Texed notes of my reading course last quarter advised by Yifeng Liu, with fellow participants Grisha Kondyrev and Jora Belousov. Our goal this reading course was understand some of Scholze’s recent work with perfectoid space techniques, in particular the proof of the monodromy weight conjecture.

Attached are only the notes I kept of my lectures, sometimes just outlines of what to talk about. I have had a few requests for these notes, and they are incomplete but I think they might still be helpful for others starting out in p-adic geometry.

During the reading course, I got the increasing feeling that we were just studying $G_Q$ as fast as our little legs could take us — our little legs being our knowledge of varieties (over various non-archimedian fields like $Q_p$ and $F_p((t))$. So, I finished with a talk on the Grothendieck-Teichmüller group — another approach to $G_Q$.

## Complex Analysis: Poles, Residues, and Child’s Drawings

Thanks to Laurens Gunnarsen for his superb pedagogy and for this amazing explanation on the incredible depth of connections springing from the Sperner lemma. All errors are mine not his. This started with a chain of events, sitting in on number theory seminars and encountering Abel’s differentials of the first and second kind, interest in the dessin, and led up to asking Laurens:

## How do I understand poles and residues?

By understanding Riemann-Roch. But, first of all, you should know the zeroes and the poles of an analytic function, together with the residues of the function at each, pretty much tell you all you need to know about the function.

This is a little bit like saying that if you know the zeroes of a rational function — which is to say, the zeros of the polynomial that is its numerator — and the poles of a rational function — which is to say, the zeroes of the polynomial that is its denominator — then you basically know the rational function.  Continue reading Complex Analysis: Poles, Residues, and Child’s Drawings