Example of what I have in mind
Consider a 2-dimensional curve with an equation:
$\ f(x)= ax^3 + bx^2 + cx +d$
And assume a transform exists such that it changes this curve to a logarithmic curve, or any polynomial curve, or 3-dimensional surface, etc.
What is my question?
which part of math describes the transformer and its behavior?
From where is it better to study about them?
Can you name some good sources?