## Saturday, 19 January 2013

### A First Post and a Problematic Problem

I made my first MathOverflow post today.

I really should be posting MathOverflow a lot more. I mean, I check it out every so often, usually when I'm Googling for a solution to a problem that I can't be bothered solving. In fact, that's pretty much how I was motivated into responding to a two year old unanswered question about solving Beltrami equations.

The problem is: given a complex-valued function $$\mu:U\rightarrow\mathbb{C}$$, can you find a solution $$f:U\rightarrow\mathbb{C}$$ to the Beltrami equation:\begin{align*}\frac{\partial f}{\partial\bar{z}}=\mu\frac{\partial f}{\partial z}?\end{align*}
According to Wikipedia, Gauss has a way of doing it for a nice neighbourhood around $$z=0$$ if $$\mu$$ can be written as a power series in $$z$$ and $$\bar{z}$$, although for the life of me I can't get their algorithm to work - in fact, I'm fairly convinced that the current Wikipedia entry is wrong. So, I sat down, tried to bash things out by writing down power series and stuff and saw that the following construction should work:

Step 1: Let's write $$\mu=\mu(z,\bar{z})$$ and mean that it's a power series in $$z$$ and $$\bar{z}$$. Then try to solve for the following differential equation (which isn't too bad, or rather, you kinda know that a solution exists):$\frac{\partial F}{\partial z}(z,w)=-\mu(w, F(z,w)),\;\text{ where } F(0,w)=w.$

Step 2: Find $$G(z,w)$$ such that $$G(z,F(z,w))=w$$. That is, the functions$(z,w)\mapsto (z,F(z,w))\text{ and }(z,w)\mapsto (z,G(z,w))$are inverse to each other.

Step 3: Take $$f(z)=G(\bar{z},z),$$ and it should work.

Note that the construction of the above $$F(z,w)$$ and $$G(z,w)$$ is strongly motivated by the Wikipedia entry on how Gauss solved this problem.

The annoying thing is that you get a (local) solution to the problem using this construction, but in the game of solving Beltrami equations, we're really looking for quasiconformal maps - that is: homeomorphisms of $$U$$ which are quasi-angle-preserving (in the sense that the angles are off, but the ratio of the "off-ness" is bounded).

On a slightly lighter note, check out this quotation taken from the shortest published paper (which was later retracted for reasons that will be obvious to any sane person) I've ever seen. Go read it, it's only four paragraphs.