We study the problem of finding a temporal hybridization network for a set of phylogenetic trees that minimizes the number of reticulations. First, we introduce an FPT algorithm for this problem on an arbitrary set of t binary trees with n leaves each with a running time of O(5^k*n*m) where k is the minimum temporal hybridization number. We also present the concept of temporal distance, which is a measure for how close a tree-child network is to being temporal. Then we introduce an algorithm for computing a tree-child network with temporal distance at most p and at most k reticulations in O((8k)^p*5^k*n*m). Lastly, we introduce a O(6^k*k!*n) algorithm for computing a minimum temporal hybridization network for a set of two nonbinary trees.

The cap set problem consists of finding the maximum size cap sets, i.e. sets without a 3-term arithmetic progression in F₃. In this thesis several known results on the behavior of this number as n → ∞ are presented. In particular we discuss a reformulation by Terence Tao and Will Sawin of a proof found by Dion Gijswijt and Jordan Ellenberg. It uses the slice rank, a rank that is defined for elements of tensor products, to give upper bounds on the size of the cap sets. In this report we will explain the slice rank and how it is related to the size of cap sets. We will also explore whether the slice rank might be used for bounding the size of arithmetic progression-free sets in F_q for q ≠ 3. We show that we can not use the slice rank to give a non-trivial upper bound on the size of n-term progression-free sets for n ≥ 7. This was already known for n ≥ 8.