You can compose THESE Riemann Surfaces as equivalence classes. And it's really no different than composing multivalued functions. At least, as I understand it.
Consider the Riemann Surface derived by the differential equation,
\(
\frac{d}{ds} y(s,z) = y^3\\
y(a,z) = z\\
\)
Then this thing just looks like,
\(
y(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + a-s}}\\
\)
Which is clearly a Riemann surface. (classifying it would be hard, I don't know how.)
Now consider the Riemann surface derived by the differential equation,
\(
\frac{d}{ds} u(s,z) = u^3\\
u(b,z) = z\\
\)
This thing just looks like,
\(
u(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + b-s}}\\
\)
And there composition \( w = y \bullet u \bullet z \) is just the Riemann surface,
\(
w(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + b+a-2s}}\\
\)
The idea is that we can always project from the Riemann surface to a local coordinate; and we're just composing the local coordinate in z; and pulling back to make a new surface. This is something I did frantically in that paper; but I always stuck to the idea of local behaviour. It's important to remember, we are treating one variable in the composition, and treating this as an equivalence class. And these are very specific Riemann surfaces.
If you gave me two Riemann surfaces; I do not think you can compose them generally. If you give me two Riemann surfaces generated by a first order differential equation; sure can compose them; as long as we think about it locally. This stuff always hurt my head though, so I never bothered to go deep into it. But I believe it would work as such.
Consider the Riemann Surface derived by the differential equation,
\(
\frac{d}{ds} y(s,z) = y^3\\
y(a,z) = z\\
\)
Then this thing just looks like,
\(
y(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + a-s}}\\
\)
Which is clearly a Riemann surface. (classifying it would be hard, I don't know how.)
Now consider the Riemann surface derived by the differential equation,
\(
\frac{d}{ds} u(s,z) = u^3\\
u(b,z) = z\\
\)
This thing just looks like,
\(
u(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + b-s}}\\
\)
And there composition \( w = y \bullet u \bullet z \) is just the Riemann surface,
\(
w(s,z) = \frac{1}{\sqrt{\frac{1}{z^2} + b+a-2s}}\\
\)
The idea is that we can always project from the Riemann surface to a local coordinate; and we're just composing the local coordinate in z; and pulling back to make a new surface. This is something I did frantically in that paper; but I always stuck to the idea of local behaviour. It's important to remember, we are treating one variable in the composition, and treating this as an equivalence class. And these are very specific Riemann surfaces.
If you gave me two Riemann surfaces; I do not think you can compose them generally. If you give me two Riemann surfaces generated by a first order differential equation; sure can compose them; as long as we think about it locally. This stuff always hurt my head though, so I never bothered to go deep into it. But I believe it would work as such.