Starting with the three axioms of projective geometry, this paper explores concepts such as perspectives, projective maps, and harmonic additions that are unique to this geometry. In addition, important theorems such as Pappus's Theorem, Desargue's Theorem, and the Fundamental Property are mentioned and worked with extensively.

In this thesis, two topics are studied: mathematical inequalities and non-linear quantum entanglement witnesses. First, various inequalities, like the Cauchy-Schwarz inequality (on finite dimensional vector spaces) and Jensen's inequality, along with their extensions and generalisations, are proved and discussed. The intimate relationship between these inequalities is studied. Because this thesis was restricted to finite dimensional vector spaces, the consequences of generalising the results to infinite dimensional vector spaces are finally determined. Secondly, the topic of entanglement detection is discussed - specifically, non-linear entanglement witnesses are considered. A bipartite and multipartite entanglement criterion based on the previously discussed inequalities are introduced and assessed extensively by considering their optimality, how they relate to other criteria as well as their limitations.