Here I keep track some of the texts I have written. I am always happy to receive feedback.
Below are abstracts for my bachelor's and master's thesis. Feel free to reach out if you would like to see a copy of the full text.
(Master's Thesis, supervised by Ben Moonen:) The Lang-Néron theorem on rational points of abelian varieties.
One of the most fundamental theorems in arithmetic algebraic geometry is the Mordell-Weil theorem. It states that if A is an abelian variety over a number field K, the group of rational points A(K) is finitely generated. While this statement is not true for arbitrary fields, there exists a more general result by Lang and Néron: it states that if K/k is a finitely generated regular extension of fields and A is an abelian variety over K, then there exists a 'maximal abelian subvariety' Tr(A) over k, called Chow's K/k-trace, and then A(K)/Tr(A)(k) is finitely generated. In the thesis we go through two different proofs of this theorem. One of these proofs is a more direct adaptation of a well-known proof of the Mordell-Weil theorem, where the main result follows from the existence of a so-called height function. The other proof takes a different approach and uses the theory of Chow groups and intersection theory.
(Bachelor's Thesis, supervised by Magdalena Kędziorek:) The projective model structure on the category of non-negatively graded chain complexes.
The theory of model categories provides a framework to do homotopy theory. Notable examples where we can apply this theory are the categories of topological spaces and of chain complexes of modules over some ring. In the thesis we prove that the so-called projective model structure, defined on the category of non-negatively graded chain complexes, satisfies the axioms of a model category.