Week 1

Much of my first week was spent understanding the "developing map" \( f : S \setminus \Sigma \rightarrow \bar{C} \) we get by specifying a base-point on \(S \setminus \Sigma\), that a Riemann surface of constant Gaussian curvature 1 with a discrete set of (cone) singularities removed, as well as this map's associated monodromy.

This was somewhat nontrivial; my knowledge of Riemann surfaces prior to some recent reading was not particularly good, and so there was a lot to do. I, however, found the appropriate things to read and canonized them in a TeX-ed pdf which is available below.

On Friday, I began the endeavor of using my new knowledge to really understand Eremenko's survey, which I had only previously skimmed. I anticipate this will take me some time and likely a Sunday, so I'll write an addendum after that and before next week!

Read about monodromy here.

Week 2

A week of disjointed reading of Eremenko's survey and Ahlfors' chapter of complex differential equations. Nothing remarkable, albeit it was nonetheless necessary.

Week 3

This week was a big one (in terms of reading and comprehension). The goal was to understand (reasonably) the quite important Chapter VIII of the book "Uniformisation des surfaces de Riemann" due to Saint-Gervais, a Bourbaki-type pseudonym representing a large group of authors.

It is a hard chapter, as there is remarkable interplay between some algebraic, differential, and complex geometric ideas within its pages. That being I said, I got through it, and that is very important.

In particualar, that is important because this chapter holds the holy grail for us, the equation \[\Delta_\mathbb{H}(\psi) = 2\psi\] which arises naturally in the effort to establish a transversality lemma in the context of proving uniformization. The \(\mathbb{H}\) of course denotes the hyperbolic setting we are working in; if we could consider an analogous equation in a spherical setting and make sense of its solutions in as a effective of a way as is done in this chapter in the hyperbolic setting, we would effectively get the result we want.

It follows the importance of this chapter here is clear, and so I dedicated a lot of time to it. My work culminated in over 20 pages of TeX and a ~2 hour presentation on Friday, but I'm quite satisfied with the sliver of knowledge I've gotten at this point; certainly I am still feeling this problem is very hard, but I feel less anxious about it, and less concerned about my comprehension.

Week 4

This week was largely spent formalizing the proof offered in Saint-Gervais and making it more digestible. I additionally encountered some nuances that make our problem fairly different than the one considered in Saint-Gervais, but all of these seem overcomable.

Week 5

More and more reading and attempts to write up the partial result we have. It seems now the goal is to pivot the strategy to attempt to get a more general result, using say some results on the structure of the monodromy map due to Earle. This is a huge commitment in terms of reading.

I also spent some time this week working on some problems posed by Neil Sloane. This earned me citation is the OEIS entries A140064 and A383464.

Week 6

Following my comments from last week, a lot of my time here was spent trying to understand Earle's seminal work. This is complicated by the age of the main paper in question, which predates LaTeX.

I spent some more time generalizing a result related to a problem of Neil Sloane and got a very general upper bound which seems interesting. I also thought about generalizations of his problems to a more differential-geometric context.