The set of certain objects we want to study or classify can be considered as a whole and form a ‘map’. Moduli spaces are magical maps that acquire extra non-trivial geometric structures (usually in an algebraic geometric sense) and are used to study the objects in a more geometric and systematic way.
Let’s say we have
a category \(\mathcal{C}\) consisting of objects we are interested in, and
a notion of what is a family of objects \(\pi:\mathcal{X}\to B\) of \(\mathcal{C}\) over any base space \(B\).
We might try to find a family \(\pi:\mathcal{X}\to B\)
whose fibres contains all possible objects (up to isomorphism) in \(\mathcal{C}\)
a notion of equivalence in \(B\) such that \(b_1\sim b_2\) if and only if the fibres \(\mathcal{X}_{b_1}\cong \mathcal{X}_{b_2}\).
Then \(\mathcal{M}:=B/\sim\) is the moduli space of objects in \(\mathcal{C}\) we are looking for, and $
The best case is that we have some invariants that partition the moduli space into finite type spaces
\[ \mathcal{M}= \bigsqcup_{i\in I} \mathcal{M}_i \]
for example the space of smooth projective curves has a discrete invariant genus, and the moduli space of curves is
\[ \mathcal{M}= \bigsqcup_{g\geq 0} \mathcal{M}_g \]
where \(\mathcal{M}_g\) is the moduli space of curves of genus \(g\), each of which has \(3g-3\) parameters.
Existence and construction of moduli spaces
This include things like stacks, GIT, Hilbert schemes, fine moduli spaces, coarse moduli spaces, etc.
Compactification of moduli spaces
Semi-stable reduction, wall-crossing, toroidal compactification, etc.
Applications of moduli spaces
Enumerative geometry, invariant computation, mirror symmetry, representation theory, combinatorics, arithmetic statistics, etc.