Gabriel Ribeiro

Here are some notes I wrote during my undergraduate and master's studies. I am sharing them in the hope they might be useful to others.

Expository notes

Exponential sums: a walk through number theory. (2022)
Here I explain what exponential sums are, how they arise naturally in number theory, and why $\ell$-adic cohomology is a powerful tool for studying them. Some familiarity with algebraic geometry may be helpful, but no knowledge of étale cohomology is assumed.
The Hodge decomposition. (2022)
The Hodge decomposition is an direct consequence of a fundamental theorem on elliptic differential operators, that is often used as a black box. The purpose of these notes is to unpack that theorem and present a surprisingly elegant proof.
Adjoints can be defined on objects. (2022)
These notes present a perhaps underappreciated consequence of the Yoneda lemma: adjoint functors can be defined on objects. We then use this to show that resolutions in homological algebra can be chosen functorially, a fundamental fact often neglected in standard expositions.
Homological algebra. (2021)
I wrote these notes while first learning homological algebra. They don't go very far, but they reflect an approach I still rarely see: abelian categories are developed abstractly (without elements), and derived categories are introduced from the outset rather than as a secondary approach.
Wedderburn's little theorem. (2021)
In these notes, we present a beautiful proof of the Chevalley–Warning theorem. As an application, we deduce Wedderburn's little theorem: every finite division ring is commutative.
Masur–Veech volumes of the moduli space of abelian differentials. (2019, in french)
This is an undergraduate report, written under the supervision of Carlos Matheus Silva Santos, on Masur–Veech volumes. Its goal is to provide an accessible introduction to Eskin–Okounkov's groundbreaking Inventiones paper "Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials".