Concrete Problems that can be solved by appealing to a Moduli Space

I have always enjoyed the idea of creating "parameter spaces" or "moduli spaces," but it is only recently that I have seen very concrete applications of studying the moduli space. Because of how pervasive this theory is, I was hoping that

Notable examples I have come across:

1. enumerative geometry in the sense of trying to solve a geometric problem by replacing it with intersections of submanifolds (varieties) of the moduli space. This is pretty similar to an idea I had here although I don't know if this approach is fruitful at all.

2. Complex dynamics. In particular how you can figure out some properties of a quadratic polynomial by looking at the mandelbrot set.

3. Vector Bundles Defining the Euler class via the interpretation of $\mathbb RP^1$ as the moduli space of lines in the plane.

Are there other applications of moduli space that solved a concrete problem? I included (3) mostly to exhaust my knowledge of the subject, but I view that as quasi-concrete in the sense that understanding the topology of the moduli space can be used in a serious way to classify line bundles.

Solution 1:

Let's define a K3 surface over $\mathbb{C}$ to be a minimal algebraic surface $X$ of Kodaira dimension $0$ with $H^0(X,K_X)=1$ and $H^1(X,\mathcal{O}_X)=0$. A concrete question to ask about these things is if they are simply connected. We expect this is possible because one can compute that the Betti numbers $b_1=b_3=0$. Now I know how to show that one K3 surface I know is simply connected. Namely, a quartic hypersurface $S$ in $\mathbb{P}^3$. Just apply the Lefschetz hyperplane theorem. In fact this shows any hypersurface in $\mathbb{P}^3$ is simply connected. Anyways, now we know that there is a $K3$ surface that is simply connected. We also know that there is a fine moduli space of $K3$ surfaces that is connected. Now we employ a theorem of Ehresmann to get that all K3 surfaces are diffeomorphic by looking at the fibers of the universal family over the moduli space. Thus by knowing that one is simply connected, we know they all are simply connected.