@Hippalectryon Just returned from jogging. Hard days here. I just defined a new class of integrals (being very special for some reasons, unique in the literature).
A charged metal ball with radius $R$ has the constant potential $V_0$. The potential $V(r)$ is continuous at the surface of the ball and satisfies Laplace's equation in the surrounding space. Determine $V(r)$
Due to spherically symmetry, the solution is $V(r) = -\dfrac{a}{r}+b$. The mean value theorem gives $V_0 = \dfrac{1}{4\pi R^2} \iint_{r=R} \dfrac{-a}{r} + b \: dS \rightarrow- \dfrac{a}{R}+b = V_0$
I don't think I had LaTeX the last time I taught Riemann surfaces, so I probably don't have exercises. (There are some in the complex geometry stuff I sent you, but ...)
1. Prove that the automorphisms of CP^1 are the Mobius transformations and use that to determine the automorphisms of CP^1 with some punctures (1,2,3 punctures, and some discussion of the 4 puncture case...)
I found the 4+ puncture situation really interesting
Arctic found nice examples with 4 punctures where you still get a reasonably big automorphism group (half of the a priori maximum, i.e. the alternating group on 4 elements)
@PVAL-inactive Right. So for instance for $\Bbb C\setminus\{0,1\}$ you get exactly 6 autormophisms
now I spent some time with arctic thinkign about what if you take out 4 points---how many automorphisms do you get? How does it depend on the choice of points?
The permutation $(0,1,\infty,z)\mapsto (z,\infty,1,0)$ always actually works (wlog you start with taking out $0,1,\infty,z$) so you get at least 1 non-trivial automorphism.
I don't know how to think about it in general. As I said, arctic used symmetry to find some examples of choices of $z$ where you get a reasonably big group of automorphisms
Anyhow, @Danu, at some point in the mid-morning, I will try to text you and see if my phone actually works in Europe. If not, I'll have other things to worry about, too. :P
@Semiclassical do you have a good reference for that method? We used it to calculate the Poisson kernel and the Green function for the ball in $\Bbb R^3$ but it's not very clear to me
@Semiclassical The question is like "Given a gravitational field $\mathbf{G} = - \nabla \phi$ with $\nabla^2 \phi = \gamma \rho$ where $\gamma$ is a constant and $\rho$is the mass density, determine $\mathbf{G}$ for earth (a sphere with radius $R$ and constant mass density $\rho_0$)"
I did that, too, when I taught RS, @PVAL. But in complex geometry I talked about my favorite proof, Griffiths's proof in terms of geometry of the canonical curve.
Alo, stupid question about Pearson correlation coefficient (wikimedia.org/api/rest_v1/media/math/render/svg/…). In shorter form is "-nxy" part inside loop or outside? E-notation without brackets always confuses me....
@TedShifrin Hartshorne spends about 1/4 of the main book (outside ch.1 which is a completely different book) talking about curves and 1/4 of the main book talking about surfaces
so it's pretty concrete just also done in lots of generality.
Right. The potential presumably shouldn't diverge, since that'd imply that it takes an infinite amount of energy to reach the center of a uniform sphere.
