A soft dive into moduli spaces: All triangles form a triangle.
We mathematicians like to measure, understand and group things according to them sharing similar properties. Understanding structure is a big part of understanding maths.
Hence an immediate reason for the birth of what is called a moduli space. Very roughly speaking, a moduli space is a geometric space where points correspond to different groups of something, usually algebro-geometric objects or classes of such.
As I just explained mathematicians love to group things with similar properties. Let me first dive into the notion of what we mean by similar properties.
An example which we have all encountered at school is the notion of congruent triangles. Those are triangles that have exactly the same structure. However, we usually don’t know that immediately. Hence why, in all those classes in maths, we were told how to prove that two triangles are congruent by using a fixed set of rules.
In fact, congruency appears in other settings too. No matter the context, two objects that are congruent are thought to be the same.
In fact, congruency is an example of an equivalence relation. This is a logical relation satisfying three rules:
1) Each object is equivalent to itself (reflexivity).
2) If object A is equivalent to object B, then object B is equivalent to object A (symmetry).
3) If object A is equivalent to object B and object B is equivalent to object C, then object A is equivalent to object C (transitivity).
There is an even simpler example of an equivalence relation which we all know — equality of numbers. It is reflexive (any number equals itself), symmetric (if a number x = y, then y = x too) and transitive (if for numbers x, y, and z, we know that x=y and y=z, then x=z).
The idea of defining equivalence relations enables us to group alike objects consistently — the members of each group are all equivalent to each other. Those groups are calles equivalence classes.
Another example of an equivalence relation involving triangles is similarity. Roughly speaking, two triangles are similar either if one an enlargement of the other. Intuitively, enlargement means that we…