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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.

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    $\begingroup$ Donaldson invariants are topological invariants of 4-manifolds defined by studying the moduli space of instantons. They have interesting applications, like the h-cobordism theorem in dim=4. $\endgroup$ – user204299 Aug 6 '18 at 21:13
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    $\begingroup$ The study of the moduli spaces related to solutions of the Floer equation allows it to establish Floer homology, which among other things proves Arnold's conjecture (all of those under suitable assumptions). $\endgroup$ – Aloizio Macedo Aug 6 '18 at 21:38
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    $\begingroup$ I spend most of my writing days thinking the word "moduli space of instantons", so I can write something about the topological applications of "spaces of soluions to PDEs" in dimensions 3 and 4. In dimension 4, the Donaldson invariants analagous to the definition of the Euler characteristic by point-counts of zeroes of a tangent vector field with sign; in dimension 3 the Floer homology is analagous to the definition of Morse homology, counting gradient flowlines between those zeroes. However, I am not sure if this counts as "concrete", since the theory takes some heavy lifting to set up. $\endgroup$ – user98602 Aug 6 '18 at 22:09
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    $\begingroup$ Heegner points are special points in moduli spaces of (roughly said) pairs of elliptic curves with an $N$-isogeny, $(E\overset\phi\longrightarrow E')$, an old reference that i like would be projecteuclid.org/download/pdf_1/euclid.jmsj/1230130446 (Benedict H. Gross, Heegner points and the modular curve of prime level). The history and the literature, the proven theorems and the opened directions, make them good actors among all modular spacial movies. $\endgroup$ – dan_fulea Aug 7 '18 at 0:09
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    $\begingroup$ For a complex polynomial $p$, it turns out that if you take any tree in $\Bbb{C}$ subject to a few conditions related to the critical values of $p$ and lift this tree iteratively, after finitely many steps you’ll have the Hubbard tree of $p$. This amounts to proving that this action of $p$ on the space of metric trees in $\Bbb{C}$ is contracting, with a fixed point at the Hubbard tree. This isn’t strictly true — there are some subtleties where there can be cycles — but it is an interesting application of moduli spaces. $\endgroup$ – Santana Afton Aug 7 '18 at 2:29
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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.

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    $\begingroup$ Minor nitpick: there is no "fine moduli space" of K3 surfaces. However, there are universal families of K3 surfaces with additional structure. That's good enough for the purposes of showing all K3 surfaces are simply connected. $\endgroup$ – Ariyan Javanpeykar Aug 7 '18 at 7:59
  • $\begingroup$ What exactly is a "moduli space of K3 surfaces", fine or otherwise? $\endgroup$ – Tyrone Aug 7 '18 at 10:07
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    $\begingroup$ Yes you are correct. I am being loose. For example one can construct the moduli space of marked K3s. $\endgroup$ – Samir Canning Aug 7 '18 at 14:55
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    $\begingroup$ @tyrone Roughly a complex analytic space or scheme whose points correspond to isomorphism classes of (marked) K3 surfaces. $\endgroup$ – Samir Canning Aug 7 '18 at 16:10

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