This project focuses on mathematical logic, specifically within set theory, and explores the relationship between forcing theory and the combinatorics of the real line. Forcing serves as a crucial tool in establishing consistency results in mathematics, that is, to prove that certain mathematical statements do not introduce contradictions in ZFC, the standard axiomatic system in which current mathematics is usually formalized. 

The project is structured around three main general aims, which point in a common direction: investigating singular cardinals in the context of cardinal invariants of the continuum. The first general aim addresses the identification of new cardinal invariants that may have countable cofinality, such as the well-known cases of the covering number of the null ideal and the almost-disjointness number. Throughout this project, we will concentrate on three specific cardinals: the covering number of the strong measure zero ideal and the so-called independence and evasion numbers.

The second general aim is focused on the possibility of adding new singular values in Cicho\'n's diagram. For this, we tackle several problems that involve incorporating singular cardinal characteristics into Cicho\'n's diagram. For example, one goal is to separate the left side of Cicho\'n's diagram with both the covering number of the null ideal and the uniformity number of the meager ideal being singular. Furthermore, we aim to force Cicho\'n's maximum with the uniformity number of the meager ideal being singular; and Cicho\'n's maximum with the covering of the null ideal being singular. While the main forcing techniques we plan to use include very powerful tools such as finite support iterations of ccc forcing notions, iterations using finitely additive measures, and iterations with templates, it will likely be necessary to develop new forcing techniques or refine existing ones to tackle the objectives. For this reason, the third General Aim of this project is focused on the development of new suitable methods allowing to force singular cardinal invariants.